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袁萌  陈启清   2月10日

附件：堆垒素数理论引论原文：

AN INVITATION TO ADDITIVE PRIME NUMBER THEORY
A. V. Kumchev, D. I. Tolev
Communicated by V. Drensky

Abstract. The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the  eld, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
1. Introduction. Additive number theory is the branch of number theory that studies the representations of natural numbers as sums of integers subject to various arithmetic restrictions. For example, given a sequence of integers
A = {a1 < a2 < a3 <    } 2000 Mathematics Subject Classi cation: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55. Key words: Goldbach problems, additive problems, circle method, sieve methods, prime numbers.
2 A. V. Kumchev, D. I. Tolev
one often asks what natural numbers can be represented as sums of a  xed number of elements of A; that is, for any  xed s ∈ N, one wants to  nd the natural numbers n such that the diophantine equation x1 +   + xs = n(1.1) has a solution in x1,...,xs ∈ A. The sequence A may be described in some generality (say, one may assume that A contains “many” integers), or it may be a particular sequence of some arithmetic interest (say, A may be the sequence of kth powers, the sequence of prime numbers, the values taken by a polynomial F(X) ∈ Z[X] at the positive integers or at the primes, etc.). In this survey, we discuss almost exclusively problems of the latter kind. The main focus will be on two questions,
known as Goldbach’s problem and the Waring–Goldbach problem, which are concerned with representations as sums of primes and powers of primes, respectively.
1.1.    Goldbach’s problem.
Goldbach’s problem appeared for the  rst time in 1742 in the correspondence between Goldbach and Euler. In modern language, it can be stated as follows. Goldbach Conjecture. Every even integer n ≥ 4 is the sum of two primes, and every odd integer n≥ 7 is the sum of three primes. The two parts of this conjecture are known as the binary Goldbach problem and the ternary Goldbach problem, respectively. Clearly, the binary conjecture is the stronger one. It is also much more di cult. The  rst theoretical evidence in support of Goldbach’s conjecture was obtained by Brun [27], who showed that every large even integer is the sum of two integers having at most nine prime factors. Brun also obtained an upper bound of the correct order for the number of representations of a large even integer as the sum of two primes. During the early 1920s Hardy and Littlewood [67]–[72] developed the ideas in an earlier paper by Hardy and Ramanujan [73] into a new analytic method in additive number theory. Their method is known as the circle method. In 1923 Hardy and Littlewood [69, 71] applied the circle method to Goldbach’s problem. Assuming the Generalized Riemann Hypothesis1 (GRH), they proved that all but  nitely many odd integers are sums of three primes and that all but Ox1/2+ε even integers n ≤ x are sums of two primes. (Henceforth, ε denotes a positive number which can be chosen arbitrarily small if the implied constant is allowed to depend on ε.)
1 An important conjecture about certain Dirichlet series; see §2.2 for details.
An invitation to additive prime number theory 3
During the 1930s Schnirelmann [201] developed a probabilistic approach towards problems in additive number theory. Using his method and Brun’s results, he was able to prove unconditionally that there exists a positive integer s such that every su ciently large integer is the sum of at most s primes. Although the value of s arising from this approach is much larger than the conjectured s = 3, Schnirelmann’s result represented a signi cant achievement, as it defeated the popular belief at the time that the solution of Goldbach’s problem must depend on GRH. (Since its  rst appearance, Schnirelmann’s method has been polished signi cantly. In particular, the best result to date obtained in this fashion by Ramare [193] states that one can take s = 7.) In 1937 I. M. Vinogradov [236] found an ingenious new method for estimating sums over primes, which he applied to the exponential sum f(α) =X p≤n e(αp),(1.2) where α is real, p denotes a prime, and e(α) = exp(2πiα). Using his estimate for f(α), Vinogradov was able to give a new, unconditional proof of the result of Hardy and Littlewood on the ternary Goldbach problem. His result is known as Vinogradov’s three prime theorem.
Theorem 1 (Vinogradov, 1937). For a positive integer n, let R(n) denote the number of representations of n as the sum of three primes. Then
R(n) =
n2 2(log n)3 S(n) + On2(logn) 4,(1.3) where S(n) =Y p|n1  1 (p 1)2Y p-n1 + 1 (p 1)3.(1.4) In particular, every su ciently large odd integer is the sum of three primes. The products in (1.4) are over the primes dividing n and over those not dividing n, respectively. In particular, when n is even, we have S(n) = 0, making (1.3) trivial. On the other hand, when n is odd, we have S(n)≥ 1. We describe the proof of Theorem 1 in §3.1. It should be noted that the independence of GRH in Theorem 1 comes at the price of a mind-boggling implied constant. If one avoids O-notation and makes all the constants explicit, one  nds that the original (GRH-dependent) work of Hardy and Littlewood establishes the ternary Goldbach conjecture for
4 A. V. Kumchev, D. I. Tolev
n ≥ 1050, whereas Vinogradov’s method requires n ≥ 106800000 and even its most re ned version available today (see Liu and Wang [163]) requires n ≥ 101346. To put these numbers in perspective, we remark that even the bound 1050 is beyond hope of “checking the remaining cases by a computer”. In fact, only recently have Deshouillers et al. [51] proved that if GRH is true, the ternary Goldbach conjecture holds for all odd n ≥7. In 1938, using Vinogradov’s method, Chudakov [42], van der Corput [43], and Estermann [54] each showed that almost all even integers n ≤ x are sums of two primes. More precisely, they proved that for any A > 0 we have E(x) = Ox(logx) A,(1.5) where E(x) denotes the number of even integers n ≤ x that cannot be represented as the sum of two primes. The  rst improvement on (1.5) was obtained by Vaughan [220]. It was followed by a celebrated work by Montgomery and Vaughan [173] from 1975, in which they established the existence of an absolute constant δ > 0 such that E(x) = Ox1 δ.(1.6) The  rst to compute an explicit numerical value for δ were Chen and Pan [36]. They showed that the method of Montgomery and Vaughan yields (1.6) with δ = 0.01. Subsequently, this result has been sharpened by several authors and currently (1.6) is known to hold with δ = 0.086 (see Li [136]). In June 2004, Pintz [186] announced a further improvement on (1.6). He has established the above bound with δ = 1 3 and can also show that for all but O(x3/5+ε) even integers n ≤ x either n or n 2 is the sum of two primes. One may also think of the binary Goldbach conjecture as a claim about the primes in the sequence
A = A(n) = {n p : p prime number, 2 < p < n},(1.7) namely, that such primes exist for all even n ≥ 6. Denote by Pr an integer having at most r prime factors, counted with their multiplicities, and refer to such a number as an almost prime of order r (thus, Brun’s result mentioned above asserts that every large even n can be represented in the form n = P9 + P0 9). In 1947 R′enyi [195] proved that there is a  xed integer r such that the sequence A contains a Pr-number when n is su ciently large. Subsequent work by many mathematicians reduced the value of r in R′enyi’s result almost to the possible
An invitation to additive prime number theory 5
limit and fell just short of proving the binary Goldbach conjecture. The best result to date was obtained by Chen [35].
Theorem 2 (Chen, 1973). For an even integer n, let r(n) denote the number of representations of n in the form n = p + P2, where p is a prime and P2 is an almost prime of order 2. There exists an absolute constant n0 such that if n ≥ n0, then r(n) > 0.67Y p>21  1 (p 1)2Y p>2 p|n p 1 p 2n (logn)2 .
In particular, every su ciently large even integer n can be represented in the form n = p + P2.
1.2. Waring’s problem. Before proceeding with the Waring–Goldbach problem, we will make a detour to present the most important results in Waring’s problem, as those results and the work on Goldbach’s problem have been the main motivation behind the Waring–Golbach problem. It was probably the ancient Greeks who  rst observed that every positive integer is the sum of four integer squares, but it was not until 1770 that a complete proof of this remarkable fact was given by Lagrange. Also in 1770, Waring proposed a generalization of the four square theorem that became known as Waring’s problem and arguably led to the emergence of additive number theory. In modern terminology, Waring’s conjecture states that for every integer k ≥ 2 there exists an integer s = s(k) such that every natural number n is the sum of at most s kth powers of natural numbers. Several special cases of this conjecture were settled during the 19th century, but the complete solution eluded mathematicians until 1909, when Hilbert [95] proved the existence of such an s for all k by means of a di cult combinatorial argument. Let g(k) denote the least possible s as above. Hilbert’s method produced a very poor bound for g(k). Using the circle method, Hardy and Littlewood were able to improve greatly on Hilbert’s bound for g(k). In fact, through the e orts of many mathematicians, the circle method in conjunction with elementary and computational arguments has led to a nearly complete evaluation of g(k). In particular, we know that g(k) is determined by certain special integers n < 4k that can only be represented as sums of a large number of kth powers of 1, 2 and 3 (see [228, §1.1] for further details on g(k)). A much more di cult question, and one that leads to a much deeper understanding of the additive properties of kth powers, is that of estimating the
6 A. V. Kumchev, D. I. Tolev
function G(k), de ned as the least s such that every su ciently large positive integer n is the sum of s kth powers. This function was introduced by Hardy and Littlewood [70], who obtained the bound G(k) ≤(k 2)2k 1 + 5.(1.8) In fact, they proved more than that. Let Ik,s(n) denote the number of solutions of the diophantine equation xk 1 + xk 2 +   + xk s = n(1.9) in x1,...,xs ∈ N. Hardy and Littlewood showed that if s ≥ (k 2)2k 1 +5, then Ik,s(n)～ Γs1 + 1 kΓ s k Sk,s(n)ns/k 1 as n →∞,(1.10) where Γ stands for Euler’s gamma-function and Sk,s(n) is an absolutely convergent in nite series, called the singular series, such that Sk,s(n) ≥ c1(k,s) > 0. While the upper bound (1.8) represents a tremendous improvement over Hilbert’s result, it is still quite larger than the trivial lower bound G(k) ≥ k+1.2 During the mid-1930s I. M. Vinogradov introduced several re nements of the circle method that allowed him to obtain a series of improvements on (1.8) for large k. In their most elaborate version, Vinogradov’s methods yield a bound of the form3 G(k) ≤ 2k(logk + O(loglogk)). First published by Vinogradov [240] in 1959, this bound withstood any signi cant improvement until 1992, when Wooley [245] proved that G(k) ≤ k(logk + loglogk + O(1)). The latter is the sharpest bound to date for G(k) when k is large. For smaller k, one can obtain better results by using more specialized techniques (usually re nements of the circle method). The best known bounds for G(k), 3≤ k ≤ 20, are of the form G(k) ≤ F(k), with F(k) given by Table 1 below. 2Let X be large. If n ≤ X, any solution of (1.9) must satisfy 1 ≤ x1,...,xs ≤ X1/k. There at most Xs/k such s-tuples, which yield at most (1/s! + o(1))Xs/k distinct sums xk 1 +     + xk s. Thus, when s ≤ k, there are not enough sums of s kth powers to represent all the integers. 3In this and similar results appearing later, one can obtain an explicit expressions in place of the O-terms, but those are too complicated to state here.
An invitation to additive prime number theory 7
k 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
F(k) 7 16 17 24 33 42 50 59 67 76 84 92 100 109 117 125 134 142 Table 1. Bounds for G(k), 3 ≤k ≤ 20. With the exception of the bound G(3) ≤ 7, all of these results have been obtained by an iterative version of the circle method that originated in the work of Davenport [46, 48] and Davenport and Erd¨os [50]. The bound for G(3) was established  rst by Linnik [141] and until recently lay beyond the reach of the circle method. The result on G(4) is due to Davenport [47], and in fact states that G(4) = 16. This is because 16 biquadrates are needed to represent integers of the form n = 31 16r, r ∈ N. Other than Lagrange’s four squares theorem, this is the only instance in which the exact value of G(k) is known. However, Davenport [47] also proved that if s ≥ 14, all su ciently large integers n ≡ r (mod 16), 1 ≤ r ≤ s, can be written as the sum of s biquadrates; Kawada and Wooley [120] obtained a similar result for as few as 11 biquadrates. The remaining bounds in Table 1 appear in a series of recent papers by Vaughan and Wooley [229]–[232]. A great deal of e ort has also been dedicated to estimating the function   G(k), which represents the least s for which the asymptotic formula (1.10) holds. For large k, Ford [57] showed that
  G(k) ≤ k2(logk + loglogk + O(1)),(1.11) thus improving on earlier work by Vinogradov [238], Hua [101], and Wooley [246]. Furthermore, Vaughan [226, 227] and Boklan [18] obtained the bounds
  G(k) ≤ 2k (k ≥ 3) and   G(k) ≤ 7 8  2k (k ≥ 6), which supersede (1.11) when k ≤ 8. The work on Waring’s problem has inspired research on several other questions concerned with the additive properties of kth powers (and of more general polynomial sequences). Such matters, however, are beyond the scope of this survey. The reader interested in a more comprehensive introduction to Waring’s problem should refer to the monographs [4, 228] or to a recent survey article by Vaughan and Wooley [233] (the latter also provides an excellent account of the history of Waring’s problem).
1.3. The Waring–Goldbach problem. Vinogradov’s proof of the three prime theorem provided a blueprint for subsequent applications of the Hardy– Littlewood circle method to additive problems involving primes. Shortly after the
8 A. V. Kumchev, D. I. Tolev
publication of Theorem 3, Vinogradov himself [237] and Hua [100] began studying Waring’s problem with prime variables, known nowadays as the Waring–Goldbach problem. They were able to generalize the asymptotic formula (1.3) to kth powers for all k ≥ 1 and ultimately their e orts led to the proof of Theorem 3 below. In order to describe the current knowledge about the Waring–Goldbach problem, we  rst need to introduce some notation. Let k be a positive integer and p a prime. We denote by θ = θ(k,p) the (unique) integer such that pθ | k and pθ+1 - k, and then de ne γ = γ(k,p) =(θ + 2, if p = 2, 2 | k, θ + 1, otherwise, K(k) = Y (p 1)|k pγ.(1.12)
In particular, we have K(1) = 2. It is not di cult to show that if an integer n is the sum of s kth powers of primes greater than k + 1, then n must satisfy the congruence condition n ≡ s (mod K(k)). Furthermore, de ne
S(q,a) =
q X h=1 (h,q)=1
eahk q , S k,s(n) = ∞ X q=1
q X a=1 (a,q)=1
S(q,a)s φ(q)s
e an q ,(1.13)
where (a,q) stands for the greatest common divisor of a and q, and φ(q) is Euler’s totient function, that is, the number of positive integers n ≤ q which are relatively prime to q. The following result will be established in §3.3 and §3.4. Theorem 3. Let k,s and n be positive integers, and let R k,s(n) denote the number of solutions of the diophantine equation
pk 1 + pk 2 +   + pk s = n(1.14) in primes p1,...,ps. Suppose that s ≥          2k + 1, if 1 ≤ k ≤ 5, 7 8  2k + 1, if 6 ≤ k ≤ 8, k2(logk + loglogk + O(1)), if k > 8. Then
R k,s(n) ～
Γs1 + 1 kΓ s k
S k,s(n) ns/k 1 (logn)s as n →∞,(1.15)
An invitation to additive prime number theory 9
where S k,s(n) is de ned by (1.13). Furthermore, the singular series S k,s(n) is absolutely convergent, and if n ≡ s (mod K(k)), then S k,s(n) ≥ c2(k,s) > 0. In particular, we have the following corollaries to Theorem 3. Corollary 3.1. Every su ciently large integer n ≡ 5 (mod 24) can be represented as the sum of  ve squares of primes.
Corollary 3.2. Every su ciently large odd integer can be represented as the sum of nine cubes of primes. Hua introduced a function H(k) similar to the function G(k) in Waring’s problem. H(k) is de ned as the least integer s such that equation (1.14) has a solution in primes p1,...,ps for all su ciently large n ≡ s (mod K(k)). It is conjectured that H(k) = k + 1 for all k ≥ 1, but this conjecture has not been proved for any value of k yet. When k ≤ 3, the sharpest known upper bounds for H(k) are those given by Theorem 3, that is, H(1) ≤ 3, H(2) ≤ 5, H(3) ≤ 9. When k ≥ 4, the best results in the literature are as follows. Theorem 4. Let k ≥ 4 be an integer, and let H(k) be as above. Then H(k) ≤(F(k), if 4 ≤ k ≤ 10, k(4log k + 2loglogk + O(1)), if k > 10,
where F(k) is given by the following table.
k 4 5 6 7 8 9 10
F(k) 14 21 33 46 63 83 107 Table 2. Bounds for H(k), 4≤ k ≤10. The cases k = 6 and 8 ≤ k ≤ 10 of Theorem 4 are due to Thanigasalam [211], and the cases k = 4,5 and 7 are recent results of Kawada and Wooley [121] and Kumchev [127], respectively. The bound for k > 10 is an old result of Hua, whose proof can be found in Hua’s book [102]. To the best of our knowledge, this is the strongest published result for large k, although it is wellknown to experts in the  eld that better results are within the reach of Wooley’s re nement of Vinogradov’s methods. In particular, by inserting Theorem 1 in Wooley [247] into the machinery developed in Hua’s monograph, one obtains H(k) ≤ k(3 2 logk + O(loglogk)) for k →∞.
10 A. V. Kumchev, D. I. Tolev
1.4. Other additive problems involving primes. There are several variants and generalizations of the Waring–Goldbach problem that have attracted a lot of attention over the years. For example, one may consider the diophantine equation
a1pk 1 + a2pk 2 +   + aspk s = n,(1.16) where n,a1,...,as are  xed, not necessarily positive, integers. There are several questions that we can ask about equations of this form. The main question, of course, is that of solubility. Furthermore, in cases where we do know that (1.16) is soluble, we may want to count the solutions with p1,...,ps ≤ X, where X is a large parameter. A famous problem of this type is the twin-prime conjecture: there exist in nitely many primes p such that p + 2 is also prime, that is, the equation p1  p2 = 2 has in nitely many solutions. It is believed that this conjecture is of the same di culty as the binary Goldbach problem, and in fact, the two problems share a lot of common history. In particular, while the twin-prime conjecture is still open, Chen’s proof of Theorem 2 can be easily modi ed to establish that there exist in nitely many primes p such that p + 2 = P2. Other variants of the Waring–Goldbach problem consider more general diophantine equations of the form f(p1) + f(p2) +   + f(ps) = n, where f(X) ∈ Z[X], or systems of equations of the types (1.1) or (1.16). For example, Chapters 10 and 11 in Hua’s monograph [102] deal with the system pj 1 + pj 2 +   + pj s = nj (1 ≤ j ≤ k). The number of solutions of this system satis es an asymptotic formula similar to (1.15), but the main term in that asymptotic formula is less understood than the main term in (1.15) (see [3, 41, 102, 170] for further details). Another classical problem in which a system of diophantine equations arises naturally concerns the existence of non-trivial arithmetic progressions consisting of r primes. It has been conjectured that for every integer r ≥ 3 there are in nitely many such arithmetic progressions. In other words, the linear system pi  2pi+1 + pi+2 = 0 (1 ≤ i≤ r 2)
An invitation to additive prime number theory 11
has in nitely many solutions in distinct primes p1,...,pr. In the case r = 3 this can be established by a variant of Vinogardov’s proof of the three primes theorem, but when r > 3 the above system lies beyond the reach of the circle method. In fact, until recently the most signi cant insight into progressions of more than three primes were the following two results:   Heath-Brown [83] succeeded to prove that there exist in nitely many arithmetic progressions of three primes and a P2-number.   Balog [11] proved that for any r there are r distinct primes p1,...,pr such that all the averages 1 2(pi + pj) are prime.
Thus, the specialists in the  eld were stunned when Green and Tao [64] announced their amazing proof of the full conjecture. The reader will  nd a brief description of their ideas and of some related recent work in the last section. Finally, instead of (1.1), one may study the inequality |x1 +   + xs  α| < ε, where α is a real number, ε is a small positive number and x1,...,xs are real variables taking values from a given sequence (or sequences). For example, by setting xj = pc j where c > 1 in not an integer, we can generalize the Waring– Goldbach problem to fractional powers of primes. We will mention several results of this form in §5.7.
2. The distribution of primes. In this section we discuss brie y some classical results about primes, which play an important role in additive prime number theory. 2.1. The Prime Number Theorem. The  rst result on the distribution of primes is Euclid’s theorem that there are in nitely many prime numbers. In 1798 Legendre conjectured that the prime counting function π(x) (i.e., the number of primes p ≤ x) satis es the asymptotic relation
lim x→∞
π(x) x/(logx) = 1;(2.1)
this is the classical statement of the Prime Number Theorem. Later Gauss observed that the logarithmic integral lix =Z x 2 dt logt
12 A. V. Kumchev, D. I. Tolev
seemed to provide a better approximation to π(x) than the function x/(log x) appearing in (2.1), and this is indeed the case. Thus, in anticipation of versions of the Prime Number Theorem that are more precise than (2.1), we de ne the error term  (x) = π(x) lix.(2.2) The  rst step toward a proof of the Prime Number Theorem was made by Chebyshev. In the early 1850s he proved that (2.1) predicts correctly the order of π(x), that is, he established the existence of absolute constants c2 > c1 > 0 such that c1x logx ≤ π(x) ≤ c2x logx .
Chebyshev also showed that if the limit on the left side of (2.1) exists, then it must be equal to 1. In 1859 Riemann published his famous memoir [197], in which he demonstrated the intimate relation between π(x) and the function which now bears his name, that is, the Riemann zeta-function de ned by
ζ(s) =
∞ X n=1
n s =Y p 1 p s 1 (Re(s) > 1).(2.3) This and similar series had been used earlier by Euler4 and Dirichlet, but only as functions of a real variable. Riemann observed that ζ(s) is holomorphic in the half-plane Re(s) > 1 and that it can be continued analytically to a meromorphic function, whose only singularity is a simple pole at s = 1. It is not di cult to deduce from (2.3) that ζ(s) 6= 0 in the half-plane Re(s) > 1. Riemann observed that ζ(s) has in nitely many zeros in the strip 0 ≤ Re(s) ≤ 1 and proposed several conjectures concerning those zeros and the relation between them and the Prime Number Theorem. The most famous among those conjecture—and the only one that is still open—is known as the Riemann Hypothesis. Riemann Hypothesis (RH). All the zeros of ζ(s) with 0 ≤ Re(s) ≤ 1 lie on the line Re(s) = 1 2 . The remaining conjectures in Riemann’s paper were proved by the end of the 19th century. In particular, it was proved that the Prime Number Theorem
4In particular, Euler established the equality between ζ(s) and the in nite product in (2.3), which is known as the Euler product of ζ(s).
An invitation to additive prime number theory 13
follows from the nonvanishing of ζ(s) on the line Re(s) = 1. Thus, when in 1896 Hadamard and de la Vall′ee Poussin proved (independently) that ζ(1+it)6= 0 for all real t, the Prime Number Theorem was  nally proved. In 1899 de la Vall′ee Poussin obtained the following quantitative result5. (Henceforth, we often use Vinogradov’s notation A  B, which means that A = O(B).) Theorem 5 (de la Vall′ee Poussin, 1899). Let  (x) be de ned by (2.2). There exists an absolute constant c > 0 such that  (x)xexp cplogx. De la Vall′ee Poussin’s theorem has been improved somewhat, but not nearly as much as one would hope. The best result to date is due to I. M. Vinogradov [239] and Korobov [123], who obtained (independently) the following estimate for  (x). Theorem 6 (Vinogradov, Korobov, 1958). Let  (x) be de ned by (2.2). There exists an absolute constant c > 0 such that  (x) xexp c(log x)3/5(loglogx) 1/5. In comparison, if the Riemann Hypothesis is assumed, one has  (x)x1/2 logx,(2.4) which, apart from the power of the logarithm, is best possible. The reader can  nd further information about the Prime Number Theorem and the Riemann zeta-function in the standard texts on the subject (e.g., [49, 103, 117, 191, 212]). 2.2. Primes in arithmetic progressions. In a couple of memoirs published in 1837 and 1840, Dirichlet proved that if a and q are natural numbers with (a,q) = 1, then the arithmetic progression a mod q contains in nitely many primes. In fact, Dirichlet’s argument can be re ned as to establish the asymptotic formula X p≤x p≡a (mod q) logp p ～ 1 φ(q)X p≤x logp p as x →∞,(2.5) 5Functions of the type f(x) = exp(logx)λ, where λ is a constant, are quite common in analytic number theory. To help the reader appreciate results such as Theorems 5 and 6, we remark that as x → ∞ such a function with 0 < λ < 1 grows more rapidly than any  xed power of logx, but less rapidly than xε for any  xed ε > 0.
14 A. V. Kumchev, D. I. Tolev
valid for all a and q with (a,q) = 1. Fix q and consider the various arithmetic progressions a mod q (here φ(q) is Euler’s totient function). Since all but  nitely many primes lie in progressions with (a,q) = 1 and there are φ(q) such progressions, (2.5) suggests that each arithmetic progression a mod q, with (a,q) = 1, “captures its fair share” of prime numbers, i.e., that the primes are uniformly distributed among the (appropriate) arithmetic progressions to a given modulus q. Thus, one may expect that if (a,q) = 1, then π(x;q,a) = X p≤x p≡a (mod q) 1 ～ lix φ(q) as x →∞.(2.6)
This is the prime number theorem for arithmetic progressions. One may consider (2.6) from two di erent view points. First, one may  x a and q and ask whether (2.6) holds (allowing the convergence to depend on q and a). Posed in this form, the problem is a minor generalization of the Prime Number Theorem. In fact, shortly after proving Theorem 5, de la Vall′ee Poussin established that
 (x;q,a) = π(x;q,a) 
lix φ(q) xexp cplogx, where c = c(q,a) > 0 and the implied constant depends on q and a. The problem becomes much more di cult if one requires an estimate that is explicit in q and uniform in a. The  rst result of this kind was obtained by Page [176], who proved the existence of a (small) positive number δ such that  (x;q,a) xexp (logx)δ,(2.7) whenever 1 ≤ q ≤ (logx)2 δ and (a,q) = 1. In 1935 Siegel [208] (essentially) proved the following result known as the Siegel–Wal sz theorem.
Theorem 7 (Siegel, 1935). For any  xed A > 0, there exists a constant c = c(A) > 0 such that
π(x;q,a) =
lix φ(q)
+ Oxexp cplogx whenever q ≤ (logx)A and (a,q) = 1. Remark. While this result is clearly sharper than Page’s, it does have one signi cant drawback: it is ine ective, that is, given a particular value of
An invitation to additive prime number theory 15
A, the proof does not allow the constant c(A) or the O-implied constant to be computed. The above results have been proved using the analytic properties of a class of generalizations of the Riemann zeta-function known as Dirichlet L-functions. For each positive integer q there are φ(q) functions χ : Z → C, called Dirichlet characters mod q, with the following properties:   χ is totally multiplicative: χ(mn) = χ(m)χ(n);   χ is q-periodic;   |χ(n)| = 1 if (n,q) = 1 and χ(n) = 0 if (n,q) > 1;   if (n,q) = 1, then Xχ mod q χ(n) =(φ(q) if n ≡ 1 (mod q), 0 otherwise. For more information about the construction and properties of the Dirichlet characters we refer the reader to [49, 108, 116, 191]. Given a character χ mod q, we de ne the Dirichlet L-function
L(s,χ) =
∞ X n=1
χ(n)n s =Y p 1 χ(p)p s 1 (Re(s) > 1). Similarly to ζ(s), L(s,χ) is holomorphic in the half-plane Re(s) > 1 and can be continued analytically to a meromorphic function on C that has at most one pole, which (if present) must be a simple pole at s = 1. Furthermore, just as ζ(s), the continued L(s,χ) has in nitely many zeros in the strip 0 ≤ Re(s) ≤ 1, and the horizontal distribution of those zeros has important implications on the distribution of primes in arithmetic progressions. For example, the results of de la Vall′ee Poussin, Page and Siegel mentioned above were proved by showing that no L-function can have a zero “close” to the line Re(s) = 1. We also have the following generalization of the Riemann Hypothesis. Generalized Riemannian Hypothesis (GRH). Let L(s,χ) be a Dirichlet L-function. Then all the zeros of L(s,χ) with 0 ≤ Re(s) ≤ 1 lie on the line Re(s) = 1 2. Assuming GRH, we can deduce easily that if (a,q) = 1, then
π(x;q,a) =
lix φ(q) + Ox1/2 logx,(2.8)
16 A. V. Kumchev, D. I. Tolev
which is nontrivial when 1 ≤ q ≤ x1/2(logx) 2 ε. In many applications one only needs (2.8) to hold “on average” over the moduli q. During the 1950s and 1960s several authors obtained estimates for averages of  (x;q,a). In particular, the following quantity was studied extensively: E(x,Q) =X q≤Q max (a,q)=1 max y≤x | (y;q,a)|. The trivial bound for this quantity is E(x,Q)  xlogx. One usually focuses on  nding the largest value of Q for which one can improve on this trivial bound, even if the improvement is fairly modest. The sharpest result in this direction was established (independently) by Bombieri [19] and A. I. Vinogradov [234] in 1965. Their result is known as the Bombieri–Vinogradov theorem and (in the slightly stronger form given by Bombieri) can be stated as follows.
Theorem 8 (Bombieri, Vinogradov, 1965). For any  xed A > 0, there exists a B = B(A) > 0 such that E(x,Q)  x(logx) A,(2.9) provided that Q ≤ x1/2(logx) B. We should note that other than the value of B(A) the range for Q in this result is as long as the range we can deduce from GRH. Indeed, GRH yields B = A + 1, whereas Bombieri obtained Theorem 8 with B = 3A + 22 and more recently Vaughan [223] gave B = A + 5/2.
2.3. Primes in short intervals. Throughout this section, we write pn for the nth prime number. We are interested in estimates for the di erence pn+1  pn between two consecutive primes. Cram′er was the  rst to study this question systematically. He proved [44] that the Riemann Hypothesis implies pn+1  pn p1/2 n logpn.
Cram′er also proposed a probabilistic model of the prime numbers that leads to very precise (and very bold) predictions of the asymptotic properties of the primes. In particular, he conjectured [45] that
limsup n→∞
pn+1  pn log2 pn = 1.(2.10)
An invitation to additive prime number theory 17
A non-trivial upper bound for pn+1 pn can be obtained as a consequence of the Prime Number Theorem, but Hoheisel [96] found a much sharper result. He proved unconditionaly the asymptotic formula π(x + h) π(x) ～ h(logx) 1 as x →∞,(2.11) with h = x1 (3300) 1. There have been several improvements on Hoheisel’s result and it is now known that (2.11) holds with h = x7/12 (see Heath-Brown [86]). Furthermore, several mathematicians have shown that even shorter intervals must contain primes (without establishing an asymptotic formula for the number of primes in such intervals). The best result in this directions is due to Baker, Harman, and Pintz [9], who proved that for each n one has pn+1  pn  p0.525 n . A related problem seeks small gaps between consecutive primes. In particular, the twin-prime conjecture can be stated in the form
liminf n→∞
(pn+1  pn) = 2.
It is an exercise to show that the Prime Number Theorem implies the inequality
liminf n→∞
pn+1  pn logpn ≤ 1.
Improvements on this trivial bound, on the other hand, have proved notoriously di cult and, so far, the best result, due to Maier [165], is
liminf n→∞
pn+1  pn logpn ≤ 0.2486... .
2.4. Primes in sparse sequences. We say that an in nite sequence of primes S is sparse if π(S;x) := #p ∈S : p ≤ x    = o(π(x)) as x →∞. A classical example that has attracted a great deal of attention but has proved notoriously di cult is that of primes represented by polynomials. To this day, there is not a single example of a polynomial f(X) ∈ Z[X] of degree at least 2 which is known to take on in nitely many prime values. The closest approximation is a result of Iwaniec [104], who showed that if a,b,c are integers such
18 A. V. Kumchev, D. I. Tolev
that a > 0, (c,2) = 1, and the polynomial f(X) = aX2 + bX + c is irreducible, then f(X) takes on in nitely many P2-numbers. On the other hand, in recent years there has been some exciting progress in the direction of  nding polynomials in two variables that represent in nitely many primes. In 1998 Friedlander and Iwaniec [58] proved that the polynomial X2 + Y 4 represents in nitely many primes. We note that this polynomial takes on O(x3/4) values up to x. In 2001 Heath-Brown [89] obtained an analogous result for the polynomial X3 + 2Y 3 whose values are even sparser: it takes on O(x2/3) values up to x. Furthermore, Heath-Brown and Moroz [92] extended the latter result to general irreducible binary cubic forms in Z[X,Y ] (subject to some mild necessary conditions). Another class of sparse sequences of prime numbers arises in the context of diophantine approximation. The two best known examples of this kind are the sequences Sλ =np : p is prime with {√p} < p λo(2.12) and Pc = {p : p = [nc] for some integer n}.(2.13) Here, λ ∈ (0,1) and c > 1 are  xed real numbers, {x} denotes the fractional part of the real number x, and [x] = x {x}. The sequence Sλ was introduced by I. M. Vinogradov, who proved (see [241, Chapter 4]) that if 0 < λ < 1/10, then
π(Sλ;x) ～
x1 λ (1 λ)logx
as x →∞.
The admissible range for λ has been subsequently extended to 0 < λ < 1/4 by Balog [10] and Harman [76], while Harman and Lewis [81] showed that Sλ is in nite for 0 < λ < 0.262. The  rst to study the sequence (2.13) was Piatetski-Shapiro [185], who considered Pc as a sequence of primes represented by a “polynomial of degree c”. Piatetski-Shapiro proved that Pc is in nite when 1 < c < 12/11. The range for c has been extended several times and it is currently known (see Rivat and Wu [199]) that Pc is in nite when 1 < c < 243/205. Furthermore, it is known (see Rivat and Sargos [198]) that when 1 < c < 1.16117... , we have
π(Pc;x) ～
x1/c logx
as x →∞.
An invitation to additive prime number theory 19
3. The Hardy–Littlewood circle method. Most of the results mentioned in the Introduction have been proved by means of the Hardy–Littlewood circle method. In this section, we describe the general philosophy of the circle method, using its applications to the Goldbach and Waring–Goldbach problems to illustrate the main points. 3.1. Vinogradov’s three prime theorem. 3.1.1. Preliminaries. Using the orthogonality relation Z 1 0 e(αm)dα =(1, if m = 0, 0, if m 6= 0, (3.1) we can express R(n) as a Fourier integral. We have R(n) = X p1,p2,p3≤nZ 1 0 e(α(p1 + p2 + p3  n)) dα(3.2) =Z 1 0 f(α)3e( αn)dα, where f(α) is the exponential sum (1.2). The circle method uses (3.2) to derive an asymptotic formula for R(n) from estimates for f(α). The analysis of the right side of (3.2) rests on the observation that the behavior of f(α) depends on the distance from α to the set of fractions with “small” denominators. When α is “near” such a fraction, we expect f(α) to be “large” and to have certain asymptotic behavior. Otherwise, we can argue that the numbers e(αp) are uniformly distributed on the unit circle and hence f(α) is “small”. In order to make these observations rigorous, we need to introduce some notation. Let B be a positive constant to be chosen later and set P = (logn)B.(3.3) If a and q are integers with 1 ≤ a ≤ q ≤ P and (a,q) = 1, we de ne the major arc6 M(q,a) =a q   P qn , a q + P qn.(3.4) 6This term may seem a little peculiar at  rst, given that M(q,a) is in fact an interval. The explanation is that, in the original version of the circle method, Hardy and Littlewood used power series and Cauchy’s integral formula instead of exponential sums and (3.1) (see [228, §1.2]). In that setting, the role of M(q,a) is played by a small circular arc near the root of unity e(a/q).
20 A. V. Kumchev, D. I. Tolev
The integration in (3.2) can be taken over any interval of length one and, in particular, overPn 1,1 + Pn 1 . We partition this interval into two subsets: M = [ q≤P [ 1≤a≤q (a,q)=1 M(q,a) and m =Pn 1,1 + Pn 1 \M,(3.5)
called respectively the set of major arcs and the set of minor arcs. Then from (3.2) and (3.5) it follows that
R(n) = R(n,M) + R(n,m),(3.6)
where we have denoted
R(n,B) =ZBf(α)3e( αn)dα. In the next section we explain how, using Theorem 7 and standard results from elementary number theory, one can obtain an asymptotic formula for R(n,M) (see (3.13) below). Then in §3.1.3 and §3.1.4 we discuss how one can show that R(n,m) is of a smaller order of magnitude than the main term in that asymptotic formula (see (3.14)).
3.1.2. The major arcs. In this section we sketch the estimation of the contribution from the major arcs. The interested reader will  nd the missing details in [116, Chapter 10] or [228, Chapter 2]. It is easy to see that the major arcs M(q,a) are mutually disjoint. Thus, using (3.4) and (3.5), we can write R(n,M) =X q≤P X 1≤a≤q (a,q)=1 Z P/(qn)  P/(qn) f(a/q + β)3e (a/q + β)ndβ.(3.7) We now proceed to approximate fa/q + βby a simpler expression. To motivate our choice of the approximation, we  rst consider the case β = 0. We split the sum fa/qinto subsums according to the residue of p modulo q and take into account the de nition (2.6). We get fa q= q X h=1 X p≤n p≡h (mod q) eap q = q X h=1 eah q π(n;q,h).
An invitation to additive prime number theory 21
The contribution of the terms with (h,q) > 1 is negligible (at most q). If (h,q) = 1, our choice (3.3) of the parameter P ensures that we can appeal to Theorem 7 to approximate π(n;q,h) by φ(q) 1 lin. We deduce that fa q= lin φ(q) q X h=1 (h,q)=1 eah q + OqnP 4.(3.8)
The exponential sum on the right side of (3.8) is known as the Ramanujan sum and is usually denoted by cq(a). Its value is known for every pair of integers a and q (see [74, Theorem 271]). In particular, when (a,q) = 1 we have cq(a) = μ(q), where μ is the M¨obius function μ(n) =          1, if n = 1, ( 1)k, if n = p1   pk is the product of k distinct primes, 0, otherwise. (3.9) The situation does not change much if instead of α = a/q we consider α = a/q + β ∈ M(q,a). In this case we  nd that fa q + β= μ(q) φ(q)  v(β) + OnP 3,(3.10) where v(β) =Z n 2 e(βu) logu du. Raising (3.10) to the third power and inserting the result into the right side of (3.7), we obtain R(n,M) =X q≤P μ(q)cq( n) φ(q)3 Z P/(qn)  P/(qn) v(β)3e( βn)dβ + On2P 1.(3.11) At this point, we extend the integration over β to the whole real line, and then the summation over q to all positive integers. The arising error terms can be controlled easily by means of well-known bounds for the functions v(β) and φ(q), and we  nd that R(n,M) = S(n)J(n) + On2P 1,(3.12)
22 A. V. Kumchev, D. I. Tolev
where S(n) and J(n) are the singular series and the singular integral de ned by
S(n) =
∞ X q=1
μ(q)cq( n) φ(q)3
, J(n) =Z ∞  ∞
v(β)3e( βn)dβ.
The series S(n) actually satis es (1.4). Indeed, the function g(q) = μ(q)cq( n)φ(q) 3 is multiplicative in q, that is, g(q1q2) = g(q1)g(q2) whenever (q1,q2) = 1. Hence, using the absolute convergence of S(n) and the elementary properties of the arithmetic functions involved in the de nition of g(q), we can represent the singular series as an Euler product:
S(n) =
∞ X q=1
g(q) =Y p 1 + g(p) + g(p2) +    =Y p|n1  1 (p 1)2Y p-n1 + 1 (p 1)3. Also, an application of Fourier’s inversion formula and some calculus reveal that
J(n) =
n2 2(logn)3
+ On2(logn) 4. Therefore, if B ≥ 4 we can conclude that
R(n,M) =
n2 2(logn)3 S(n) + On2(logn) 4.(3.13) 3.1.3. The minor arcs. In view of (3.6) and (3.13), it su ces to prove that (for some B ≥ 4) R(n,m)n2(logn) 4.(3.14) We have |R(n,m)|≤Zm|f(α)|3 dα ≤sup m |f(α)|Z 1 0 |f(α)|2 dα.(3.15) By Parseval’s identity and the Prime Number Theorem, Z 1 0 |f(α)|2 dα =X p≤n 1 n(logn) 1.(3.16)
An invitation to additive prime number theory 23
Thus, (3.14) will follow from (3.15), if we show that
sup m |f(α)|n(logn) 3.(3.17)
We note that the trivial estimate for f(α) is f(α) X p≤n 1 n(logn) 1, so in order to establish (3.17), we have to save a power of logn over this trivial estimate (uniformly with respect to α ∈ m). We can do this using the following lemma, which provides such a saving under the assumption that α can be approximated by a reduced fraction whose denominator q is “neither too small, nor too large.”
Lemma 3.1. Let α be real and let a and q be integers satisfying 1 ≤ q ≤ n, (a,q) = 1, |qα a|≤ q 1.
Then
f(α) (logn)3nq 1/2 + n4/5 + n1/2q1/2.
This is the sharpest known version of the estimate for f(α) established by I. M. Vinogradov [236] in 1937. As we mentioned in the Introduction, that result was the main innovation in Vinogradov’s proof of Theorem 1. The above version is due to Vaughan [225]. We shall explain the proof of Lemma 3.1 in the next section and now we shall use it to establish (3.17). To this end we need also the following lemma, known as Dirichlet’s theorem on diophantine approximation; its proof is elementary and can be found in [228, Lemma 2.1]. Lemma 3.2 (Dirichlet). Let α and Q be real and Q ≥ 1. There exist integers a and q such that 1 ≤ q ≤ Q, (a,q) = 1, |qα a| < Q 1.
Let α ∈ m. By (3.5) and Lemma 3.2 with Q = nP 1, there are integers a and q such that P < q ≤ nP 1, (a,q) = 1, |qα a| < Pn 1 ≤ q 1.
24 A. V. Kumchev, D. I. Tolev
Hence, an appeal to (3.3) and Lemma 3.1 gives f(α) (logn)3nP 1/2 + n4/5 n(logn)3 B/2.(3.18) and (3.17) follows on choosing B ≥ 12. This completes the proof of Theorem 1. The above proof of Vinogradov’s theorem employs the Siegel–Wal sz theorem and, therefore, is ine ective (recall the remark following the statement of Theorem 7). The interested reader can  nd an e ective proof (with a slightly weaker error term) in [116, Chapter 10].
3.1.4. The estimation of f(α). The main tool in the proof of Lemma 3.1 are estimates for bilinear sums of the form S = X X<x≤2X X Y <y≤2Y xy≤n ξxηye(αxy).(3.19)
We need to control two kinds of such sums, known as type I sums and type II sums. For simplicity, we describe these two types of sums in the simplest cases, noting that the more general sums arising in the actual proof of Lemma 3.1 can be reduced to these special cases using standard trickery:
  type I sums: sums (3.19) with |ξx|≤ 1, ηy = 1 for all y, and X is “not too large”;
  type II sums: sums (3.19) with |ξx| ≤ 1, |ηy| ≤ 1, and X,Y are “neither large, nor small”.
Vinogradov reduced the estimation of f(α) to the estimation of type I and type II sums by means of an intricate combinatorial argument. Nowadays we can achieve the same result almost instantaneously by referring to the combinatorial identities of Vaughan [223, 225] or Heath-Brown [84]. Let Λ(k) denote von Mangoldt’s function, whose value is logp or 0 according as k is a power of a prime p or not. Vaughan’s identity states that if U and V are real parameters exceeding 1, then Λ(k) = X dm=k 1≤d≤V μ(d)log m  X dlm=k 1≤d≤V 1≤m≤U μ(d)Λ(m)  X dlm=k 1≤d≤V m>U,dl>V μ(d)Λ(m).(3.20)
An invitation to additive prime number theory 25
Heath-Brown’s identity states that if k ≤ x and J is a positive integer, then
Λ(k) =
J X j=1J j( 1)j 1 X m1   m2j=k m1,...,mj≤x1/J
μ(m1)   μ(mj)logm2j,
where μ(m) is the M¨obius function. Both identities can be used to reduce f(α) to type I and type II sums with equal success. Here, we apply Vaughan’s identity with U = V = n2/5. We obtain X k≤n Λ(k)e(αk) = W1  W2  W3,(3.21) with Wj =X k≤n aj(k)e(αk) (1 ≤ j ≤ 3) where aj(k) denotes the jth sum on the right side of (3.20). The estimation of the sum on the left side of (3.21) is essentially equivalent to that of f(α). The sums W1 and W2 on the right side of (3.21) can be reduced to type I sums with X  n4/5; W3 can be reduced to type II sums with n2/5 X,Y  n3/5. The reader can  nd all the details in the proof of [228, Theorem 3.1]. Here we will be content with a brief description of the estimation of the type I and type II sums. Consider a type I sum S1. We have |S1|≤ X X<x≤2X
 
 
 X Y <y≤Y 0 e(αxy)

,(3.22) where Y 0 = min(2Y,n/x). We can estimate the inner sum in (3.22) by means of the elementary bound

X a<y≤b e(αy)
 
 
≤ minb a + 1,kαk 1,(3.23) where kαk denotes the distance from α to the nearest integer. This inequality follows on noting that the sum on the left is the sum of a geometric progression. We obtain |S1|≤ X x≤2X minY,kαxk 1= T(α), say.(3.24)
26 A. V. Kumchev, D. I. Tolev
Obviously, the trivial estimate for T(α) is
T(α)  XY. However, under the hypotheses of Lemma 3.1, one can establish by elementary methods that (see [228, Lemma 2.2]) T(α) XY1 q + 1 Y + q XYlog(2XY q).(3.25) Inserting this bound into the right side of (3.24), we obtain a satisfactory bound for S1. To estimate a type II sum S2, we  rst apply Cauchy’s inequality and get |S2|2 Y X Y <y≤2Y
 
 
 X X<x≤X0 ξxe(αxy)

2 , where X0 = min(2X,n/y). Squaring out and interchanging the order of summation, we deduce |S2|2 Y X Y <y≤2Y X X<x1,x2≤X0 ξx1ξx2e(α(x1  x2)y) Y X X<x1,x2≤2X
 
 
 X Y <y≤Y 0 e(α(x1  x2)y)

Y X X<x≤2X X |h|<X
 
 
 X Y <y≤Y 0 e(αhy)

, where Y < Y 0 ≤ 2Y . We remark that the innermost sum is now free of “unknown” weights and can be estimated by means of (3.23). We get |S2|2 XY 2 + XY T(α),(3.26) and (3.25) again leads to a satisfactory bound for S2.
3.2. The exceptional set in Goldbach’s problem. We now sketch the proof of (1.5). We will not discuss the proof of the more sophisticated results of Montgomery and Vaughan [173] and Pintz [186], since they require knowledge of the properties of Dirichlet L-functions far beyond the scope of this survey. The reader can  nd excellent expositions of the Montgomery–Vaughan result in their original paper and also in the monograph [177].
An invitation to additive prime number theory 27
For an even integer n, let r(n) denote the number of representations of n as the sum of two primes, let Z(N) denote the set of even integers n ∈ (N,2N] with r(n) = 0, and write Z(N) = |Z(N)|. Since
E(x) =
∞ X j=1
Z(x2 j),
it su ces to bound Z(N) for large N. De ne f(α), M, and m as before, with N in place of n. When n is an even integer in (N,2N], a variant of the method in §3.1.2 gives ZMf(α)2e( αn)dα = S2(n) n (logn)2 + ON (logN)3, where S2(n) =Y p-n1  1 (p 1)2Y p|np p 1 is the singular series. In particular, we have S2(n) ≥ 1 for even n. Thus, for n ∈Z(N), we have

Zmf(α)2e( αn)dα

=

ZMf(α)2e( αn)dα

N(logN) 2,(3.27) whence Z(N) N 2(logN)4 X n∈Z(N)

Zmf(α)2e( αn)dα

2 .(3.28) On the other hand, by Bessel’s inequality, Xn ∈Z(N)

Zmf(α)2e( αn)dα

2 ≤Zm|f(α)|4 dα,(3.29) and (3.16) and (3.18) yield Zm|f(α)|4 dα ≤sup α∈m|f(α)|2Z 1 0 |f(α)|2 dα  N3P 1(logN)5.(3.30) Combining (3.28)–(3.30), we conclude that Z(N) NP 1(logN)9 N(logN) A,
28 A. V. Kumchev, D. I. Tolev
on choosing, say, P = (logN)A+9. This completes the proof of (1.5).
3.3. The circle method in the Waring–Goldbach problem. We now turn our attention to Theorems 3 and 4. Much of the discussion in §3.1 can be generalized to kth powers (k ≥ 2). Using (3.1), we can write R k,s(n) as R k,s(n) =Z 1 0 f(α)se( αn)dα, where now f(α) =X p≤N eαpk, N = n1/k. De ne the sets of major and minor arcs as before (that is, by (3.4) and (3.5), with P = (logN)B and B = B(k,s) to be chosen later). The machinery in §3.1.2 generalizes to kth powers with little extra e ort. The argument leading to (3.10) gives fa q + β= φ(q) 1S(q,a)v(β) + error term,(3.31) where S(q,a) is de ned by (1.13) and v(β) =Z N 2 eβuk logu du. We now raise (3.31) to the sth power and integrate the resulting approximation for f(α)s over M. Using known estimates for v(β) and S(q,a), we  nd that when s ≥ k + 1, ZMf(α)se( αn)dα = S k,s(n)J k,s(n) + ONs kP 1/k+ε,(3.32) where S k,s(n) is de ned by (1.13) and J k,s(n) is the singular integral J k,s(n) =Z ∞  ∞ v(β)se( βn)dβ = Γs1 + 1 kΓ s k ns/k 1 (logn)s + Ons/k 1(logn) s 1.
An invitation to additive prime number theory 29
This reduces the proof of Theorem 3 to the estimate Zmf(α)se( αn)dα  Ns k(logN) s 1.(3.33) Notice that when k = 1 and s = 3, (3.33) turns into (3.14). Thus, it is natural to try to obtain variants of (3.16) and (3.17) for f(α) when k ≥ 2. To estimate the maximum of f(α) on the minor arcs, we use the same tools as in §3.1.4, that is:   Heath-Brown’s or Vaughan’s identity to reduce the estimation of f(α) to the estimation of bilinear sums X X<x≤2X X Y <y≤2Y xy≤N ξxηyeα(xy)k;
  Cauchy’s inequality to bound those bilinear sums in terms of the quantity T(α) appearing in (3.24). The following result due to Harman [75] is the analogue of Lemma 3.1 for k ≥ 2. Lemma 3.3. Let k ≥ 2, let α ∈ R, and suppose that a and q are integers satisfying 1 ≤ q ≤ Nk, (a,q) = 1, |qα a| < q 1. There is a constant c = c(k) > 0 such that f(α)N(logN)cq 1 + N 1/2 + qN k41 k .
On choosing the constant B (in the de nition of m) su ciently large, one can use Lemmas 3.2 and 3.3 to show that, for any  xed A > 0,
sup α∈m|f(α)|N(logN) A.
Hence, if s = 2r + 1, one has Zm|f(α)|s dα ≤ sup α∈m|f(α)|Z 1 0 |f(α)|2r dα N(logN) AZ 1 0 |f(α)|2r dα,
30 A. V. Kumchev, D. I. Tolev
and it su ces to establish the estimate Ir(N) :=Z 1 0 |f(α)|2rdα N2r k(logN)c,(3.34) with c = c(k,r).
3.4. Mean-value estimates for exponential sums. We now turn to the proof of (3.34). By (3.1), Ir(N) represents the number of solutions of the diophantine equation (xk 1 +   + xk r = xk r+1 +   + xk 2r, 1 ≤ x1,...,x2r ≤ N (3.35) in primes x1,...,x2r, and therefore, Ir(N) does not exceed the number of solutions of (3.35) in integers x1,...,x2r. Using (3.1) to write the latter quantity as a Fourier integral, we conclude that Ir(N) ≤Z 1 0 |g(α)|2rdα, g(α) = X x≤N eαxk.(3.36) This reduces the estimation of the even moments of f(α) to the estimation of the respective moments of the exponential sum g(α), whose analysis is much easier. In particular, we have the following two results. Lemma 3.4 (Hua’s lemma). Suppose that k ≥ 1, and let g(α) be de ned by (3.36). There exists a constant c = c(k) ≥ 0 such that Z 1 0 |g(α)|2kdα N2k k(logN)c.(3.37)
Lemma 3.5. Suppose that k ≥ 11 and g(α) is de ned by (3.36). There exists a constant c = c(k) > 0 such that for r > 1 2 k2(logk + loglogk + c), Z 1 0 |g(α)|2rdα N2r k.(3.38)
These lemmas are, in fact, rather deep and important results in the theory of Waring’s problem. Unfortunately, their proofs are too complicated to include
An invitation to additive prime number theory 31
in this survey in any meaningful way. The reader will  nd a proof of a somewhat weaker version of Hua’s lemma (with a factor of Nε in place of (logN)c) in [228, Lemma 2.5] and a complete proof in [102, Theorem 4]. Results somewhat weaker than Lemma 3.5 are classical and go back to Vinogradov’s work on Waring’s problem (see [102, Lemma 7.13] or [228, Theorem 7.4]). Lemma 3.5 itself follows from the results in Ford [57] (in particular, see [57, (5.4)]). Combining (3.36) and Lemmas 3.4 and 3.5, we get (3.34) with r =(2k 1, if k ≤ 10, 1 2k2(logk + loglogk + c) + 1, if k ≥ 11. Clearly, this completes the proof of Theorem 3, except for the case 6 ≤ k ≤ 8, which we will skip in order to avoid the discussion of certain technical details.
3.5. Diminishing ranges. In this section, we describe the main new idea that leads to the bounds for H(k) in Theorem 4. This idea, known as the method of diminishing ranges, appeared for the  rst time in the work of Hardy and Littlewood on Waring’s problem and later was developed into a powerfull technique by Davenport. The limit of the method employed in §3.3 is set by the mean-value estimates in Lemmas 3.4 and 3.5. The key observation in the method of diminishing ranges is that it can be much easier to count the solutions of the equation in (3.35) if the unknowns x1,...,x2r are restricted to proper subsets of [1,N]. For example, the simplest version of the method that goes back to Hardy and Littlewood uses that when N2,...,Nr are de ned recursively by Nj = k 1N1 1/k j 1 (2 ≤ j ≤ r), the equation (xk 1 +   + xk r = xk r+1 +   + xk 2r, Nj < xj,xr+j ≤ 2Nj (1 ≤ j ≤ r), (3.35*) has only “diagonal” solutions with xr+j = xj, j = 1,...,r. Thus, the number of solutions of (  *) is bounded above by N1   Nr N2 λ 1 (N2   Nr)2 where λ = 1 +1  1 k+   +1  1 kr 1 ≥ k ke r/k.
32 A. V. Kumchev, D. I. Tolev
That is, we have the bound Z 1 0  
 
g1(α)g2(α)   gr(α)

2dα N2 λ 1 (N2   Nr)2,(3.39)
where
gj(α) = X Nj<x≤2Nj
eαxk (1 ≤ j ≤ r). We can use (3.39) as a replacement for the mean-value estimates in §3.4. Let Tk,s(n) denote the number of solutions of
pk 1 + pk 2 +   + pk s = n in primes p1,...,ps subject to Nj < pj,pr+j ≤ 2Nj (1 ≤ j ≤ r), N1 < p2r+1,...,ps ≤ 2N1. Then Tk,r(n) =Z 1 0 f1(α)s 2r+2f2(α)2   fr(α)2e( αn)dα,(3.40) where fj(α) = X Nj<p≤2Nj eαpk (1 ≤ j ≤ r). When r ～ ck logk, we can use (3.39) to derive a bound of the form Z 1 0  
 
f1(α)2f2(α)   fr(α)
 
 2dα  N4 k 1 (N2   Nr)2. Furthermore, assuming that s is just slightly larger than 2r (it su ces to assume that s ≥ 2r + 3, for example), we can then obtain an asymptotic formula for the right side of (3.40) by the methods sketched in §3.3. This is (essentially) how one proves Theorem 4 for k ≥ 11. The proof for k ≤ 10 follows the same general approach, except that we use more elaborate choices of the parameters N1,...,Nr in (  *).
3.6. Kloosterman’s re nement of the circle method. Consider again equation (1.9) with k = 2. The Hardy–Littlewood method in its original
An invitation to additive prime number theory 33
form establishes the asymptotic formula (1.10) for s > 4, but it fails to prove Lagrange’s four squares theorem. In 1926 Kloosterman [122] proposed a variant of the circle method, known today as Kloosterman’s re nement, which he used to prove an asymptotic formula for the number of solutions of the equation a1x2 1 +   + a4x2 4 = n,(3.41) where ai are  xed positive integers. Denote by I(n) the number of solutions of (3.41) in positive integers xi. By (3.1), I(n) =Z 1 0 H(α)e( αn)dα,(3.42) where H(α) = h(a1α)   h(a4α), h(α) = X x≤N eαx2, N = n1/2. A “classical” Hardy–Littlewood decomposition of the right side of (3.42) into integrals over major and minor arcs is of little use here, since we cannot prove that the contribution from the minor arcs is smaller than the expected main term. Kloosterman’s idea is to eliminate the minor arcs altogether. The elimination of the minor arcs requires greater care in the handling of the major arcs. Let X be the integer with X  1 < N ≤ X. It is clear that the integration in (3.42) can be taken over the intervalX 1,1 + X 1 , which can be represented as a union of disjoint intervals X 1,1 + X 1 = [ q≤N [ 1≤a≤q (a,q)=1 a q   1 qq1 , a q + 1 qq2,(3.43) where for each pair q,a in the union, the positive integers q1 = q1(q,a) and q2 = q2(q,a) are uniquely determined and satisfy the conditions N < q1,q2 ≤ 2N, aq1 ≡ 1 (mod q), aq2 ≡ 1 (mod q).(3.44) The decomposition (3.43) is known as the Farey decomposition and provides a natural way of partitioning of the unit interval into non-overlapping major arcs (see Hardy and Wright [74, Section 3.8]). Let M(q,a) denote the interval in the Farey decomposition “centered” at a/q. We have I(n) =X q≤N X 1≤a≤q (a,q)=1 e an q ZB(q,a) Ha q + βe( βn)dβ,(3.45)
34 A. V. Kumchev, D. I. Tolev
where B(q,a) is de ned by
B(q,a) = {β ∈R : a/q + β ∈ M(q,a)}.(3.46) We can  nd an asymptotic formula for the integrand on the right side of (3.45). The contribution of the main term in that asymptotic formula produces the expected main term in the asymptotic formula for I(n). However, in order to obtain a satisfactory bound for the contribution of the error term, we have to take into account the cancellation among terms corresponding to di erent Farey fractions a/q with the same denominator. To this end, we want to interchange the order of integration and summation over a in (3.45). Since the endpoints of B(q,a) depend on a, the total contribution of the error terms can be expressed as X q≤NZ 1/(qN)  1/(qN)X(β) 1≤a≤q (a,q)=1 Ea q + βe an q e( βn)dβ,(3.47)
where E(a/q+β) is the error term in the major arc approximation to H(a/q+β) and the superscript inP(β) indicates that the summation is restricted to those a for which B(q,a) 3 β. Using (3.44) and (3.46), we can transform the latter constraint on a into a condition about the multiplicative inverse of a modulo q, that is, the unique residue class ˉ a modulo q with ˉaa ≡ 1 (mod q). Thus, a special kind of exponential sums enter the scene: the Kloosterman sums
K(q;m,n) =
q X x=1 (x,q)=1
emx + nˉ x q .
There also other (in fact, more substantial) reasons for the Kloosterman sums to appear, but those are too technical to include here. The success of Kloosterman’s method hinges on the existence of su ciently sharp estimates for K(q;m,n). The  rst such estimate was found by Kloosterman himself and later his result has been improved. Today it is known that |K(q;m,n)|≤ τ(q)q1/2 (m,n,q)1/2,(3.48) where (m,n,q) is the greatest common divisor of m,n,q and τ(q) is the number of positive divisors of q. In 1948 A. Weil [243] proved (3.48) in the most important
An invitation to additive prime number theory 35
case: when q is a prime. In the general case (3.48) was established by Estermann [55]. This estimate plays an important role not only in the Kloosterman re nement of the circle method, but in many other problems in number theory. Kloosterman’s method has been applied to several additive problems, and in particular, to problems with primes and almost primes. We refer the reader, for example, to Estermann [56], Hooley [99], Heath-Brown [85, 87, 88], Bru¨dern and Fouvry [24], Heath-Brown and Tolev [94].
4. Sieve methods. In this section we describe the so-called sieve methods, which are an important tool in analytic number theory and, in particular, in the proof of Chen’s theorem (Theorem 2 in the Introduction). We start with a brief account of the main idea of the method (§4.1 and §4.2). This allows us in §4.3 to present a proof of a slightly weaker (but much simpler) version of Chen’s result, in which P2-numbers are replaced by P4-numbers. We conclude the section by sketching some of the new ideas needed to obtain Chen’s theorem in its full strength (§4.4) and of some further work on sieve methods (§4.5). 4.1. The sieve of Eratosthenes. Let A be a  nite integer sequence. We will be concerned with the existence of elements ofAthat are primes or, more generally, almost primes Pr, with r bounded. In general, sieve methods reduce such a question to counting the elements a ∈A not divisible by small primes p from some suitably chosen set of primes P. To be more explicit, we consider a set of prime numbers P and a real parameter z ≥ 2 and de ne the sifting function S(A,P,z) = |{a ∈A: (a,P(z)) = 1}|, P(z) = Y p<z p∈P p,(4.1) where |A| denotes the number of elements of a sequence A (not the cardinality of the underlying set). In applications, the set P is usually taken to be the set of possible prime divisors of the elements of A, so the sifting function (4.1) counts the elements of A free of prime divisors p < z. In order to bound S(A,P,z), we recall the following fundamental property of the M¨obius function (see [74, Theorem 263]): X d|k μ(d) =(1, if k = 1, 0, if k > 1. (4.2) Using this identity, we can express the sifting function in the form S(A,P,z) =X a∈A X d|(a,P(z)) μ(d).(4.3)
36 A. V. Kumchev, D. I. Tolev
We can now interchange the order of summation to get S(A,P,z) = X d|P(z) μ(d)|Ad|,(4.4) where Ad = {a ∈A: a ≡ 0 (mod d)}. To this end, we suppose that there exist a (large) parameter X and a multiplicative function ω(d) such that |Ad| can be approximated by Xω(d)/d. We write r(X,d) for the error term in this approximation, that is,
|Ad| = X
ω(d) d + r(X,d).(4.5) We expect r(X,d) to be ‘small’, at least in some average sense over d. Substituting (4.5) into the right side of (4.4), we  nd that S(A,P,z) = XV (z) + R(X,z),(4.6) where V (z) = X d|P(z) μ(d) ω(d) d , R(X,z) = X d|P(z) μ(d)r(X,d).(4.7) We would like to believe that, under ‘ideal circumstances’, (4.6) is an asymptotic formula for the sifting function S(A,P,z), XV (z) being the main term and R(X,z) the error term. However, such expectations turn out to be unrealistic, as we are about to demonstrate. Let us try to apply (4.6) to bound above the number of primes ≤ x. We choose A= {n ∈ N : n ≤ x}, P = {p : p is a prime}.(4.8) Then |Ad| =hx di= x d + r(x,d), |r(x,d)|≤ 1, that is, X = x and ω(d) = 1 for all d. Using an elementary property of multiplicative functions (see [74, Theorem 286]), we can write V (z) as V (z) =Y p<z1  ω(p) p .(4.9)
An invitation to additive prime number theory 37
When ω(p) = 1, this identity and an asymptotic formula due to Mertens (see [74, Theorem 429]) reveal that the main term in (4.6) is XV (z) = XY p<z1  1 p～ X e γ logz as z →∞;(4.10) here γ = 0.5772... is Euler’s constant. Thus, if A and P are as in (4.8) and z = x1/2, the projected ‘main term’ in (4.6) is ～ 2e γx(logx) 1 as x → ∞, whereas the true size of the sifting function on the left side is S(A,P,√x) = π(x) π(√x) + 1 ～ x logx as x →∞, by the Prime Number Theorem. Since 2e γ = 1.122... , we conclude that the ‘error term’ R(x,√x) is in this case of the same order of magnitude as the ‘main term’. Identity (4.6) is known as the sieve of Eratosthenes–Legendre. The basic idea goes back to the ancient Greeks (usually attributed to Eratosthenes), while the formal exposition above is essentially due to Legendre, who used the above argument to show that π(x)  x loglogx .
The sieve of Eratosthenes–Legendre can be extremely powerful in certain situations7, but in most cases the sum R(X,z) contains ‘too many’ terms for (4.6) to be of any practical use (e.g., in the above example, R(X,z) contains 2π(z) terms). Modern sieve methods use various clever approximations to the left side of (4.2) to overcome this problem. In the following sections, we describe one of the variants of one the existing approaches. The reader can  nd other constructions, comparisons of the various approaches, and proofs in the monographs on sieve methods [63, 66, 174] or in [90] (see also the remarks in §4.5 for other references). 4.2. The linear sieve. Let y > 0 be a parameter to be chosen later in terms of X and suppose that λ+(d) and λ (d) are real-valued functions supported on the squarefree integers d (i.e., λ±(d) = 0 if d is divisible by the square of a prime). Furthermore, suppose that |λ±(d)|≤ 1 and λ±(d) = 0 for d ≥ y,(4.11) 7For example, I. M. Vinogradov’s combinatorial argument for converting sums over primes into linear combinations of type I and type II sums is based on a variant of (4.6). See Harman [79] for other applications and further discussion.
38 A. V. Kumchev, D. I. Tolev
and that
X d|n
λ (d) ≤X d|n
μ(d) ≤X d|n λ+(d) for all n = 1,2,....(4.12) Using (4.3) and the left inequality in (4.12), we obtain S(A,P,z) ≥X a∈A X d|(a,P(z)) λ (d). We can interchange the order of summation in the right side of this inequality and apply (4.5) and (4.11) to get the bound S(A,P,z) ≥ X d|P(z) λ (d)|Ad| = X d|P(z) λ (d)Xω(d) d + r(X,d) = X X d|P(z) λ (d)ω(d) d + X d|P(z) λ (d)r(X,d) ≥ XM   R, where M± = X d|P(z) λ±(d)ω(d) d , R = X d|P(z) d<y |r(X,d)|.(4.13)
In a similar fashion, we can use the right inequality in (4.12) to estimate the sifting function from above. That is, we have XM   R≤ S(A,P,z) ≤ XM+ +R.(4.14) We are now in a position to overcome the di culty caused by the “error term” in the Eratosthenes–Legendre sieve. The sum R is similar to the error term R(X,z) de ned in (4.7), but unlike R(X,z) we can use the parameter y to control the number of terms in R. Thus, our general strategy will be to construct functions λ±(d) which satisfy (4.11) and (4.12) and for which the sumsM± are of the same order as the sum V (z) de ned in (4.7). There are various constructions of such functions λ±(d). However, since it is not our goal to give a detailed treatment of sieve theory here, we will simply state one of the modern sieves in a form suitable for an application to the binary Goldbach problem. The sieve method we will use is known as the Rosser–Iwaniec sieve. Its idea appeared for the  rst time in an unpublished manuscript by Rosser. The full edged version of this sieve was developed independently by Iwaniec [105, 106].
An invitation to additive prime number theory 39
Suppose that the multiplicative function ω in (4.5) satis es the condition Yw 1≤p<w21  ω(p) p  1 ≤logw2 logw1κ1 + K logw1 (2 ≤ w1 < w2),(4.15) where κ > 0 is an absolute constant known as the sieve dimension and K > 0 is independent of w1 and w2. This inequality is usually interpreted as an average bound for the values taken by ω(p) when p is prime, since it is consistent with the inequality ω(p) ≤ κ. In our application of the sieve to Goldbach’s problem, we will have to deal with a sequence A (given by (1.7)) for which (4.15) holds with κ = 1, so we will state the Rosser–Iwaniec sieve in this special case, in which it is known as the linear sieve. Suppose that ω(p) satis es (4.15) with κ = 1 and that
0 < ω(p) < p when p ∈ P and ω(p) = 0 when p 6∈ P.(4.16) We put λ±(1) = 1 and λ±(d) = 0 if d is not squarefree. If d > 1 is squarefree and has prime decomposition d = p1   pr, p1 > p2 >     > pr, we de ne λ+(d) =(( 1)r if p1   p2lp3 2l+1 < y whenever 0 ≤ l ≤ (r 1)/2, 0 otherwise, (4.17) λ (d) =(( 1)r if p1   p2l 1p3 2l < y whenever 1 ≤ l ≤ r/2, 0 otherwise. (4.18)
It can be shown (see [63, 105]) that these two functions satisfy conditions (4.11) and (4.12). Furthermore, if the quantities M± are de ned by (4.13) with λ±(d) given by (4.17) and (4.18), we have V (z) ≤M+ ≤ V (z)F(s) + Oe s(logy) 1/3for s≥ 1,(4.19) V (z) ≥M  ≥ V (z)f(s) + Oe s(logy) 1/3for s≥ 2,(4.20) where s = logy/logz and the functions f(s) and F(s) are the continuous solutions of a system of di erential delay equations (see [63, 105]). The analysis of that system reveals that the function F(s) is strictly decreasing for s > 0, that the function f(s) is strictly increasing for s > 2, and that
0 < f(s) < 1 < F(s) for s > 2.(4.21)
40 A. V. Kumchev, D. I. Tolev
Furthermore, both functions are very close to 1 for large s. More precisely, they satisfy F(s), f(s) = 1 + O(s s) as s→∞.(4.22) Substituting (4.19) and (4.20) into (4.14), we obtain S(A,P,z) ≤ XV (z)F(s) + O(logy) 1/3+R for s ≥ 1,(4.23) S(A,P,z) ≥ XV (z)f(s) + O(logy) 1/3 R for s ≥ 2,(4.24) where R is de ned by (4.13). We now return to our initial goal—namely, to prove that the sequence A contains almost primes. We want to use (4.24) to show that S(A,P,Xα) > 0(4.25) for some  xed α > 0. This will imply the existence of an a ∈ A all of whose prime divisors exceed Xα. If |a| Xg for all a ∈A, it will then follow that A contains a Pr-number, where r ≤ g/α. Clearly, since we want to minimize r, we would like to take α as large as possible. On the one hand, in order to derive (4.25) from (4.24), we need to ensure that the main term in (4.24) is positive and that the error term R is of a smaller order of magnitude than the main term. It is the balancing of these two requirements that determines the optimal choice for z and, ultimately, the quality of our result. In view of (4.21), the positivity of the main term in (4.24) requires choosing y slightly larger than z2. On the other hand, while in some applications the estimation of R is easier than in others, it is always the case that it imposes a restriction on how large we can choose y, and hence, how large we can choose z. In the next section, we demonstrate how this general approach works when applied to the binary Goldbach problem.
4.3. The linear sieve in the binary Goldbach problem. In this section, we apply the linear Rosser–Iwaniec sieve to the sequence A in (1.7) and the set P of odd primes that do not divide n, that is, A = A(n) = {n p : 2 < p < n} and P = {p : p > 2, p - n}. It is clear that all elements of A are odd numbers and that at most logn of them may have a common prime factor with n (for (n,n p) > 1 implies p | n, and n has at most logn odd prime factors). Thus, P is the set of “typical” prime divisors of elements of A.
An invitation to additive prime number theory 41
Next, we proceed to de ne the quantity X and the multiplicative function ω(d) in (4.5). We have |Ad| = X 2<p<n p≡n (mod d) 1 = π(n;d,n) 1,(4.26)
so the prime number theorem for arithmetic progressions suggests the choice X = lin and ω(d) =(d/φ(d) if (d,n) = 1, 0 otherwise. (4.27)
With this choice, the error terms r(X,d) de ned by (4.5) satisfy the inequality |r(X,d)|≤      1 +

π(n;d,n)  lin φ(d)

if (d,n) = 1, 1 otherwise. It then follows from the Bombieri–Vinogradov theorem (Theorem 8) that R≤ y +X d≤y max (a,d)=1

π(n;d,a)  lin φ(d)

n(logn) 3,(4.28) whenever y ≤ n1/2(logn) 6. Furthermore, we have V (z) =Y p<z p-n 1  1 p 1≥Y p<z1  1 p 1(logz) 1.(4.29)
On choosing y = n1/2(logn) 6 and z = n2/9, we have
logy logz
=
9 4
+ Ologlogn logn > 2.2, provided that n is su ciently large. Hence, we deduce from (4.21), (4.24) and (4.27)–(4.29) that S(A,P,z) n(logn) 2.(4.30) That is, there are n(logn) 2 elements of Athat have no prime divisors smaller than n2/9. Since the numbers in A do not exceed n, the elements of A counted
42 A. V. Kumchev, D. I. Tolev
on the left side of (4.30) have at most four prime divisors each, that is, the left side of (4.30) counts solutions of n p = P4. We have some freedom in our choice of parameters in the above argument. For example, we could have set z = nα, where α is any  xed real number in the range 1/5 < α < 1/4. Of course, what we would really like to do is set z = nα, where α > 1/4. With such a choice for z, the above argument would establish the existence of in nitely many solutions to n p = P3. Unfortunately, our choice of z is restricted (via the condition s = logy/logz > 2) by the largest value of Q admissible in the Bombieri–Vinogradov theorem. In particular, in order to be able to choose z = n1/4, we would need a version of the Bombieri–Vinogradov theorem that holds for Q ≤ x1/2+ε.
4.4. Weighted sieves and Chen’s theorem. The idea of a weighted sieve was introduced by Kuhn [124] who observed that instead of the sifting function S(A,P,z) one may consider a more general sum of the type W(A,P,z) = X a∈A, (a,P(z))=1 w(a),(4.31)
where w(a) are weights at one’s disposal to choose. It is common to use weights of the form w(a) = 1  X p|a z≤p<z1 ωp,(4.32)
with suitably chosen 0 ≤ ωp < 1. With such a choice of w(a), (4.31) can be written in the form W(A,P,z) = S(A,P,z)  X z≤p<z1 ωpS(Ap,P,z).(4.33) We can now use an ordinary sieve to estimate the right side of (4.33). For example, we can appeal to (4.24) to bound S(A,P,z) from below and to (4.23) to bound each sifting function S(Ap,P,z) from above. If the resulting lower bound for the right side of (4.33) is positive, we then conclude that there exist elements a of A with w(a) > 0. Such numbers a have no prime divisors p < z and the number of their prime divisors with z ≤ p < z1 can be controlled via the choice of the ωp’s.
An invitation to additive prime number theory 43
The above idea plays an important role in improvements on the result established in§4.3. Using weighted sieves, Buchstab [29] and Richert [196] proved that every su ciently large even n can be represented as the sum of a prime and a P3-number. Richert used weights of the form w(a) = 1 θ X p|a z≤p<z1 1  logp logz1,
while Buchstab’s weights were somewhat more complicated. Chen’s proof of Theorem 2 uses weights of the form (4.32) with z = n1/10, z1 = n1/3, and
ωp =
1 2
+
1 2
δp(a),
where
δp(a) =(1 if a = pp1p2 with p1 ≥ z1, 0 otherwise.
Here, n is the even number appearing in the statement of Theorem 2 and a is an element of the sequence (1.7). With this choice of ωp, successful sifting produces numbers a ∈A with w(a) > 0 and no prime divisors p < n1/10. One can prove that any such number a must in fact be a P2-number. The reader can  nd a detailed proof of Chen’s theorem in [66, Chapter 11], [175, Chapter 10], or [178, Chapter 9].
4.5. Other sieve methods. We conclude our discussion of sieve methods with a brief account of some of the important ideas in sieve theory left out of the previous sections. Selberg’s sieve. The Rosser–Iwaniec sieve de ned by (4.17) and (4.18) is not particularly sensitive to the arithmetical nature of the sequence A that is being sifted. In fact, the only piece of information about A that the Rosser– Iwaniec sieve does take into account is its sieve dimension. Such sieves are known as combinatorial. Selberg [204] proposed another approach, which uses the multiplicative function ω(d) appearing in (4.5) to construct essentially best possible upper sieve weights λ+(d) for a given sequence A. Suppose that ρ(d) is a real function such that ρ(1) = 1. Then
X d|n
μ(d) ≤   X d|n
ρ(d)   
2
.
44 A. V. Kumchev, D. I. Tolev
We can apply this inequality to estimate S(A,P,z) as follows: S(A,P,z) ≤X a∈A    X d|n ρ(d)    2 =X a∈A X d1,d2|n ρ(d1)ρ(d2) = X d1,d2 ρ(d1)ρ(d2)
 
A[d1,d2]
 
, where |Ad| is as before and [d1,d2] is the least common multiple of d1 and d2. Using (4.5), we  nd that S(A,P,z) ≤ XW +R0, where W = X d1,d2 ρ(d1)ρ(d2) ω([d1,d2]) [d1,d2] , R0 = X d1,d2 ρ(d1)ρ(d2)r(X,[d1,d2]). In order to control the “error term” R0, we further assume that ρ(d) = 0 when d > ξ, where ξ > 0 is a parameter. The double sum W is a quadratic form in the variables ρ(d), 1 < d ≤ ξ. Selberg’s idea is to choose the values of these variables as to minimize this quadratic form. More information about Selberg’s sieve—including the techniques used to construct the lower sieve function λ (d) of Selberg’s sieve—can be found in [66, 174] and in Selberg’s collected works [205, 206]. The large sieve. The method known as the large sieve was introduced in 1941 by Linnik [140], but its systematic study did not commence until R′enyi’s work [195] on the binary Goldbach problem. The original idea of Linnik and R′enyi evolved into a general analytic principle that has penetrated analytic number theory on many levels (and perhaps does not warrant the name “sieve” anymore, but the term has survived for historical reasons). The most prominent application of the large sieve is the Bombieri–Vinogradov theorem. The reader will  nd discussion of the number-theoretic aspects of the large sieve in [20, 49, 171] and of the analytic side of the story in [49, 171, 172]. Alternative form of the error term in the sieve. Iwaniec [106] obtained a variant of the linear sieve featuring an error term that is better suited for certain applications than the error term R de ned in (4.13). It is of the form X m<M m|P(z) X n<N n|P(z) ambnr(X,mn),(4.34)
An invitation to additive prime number theory 45
where the coe cients am and bn are bounded above in absolute value and r(X,mn) are the remainder terms de ned earlier. In some applications, one can use the bilinearity of this expression to estimate the double sum when the product MN is larger than the largest value of y for which one can obtain a satisfactory bound forR. Iwaniec [104] used this idea in his proof that certain quadratic polynomials take on in nitely P2-numbers (recall §2.4). Prime detecting sieves. For a long time it was believed that sieve methods are not capable of detecting prime numbers; there are even a couple of prominent papers (see [21, 205]) that quantify the shortcomings of the classical sieve technology. In short, classical sieves are incapable of distinguishing between integers having even number of prime divisors and those having an odd number of prime divisors (this is known in sieve theory as the parity obstacle). A prime detecting sieve overcomes the parity obstacle by combining the general sieve philosophy with additional analytic information. A variant of the basic idea can be traced all the way back to Vinogradov’s work on sums over primes, but the  rst explicit uses of prime detecting sieves appeared in the late 1970s in investigations of the distribution of primes in short intervals (see [91, 107]). The method  ourished during the last decade and has been instrumental in the proofs of several of the restults mentioned in the previous sections: the result of Friedlander and Iwaniec [58] on prime values of x2 + y4; the results of Heath-Brown and Moroz [89, 92] on prime values of binary cubic forms; and the result of Baker, Harman, and Pintz [9] on primes in short intervals are just three such examples. Compared to classical sieve methods, the theory of prime detecting sieves is still in its infancy and thus the general literature on the subject is relatively scarce, but the reader eager to learn more about such matters will  nd two excellent expositions in [59] and [79].
5. Other work on the Waring–Goldbach problem. In the Introduction, we mentioned the cornerstones in the study of the Goldbach and Waring–Goldbach problems. However, as is often the case in mathematics, those results are intertwined with a myriad of other results on various aspects and variants of the two main problems. In this section, we describe some of the more important results of the latter kind. The circle method, sieve methods, or a combination of them play an essential role in the proofs of all these. 5.1. Estimates for exceptional sets. Inspired by the work of Chudakov [42] and Estermann [54] on the exceptional set in the binary Goldbach problem, Hua studied the function h(k), de ned to be the least s such that almost all integers n ≤ x, n ≡ s (mod K(k)), can be written as the sum of s kth
46 A. V. Kumchev, D. I. Tolev
powers of primes (K(k) is de ned by (1.12)). Let Ek,s(x) denote the number of exceptions, that is, the number of integers n, with n≤ x and n ≡ s (mod K(k)), for which (1.14) has no solution in primes p1,...,ps. Hua showed (essentially) that if H(k) ≤ s0(k), then Ek,s(x) = o(x) for any s ≥ 1 2 s0(k). Later, Schwarz [202] re ned Hua’s method to show that Ek,s(x)  x(logx) A(5.1) for any  xed A > 0. In recent years, motivated by the estimate (1.6) of Montgomery and Vaughan, several authors have pursued similar estimates for exceptional sets for squares and higher powers of primes. The  rst to obtain such an estimate were Leung and Liu [134], who showed that E2,3(x) x1 δ, with an absolute constant δ > 0. Explicit versions of this result were later given in [16, 80, 128, 159, 160], the best result to date being the estimate (see Harman and Kumchev [80]) E2,3(x)  x6/7+ε. Furthermore, several authors [80, 147, 149, 155, 249] obtained improvements on Hua’s bound (5.1) for E2,4(x), the most recent being the bound E2,4(x)  x5/14+ε, established by Harman and Kumchev [80]. Ren [194] studied the exceptional set for sums of  ve cubes of primes and proved that E3,s(x)  x1 (s 4)/153 (5 ≤ s ≤ 8). This estimate has since been improved by Wooley [248] and Kumchev [126]. In particular, Kumchev [126] showed that E3,5(x) x79/84, E3,6(x)  x31/35, E3,7(x)  x51/84, E3,8(x)  x23/84. Finally, Kumchev [126] has developed the necessary machinery to obtain estimates of the form Ek,s(x)  x1 δ, with explicit values of δ = δ(k,s) > 0, for all pairs of integers k ≥ 4 and s for which an estimate of the form (5.1) is known. In 1973 Ramachandra [192] considered the exceptional set for the binary Goldbach problem in short intervals. He proved that if y ≥ x7/12+ε and A > 0, then E(x + y) E(x) y(logx) A,
An invitation to additive prime number theory 47
where the implied constant depends only on A and ε. After a series of improvements on this result [8, 52, 53, 111, 112, 115, 135, 167, 182], this estimate is now known for y ≥ x7/108+ε (see Jia [115]). Lou and Yao [164, 250] were the  rst to pursue a short interval version of the estimate (1.6) of Montgomery and Vaughan. Their result was substantially improved by Peneva [180] and the best result in this direction, due to Languasco [131], states that there exists a small constant δ > 0 such that E(x + y) E(x)  y1 δ/600, whenever y ≥ x7/24+7δ. Furthermore, J. Liu and Zhan [157] and Mikawa [169] studied the quantity E2,3(x) in short intervals and the latter author showed that E2,3(x + y) E2,3(x)  y(logx) A for any  xed A > 0 and any y ≥ x1/2+ε.
5.2. The Waring–Goldbach problem with almost primes. There have also been attempts to gain further knowledge about the Waring–Goldbach problem by studying closely related but more accessible problems. The most common such variants relax the multiplicative constraint on (some of) the variables. Consider, for example, Lagrange’s equation
x2 1 + x2 2 + x2 3 + x2 4 = n.(5.2)
Greaves [62] proved that every su ciently large n 6≡ 0,1,5 (mod 8) can be represented in the form (5.2) with x1,x2 primes and x3,x4 (unrestricted) integers. Later, Plaksin [189] and Shields [207] found independently an asymptotic formula for the number of such representations. Bru¨dern and Fouvry [24] proved that every su ciently large integer n ≡ 4 (mod 24) can be written as the sum of four squares of P34-numbers. Heath-Brown and Tolev [94] established, under the same hypothesis on n, that one can solve (5.2) in one prime and three almost primes of type P101 or in four almost primes, each of type P25. Tolev [218] has recently improved the results in [94], replacing the types of the almost primes involved by P80 and P21, respectively. We must also mention the recent result by Blomer and Bru¨dern [17] that all su ciently large integers n such that n ≡ 3 (mod 24) and 5 - n are sums of three almost primes of type P521 (and of type P371 if n is also squarefree).
48 A. V. Kumchev, D. I. Tolev
In 1951 Roth [200] proved that if n is su ciently large, the equation
x3 + p3 1 +   + p3 7 = n(5.3) has solutions in primes p1,...,p7 and an integer x. Bru¨dern [22] showed that if n ≡ 4 (mod 18), then x can be taken to be a P4-number, and Kawada [118] used an idea from Chen’s proof of Theorem 2 to obtain a variant of Bru¨dern’s result for almost primes of type P3. Furthemore, Bru¨dern [23] proved that every su ciently large integer is the sum of the cubes of a prime and six almost-primes ( ve P5-numbers and a P69-number) and Kawada [119] has shown that every su ciently large integer is the sum of seven cubes of P4-numbers. Wooley [249] showed that all but O(logx)6+εintegers n≤ x, satisfying certain natural congruence conditions can be represented in the form (5.2) with prime variables x1,x2,x3 and an integer x4. Tolev [219] established a result of similar strength for the exceptional set for equation (5.2) with primes x1,x2,x3 and an almost prime x4 of type P11.
5.3. The Waring–Goldbach problem with restricted variables. Through the years, a number of authors have studied variants of the Goldbach and Waring–Goldbach problems with additional restrictions on the variables. In 1951 Haselgrove [82] announced that every su ciently large odd integer n is the sum of three primes p1,p2,p3 such that |pi n/3|≤ n63/64+ε. In other words, one can take the primes in Vinogradov’s three prime theorem to be “almost equal”. Subsequent work by several mathematicians [7, 34, 109, 110, 177, 254] tightened the range for the pi’s to |pi  n/3|≤ n4/7 (see Baker and Harman [7]). Furthermore, Bauer, Liu, and Zhan [13, 156, 158] considered the problem of representations of an integer as sums of  ve squares of almost equal primes. The best result to date is due to Liu and Zhan [158], who proved that every su ciently large integer n ≡ 5 (mod 24) can be written as n = p2 1 +   + p2 5, with primes p1,...,p5 satisfying |p2 i  n/5| < n45/46+ε. Liu and Zhan [156] also showed that the exponent 45 46 can be replaced by 19 20 on the assumption of GRH. In 1986 Wirsing [244] proved that there exist sparse sequences of primes S such that every su ciently large odd integer can be represented as the sum of three primes from S. However, his method was probabilistic and did not yield an example of such a sequence. Thus, Wirsing proposed the problem of  nding “natural” examples of arithmetic sequences having this property. The
An invitation to additive prime number theory 49
 rst explicit example was given by Balog and Friedlander [12]. They proved that the sequence of Piatetski-Shapiro primes (recall (2.13)) is admissible for 1 < c < 21/20. Jia [113] improved the range for c to 1 < c < 16/15, and Peneva [181] studied the binary problem with a Piatetski-Shapiro prime and an almost prime. Tolev [215]–[217] and Peneva [179] considered additive problems with prime variables p such that the integers p+2 are almost-primes. For example, Tolev [217] proved that every su ciently large n≡ 3 (mod 6) can be represented as the sum of primes p1,p2,p3 such that p1+2 = P2, p1+2 = P5, and p1+2 = P7. Green and Tao announced at the end of [64] that, using their method, one can prove that there are arbitrarily long non trivial arithmetic progressions consisting of primes p such that p+2 = P2. They presented in [65] a proof of this result for progressions of three primes.
5.4. Linnik’s problem and variants. In the early 1950s Linnik proposed the problem of  nding sparse sequences A such that all su ciently large integers n (possibly subject to some parity condition) can be represented as sums of two primes and an element of A. He considered two special sequences. First, he showed [143] that if GRH holds, then every su ciently large odd n is the sum of three primes p1,p2,p3 with p1  (logn)3. Montgomery and Vaughan [173] sharpened the bound on p1 to p1  (logn)2 and also obtained an unconditional result with p1  n7/72+ε; the latter bound has been subsequently improved to p1  n0.02625 (this follows by the original argument of Montgomery and Vaughan from recent results of Baker, Harman, and Pintz [9] and Jia [114]). Linnik [142, 144] was also the  rst to study additive representations as sums of two primes and a  xed number of powers of 2. He proved,  rst under GRH and later unconditionally, that there is an absolute constant r such that every su ciently large even integer n can be expressed as the sum of two primes and r powers of 2, that is, the equation p1 + p2 + 2ν1 +   + 2νr = n, has solutions in primes p1,p2 and non-negative integers ν1,...,νr. Later Gallagher [60] established the same result by a di erent method. Several authors have used Gallagher’s approach to  nd explicit values of the constant r above (see [137, 138, 150, 151, 152, 242]); in particular, Li [138] proved that r = 1906 is admissible and Wang [242] obtained r = 160 under GRH. Recently, Heath-Brown and Puchta [93] and Pintz and Ruzsa [187] made (independently) an important discovery that leads to a substantial improvement on the earlier results. Their device establishes Linnik’s result with r = 13 (see [93]) and with r = 7 under
50 A. V. Kumchev, D. I. Tolev
GRH (see [93, 187]). Furthermore, Pintz and Rusza [188] have announced an unconditional proof of the case r = 8. There is a similar approximation to the Waring–Goldbach problem for four squares of primes. J. Y. Liu, M. C. Liu, and Zhan [153, 154] proved that there exists a constant r such that every su ciently large even integer n can be expressed in the form
p2 1 + p2 2 + p2 3 + p2 4 + 2ν1 +   + 2νr = n, where p1,...,p4 are primes and ν1,...,νr are non-negative integers. J. Y. Liu and M. C. Liu [148] established this result with r = 8330 and considered also the related problem about representations of integers as sums of a prime, two squares of primes and several powers of 2.
5.5. Additive problems with mixed powers. In 1923 Hardy and Littlewood [71] used the general philosophy underlying the circle method to formulate several interesting conjectures. For example, they stated a conjectural asymptotic formula for the number of representations of a large integer n in the form
p + x2 + y2 = n,(5.4)
where p is a prime and x,y are integers. Their prediction was con rmed in the late 1950s,  rst by Hooley [97] under the assumption of GRH and then unconditionally by Linnik [145]. The reader will  nd the details of the proof in [98, 146]. In another conjecture, Hardy and Littlewood proposed an asymptotic formula for the number of representations of a large integer n as the sum of a prime and a square. While such a result appears to lie beyond the reach of present methods, Miech [166] showed that this conjecture holds for almost all integers n ≤ x. Let Ek(x), k ≥ 2, denote the number of integers n ≤ x such that the equation n = p + xk has no solution in a prime p and an integer x. Miech obtained the bound E2(x)  x(logx) A for any  xed A > 0. Subsequent work of Bru¨dern, Bru¨nner, Languasco, Mikawa, Perelli, Pintz, Polyakov, A. I. Vinogradov, and Zaccagnini [26, 28, 132, 168, 183, 190, 235, 251] extended and sharpened Miech’s estimate considerably. Here is a list of some of their results:   For any  xed k ≥ 2, we have Ek(x)  x1 δk, where δk > 0 depends at most on k; see [28, 190, 235] for the case k = 2 and [183, 251] for the general case.
An invitation to additive prime number theory 51
  Assuming GRH, we have Ek(x)  x1 δk, where δk = 1/(k2k) or δk = 1/(25k) according as 2 ≤ k ≤ 4 or k ≥ 5; see [183] and [26].   If k ≥ 2 is a  xed integer and K = 2k 2 then there exists a small absolute constant δ > 0 such that Ek(x + y) Ek(x)  y1 δ/(5K), provided that x(7/12)(1 1/k)+δ ≤ y ≤ x; see [132]. Furthermore, several mathematicians [14, 15, 26, 157] obtained variants of the above bounds in the case when the variable x is also restricted to primes, while Zaccagnini [252] studied the more general problem of representing a large integer n in the form n = p + f(x), where f(X) ∈ Z[X]. Several interesting theorems were proved by Bru¨dern and Kawada [25]. For example, one of them states that if k is an integer with 3 ≤ k ≤ 5, then all su ciently large integers n can be represented as
x + p2 1 + p3 2 + pk 3 = n,
where pi are primes and x = P2.
5.6. The Waring–Goldbach problem “with coe cients”. In this section we discuss the solubility of equations of the form (1.16), which we introduced in§1.4 as a natural generalization of the Waring–Goldbach problem. There are two substantially di erent contexts in which one can study this problem. Suppose  rst that all a1,...,as,n are all of the same sign. Then one expects that (1.16) must have solutions for su ciently large |n|. When “su ciently large” is understood as |n|≥ C(a1,...,as), with some unspeci ed constant depending on the aj’s, this is a trivial modi cation of the Waring–Goldbach problem (that can be handled using essentially the same tools). On the other hand, the problem of  nding solution when |n| is not too large compared to |a|∞ = max{|a1|,...,|as|} is signi cantly more challanging. Similarly, if a1,...,as are not all of the same sign, one wants to  nd solutions of (1.16) in primes p1,...,ps that are not too large compared to |a|∞ and |n|. Such questions were investigated  rst by Baker [5], who studied the case k = 1 and s = 3. Later, Liu and Tsang [161] showed, again for k = 1 and s = 3, that (1.16) has solutions when:   a1,a2,a3 are of the same sign and |n|  |a|A ∞ for some absolute constant A > 0;
52 A. V. Kumchev, D. I. Tolev
  a1,a2,a3 are not of the same sign and max{p1,p2,p3}|a|A 1 ∞ +|n|. In these results, the coe cients a1,a2,a3,n must satisfy also certain necessary congruence conditions (which generalize the requirement that n be odd in Vinogradov’s three primes theorem). Through the e orts of several mathematicians, the constant A has been evaluated and it is known that the value A = 38 is admissible (see Li [139]). Furthermore, if we replace the natural arithmetic conditions on the coe cients by another set of conditions, which are somewhat more restrictive but also simplify greatly the analysis, we can decrease the value of A further. In particular, Choi and Kumchev [38] have shown that A = 23/3 is admissible under such stronger hypotheses. Liu and Tsang [162] studied also the quadratic case of (1.16) in  ve variables and obtained results similar to those stated above for the linear case. In this problem, explicit values of the analogue of A above were given by Choi and Liu [39, 40], Choi and Kumchev [37], and Harman and Kumchev [80]. In particular, it is proved in [80] that (1.16) with k = 2 and s = 5 has solutions when:   a1,...,a5 are of the same sign and |n||a|15+ε ∞ ;   a1,...,a5 are not of the same sign and max{p1,...,p5}|a|7+ε ∞ +|n|1/2.
5.7. Diophantine inequalities with primes. Some variants of the Waring–Goldbach problem are stated most naturally in terms of diophantine inequalities. The best-known problem of this kind concerns the distribution of the values of the forms
λ1pk 1 +   + λspk s,(5.5) where k and s are positive integers, λ1,...,λs are nonzero real numbers, and p1,...,ps are prime variables. It is natural to conjecture that if λ1,...,λs are not all of the same sign and if λi/λj is irrational for some pair of indices i,j, then the values attained by the form (5.5) are dense in R whenever s ≥ s0(k). In other words, given any ε > 0 and α ∈ R, the inequality

λ1pk 1 +   + λspk s  α

< ε(5.6) should have a solution in primes p1,...,ps. The  rst results in this problem were obtained by Schwarz [203], who established the solvability of (5.6) under the same restrictions on s as in Theorem 3. Baker [5] and Vaughan [221, 222, 224] proposed the more di cult problem of replacing the  xed number ε on the right
An invitation to additive prime number theory 53
side of (5.6) by an explicit function of max{p1,...,ps} that approaches 0 as max{p1,...,ps}→∞. Further work has focused primarily on the case of small k. For example, Harman [78] has shown that under the above assumptions on λ1,λ2,λ3, the diophantine inequality
 
 
λ1p1 + λ2p2 + λ3p3  α
 
< max{p1,p2,p3} 1/5+εhas in nitely many solutions in primes p1,p2,p3. Baker and Harman [6] showed that on GRH the exponent 1 5 in this result can be replaced by 1 4 . Furthermore, Harman [77] proved that if λ1/λ2 is a negative irrational number, then for any real α the inequality
 
 
λ1p + λ2P3  α
 
< p 1/300has in nitely many solutions in a prime p and a P3-almost prime. (This improves on an earlier result of Vaughan [224], where the almost prime is a P4-number.) In 1952 Piatetski-Shapiro [184] considered a variant of the Waring–Goldbach problem for non-integer exponents c > 1. He showed that for any  xed c > 1, which is not an integer, there exists an integer H(c) with the following property: if s ≥ H(c), the inequality
 
 
pc 1 +   + pc s  α
 
< ε(5.7) has solutions in primes p1,...,ps for any  xed ε > 0 and α ≥ α0(ε,c). In particular, Piatetski-Shapiro showed that H(c) ≤ 5 for 1 < c < 3/2. Motivated by Vinogradov’s three prime theorem, Tolev [213] proved that H(c) ≤ 3 for 1 < c < 15/14. The range of validity of Tolev’s result was subsequently extended by several authors [32, 33, 125, 130]; in particular, Kumchev [125] has given the range 1 < c < 61/55. Furthemore, it follows from the work of Kumchev and Laporta [129, 133] that H(c) ≤ 4 for 1 < c < 6/5 and for almost all (in the sense of Lebesgue measure) 1 < c < 2, while Garaev [61] has showed that H(c) ≤5 for 1 < c < (1 +√5)/2 = 1.61.... Finally, Tolev [214] and Zhai [253] have studied systems of inequalities of the form (5.7). Several authors [1, 2, 30, 31] have studied variants of Goldbach’s problem, suggested by results about additive inequalities. For example, Arkhipov, Chen, and Chubarikov [2] proved that if λ1/λ2 is an algebraic irrationality, then all but O(x2/3+ε) positive integers n ≤ x can be represented in the form [λ1p1] + [λ2p2] = n,
54 A. V. Kumchev, D. I. Tolev
where p1,p2 are primes.
6. A new path: arithmetic progressions of primes. Finally, we should say a few words about the astonishing result of Green and Tao [64] on the existence of arbitrarily long arithmetic progressions of prime numbers. They deduce the existence of such arithmetic progressions from a generalization of a celebrated theorem of Szemer′edi [209, 210], which is itself a deep result in combinatorial number theory. Let A be a set of positive integers with positive upper density, that is,
δ(A) = limsup N→∞
#{n ∈A: n ≤ N} N
> 0.
In its original, most basic form, Szemer′edi’s theorem asserts that such a set A contains an arithmetic progression of length k for all integers k ≥ 3. From this basic statement, Green and Tao deduce the following more general result.
Theorem 9 (Szemer′edi’s theorem for pseudorandom measures). Let δ ∈ (0,1] be a  xed real number, let k ≥ 3 be a  xed integer, and let N be a large prime. Suppose that ν is a “k-pseudorandom measure8” on ZN = (Z/NZ) and f : ZN → [0,∞) is a function satisfying 0≤ f(x) ≤ ν(x) for all x ∈ ZN(6.1) and Xx ∈ZN f(x)≥ δN. Then Xx ∈ZN X r∈ZN f(x)f(x + r)...f(x + (k 1)r) N2,(6.2) the implied constant depending at most on δ and k. To relate this result to the version of Szemer′edi’s theorem stated earlier, consider the case where ν(x) = 1 for all x (this is a k-pseudorandom measure) and f(x) is the characteristic function of the set AN = A∩ [1,N] considered 8A k-pseudorandom measure on ZN is a non-negative function on ZN whose average over ZN is close to 1 and which is subject to a couple of additional constraints that are too technical to state here. See [64] for details.
An invitation to additive prime number theory  55
as a subset of ZN. Then the left side of (6.2) counts (essentially) the k-term arithmetic progressions in the set AN (the majority of which are also k-term arithmetic progressions in A∩Z). To derive the result on arithmetic progressions of primes, Green and Tao take f(x) to be a function which, in some sense (see [64] for details), approximates the characteristic function of the primes in the interval [c1N,c2N], where 0 < c1 < c2 < 1 are suitable constants. Then they construct a pseudorandom measure ν(x) such that (6.1) holds. This leads to the following theorem. Theorem 10 (Green and Tao, 2004). Let k ≥ 3 and let A be a set of prime numbers such that
limsup N→∞
#{n ∈A : n ≤ N} π(N)
> 0.
Then A contains in nitely many k-term arithmetic progresions. In particular, there are in nitely many k-term arithmetic progresions of prime numbers. We remark that the in nitude of the k-term progressions of primes is a consequence of (6.2). In fact, using the explicit form of the function f(x) to which they apply Theorem 9, Green and Tao establish the existence of N2(logN) k k-term progressions within A∩[1,N]. Several other interesing results are announced in [64]. For example, one of them asserts that there are in nitely many progressions of primes p1,...,pk such that each pi + 2 is a P2-number (a proof of this result in the case k = 3 is presented in [65]).
Conclusion. With this, our survey comes to a close. We tried to describe the central problems and the main directions of research in the additive theory of prime numbers and to introduce the reader to the classical methods. Complete success in such an undertaking is perhaps an impossibility, but hopefully we have been able to paint a representative picture of the current state of the subject and to motivate the reader to seek more information from the literature. Maybe some of our readers will one day join the ranks of the number theorists trying to turn the great conjectures mentioned above into beautiful theorems!
Acknowledgement. This paper was written while the  rst author enjoyed the bene ts of postdoctoral positions at the University of Toronto and at the University of Texas at Austin. He would like to take this opportunity to express his gratitude to these institutions for their support. The second author was supported by Plovdiv University Scienti c Fund grant 03-MM-35. Last but
56 A. V. Kumchev, D. I. Tolev
not least, the authors would like to thank Professors J. Friedlander and D.R. Heath-Brown for several valuable discussions over the years and for some useful comments about the preliminary version of the survey.
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A. V. Kumchev Department of Mathematics 1, University Station, C1200 The University of Texas at Austin Austin, TX 78712, U.S.A. e-mail: kumchev@math.utexas.edu
D. I. Tolev Department of Mathematics Plovdiv University “P. Hilendarski” 24, Tsar Asen Street 4000 Plovdiv, Bulgaria e-mail: dtolev@pu.acad.bg

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