陈景润与华罗庚学派

《堆垒素数论》是1940年（民国二十九年），国立西南联合大学的教授华罗庚在一个吊脚楼上，用八个月完成了第一部数学专着。本书于1940-1941年(1939-1941年)成书，最初提交苏联科学院出版。但由于1941-1945年的战争条件，中国科学院于1953年在苏联以俄语出版。

1954年，陈景润存在厦门大学数学系数据室担任数据员 阅读华罗庚的《堆积素数论》，深入研究塔内问题。第二年，他发表了一篇塔内问题（组合数学）的论文，引起了华罗庚的注意。1956年，陈景润被邀请到北京参加一次数学会议，报告塔内问题。
1957年9月，陈景润转入中国科学院数量研究所，跟随华罗庚研究堆垒素数论。1973年，陈景润定理发表，成为堆垒素数论的最佳代表作。从此，华罗庚V堆垒素数论闻名于世！

袁萌 陈启清 2月11日

附件：堆垒素数论中陈景润。地位可见

Serdica Math. J. 31 (2005), 1–74
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.)
1An 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 thatone 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|n 1  1 (p 1)2 Y p-n 1 + 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) = O x1 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>2 1  1 (p 1)2 Y p>2 p|n  p 1 p 2  n (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 k Γ 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
e ahk 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 k Γ 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, over Pn 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)n dβ.(3.7) We now proceed to approximate fa/q + β by a simpler expression. To motivate our choice of the approximation, we  rst consider the case β = 0. We split the sum fa/q into subsums according to the residue of p modulo q and take into account the de nition (2.6). We get f a q = q X h=1 X p≤n p≡h (mod q) e ap q  = q X h=1 e ah 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 f a q = lin φ(q) q X h=1 (h,q)=1 e ah 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 f a 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|n 1  1 (p 1)2 Y p-n 1 + 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)3 nq 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)3 nP 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 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=1 J 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 Y

,(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

≤ 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)   XY 1 q + 1 Y + q XY log(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

X X

2 , where X0 = min(2X,n/y). Squaring out and interchanging the order of summation, we deduce |S2|2   Y X Y

X Y

Y X X

X Y

, 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 + O  N (logN)3 , where S2(n) =Y p-n 1  1 (p 1)2 Y p|n  p 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 e αpk , 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 f a 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) + O Ns 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 k Γ s k  ns/k 1 (logn)s + O ns/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 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)c q 1 + N 1/2 + qN k 41 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 e αxk .(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 k r 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 e αxk  (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

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) H a q + β e( 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 E a q + β e  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
e mx + 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