SOME PROBLEMS IN ADDITIVE NUMBER THEORY

Transcription

1 SOME PROBLEMS IN ADDITIVE NUMBER THEORY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by John W. Hoffman August 205

2 Dissertation written by John W. Hoffman B.S., Youngstown State University, 2009 M.A., Kent State University, 20 Ph.D., Kent State University, 205 Approved by Dr. G. Yu, Chair, Doctoral Dissertation Committee Dr. U. Vorhauer, Member, Doctoral Dissertation Committee Dr. M. Davidson, Member, Doctoral Dissertation Committee Dr. S. Gagola, Member, Doctoral Dissertation Committee Dr. J. Maletic, Dr. J. Khan, Member, Outside Discipline Member, Graduate Representative Accepted by Dr. A. Tonge, Chair, Department of Mathematical Sciences Dr. J. Blank, Dean, College of Arts and Sciences ii

3 TABLE OF CONTENTS ACKNOWLEDGEMENTS v NOTATION vi Introduction Waring s Problem Goldbach s Problem Waring-Goldbach Problems Mixed Power Problems Thin Sets Preliminaries The Hardy-Littlewood Method Weyl s Inequality Exponential Sums over Prime Variables Hua and van der Corput Inequalities Exponent Pair Method Double Large Sieve Method Representations of n = p 2 + p2 2 + p2 3 + p2 γ + qγ Introduction The Minor Arcs The Major Arcs Estimating f γ g iii

4 3.5 Estimating Type I and Type II Sums Representations of n = [m c ] + p k The Treatment of S The Treatment of S The Error Term The Main Error Term Representations of N = m 2 + p 3 + q Exponential Sums over Squarefree Numbers Proof of Theorem Proof of Theorem Proof of Theorem Singular Series BIBLIOGRAPHY iv

5 ACKNOWLEDGEMENTS In my nine year educational journey, I have had the help of many people along the way. This dissertation would not be possible without their support. First, I would like to thank my committee members for their suggestions and corrections to my work. In particular, the number theory seminar group have been wonderful teachers, mentors and friends during my time at Kent. I want to thank Dr. Yu for his patience and wisdom throughout my research process. Every conversation with him always ends with me feeling confident and capable. He has been a great mentor, and he helped me unlock my potential as a number theorist. Also, I want to thank my friends at YSU for their encouragement and support throughout the years. My experience in Dr. Faires calculus class is the reason I decided to pursue mathematics. Similarly, my love of number theory began in Dr. Fabrykowski s number theory class. Finally, I want to thank my family and friends for their love and support. I want to thank my parents for instilling a high value in education since grade school, and my big sister (the original Dr. Hoffman) for being the best role model in the world. Lastly, I want to thank Lindsay for being my even prime and supporting me throughout my education. v

6 NOTATION Standard notation is used whenever possible. Common notation used throughout the text is explained here, while more specific notation will be given in the body of the text. k, l, m, n, M, N Natural numbers. p A prime number. α, x, y Real variables. ζ(s) Riemann s zeta function defined by ζ(s) = n= n s when Re(s) > and the analytic continuation to the whole complex plane. exp z = e z. e(x) = e 2πix. log x =ln x. t =min n Z t n. [x] The greatest integer not exceeding x. {x} = x [x]. ψ(x) = x [x] 2. Λ(n) The von Mangoldt function defined by Λ(n) = log p if n = p m and zero otherwise. µ(n) The Möbius function defined by µ(n) = ( ) k if n = p p k (the p j being distinct), µ() =, and zero otherwise. d n A sum taken over all positive divisors of n. π(x) The number of primes not exceeding x. vi

7 f(x) = O(g(x)) Means f(x) Cg(x) for x x 0 and some absolute constant C > 0. f(x) g(x) f(x) g(x) f(x) g(x) Means f(x) = O(g(x)). Means g(x) = O(f(x)). Means g(x) f(x) g(x) m M Means M < m 2M. ε, δ An arbitrarily small constant, not necessarily the same at each occurence. vii

8 CHAPTER Introduction In this chapter, we introduce the historical background of the techniques and problems investigated in the later chapters. We will begin with the classic Waring and Goldbach problems and discuss the results on these problems. In addition, we will explore variations on these classics, including Waring-Goldbach and mixed power problems. Lastly, we will introduce the history of Piatetski-Shapiro primes and the results on their existence. This chapter also includes the theorems that will be proved in the later chapters.. Waring s Problem Additive problems have a long history in mathematics. The classic example is regarding representing integers as sums of squares. This problem was studied throughout history by many different people. The well known four square theorem was first stated by Bachet in 62 (the result was most likely known by Diophantus), and its proof was claimed by Fermat. Fermat, however, did not provide the details of the proof. It wasn t until 770 when Lagrange provided a proof of the theorem by building on the earlier work of Euler. This problem can be generalized to any k-th powers. In 770 Edward Waring stated (without proof) in his book Meditationes Algebraicae that every integer is a sum of 9 cubes, 9 fourth powers, and so on. That is, there exists an s such that every natural number n can be represented as a sum of s k-th powers. Let g(k) represent the minimum such s. Waring claimed that g(3) = 9, g(4) = 9 and so on. Of course Lagrange proved g(2) = 4. In 859 Liouville gave the next Waring-type result by showing g(4) 53. The existence of g(k) for all k was done by Hilbert in 909 [40]. Hilbert s proof is complicated and provides a

9 poor bound for g(k). With the establishment of the existence of g(k), the natural problem is to determine the size of g(k). Hardy and Littlewood established an analytic method which greatly improved Hilbert s bound for g(k). Their method, also known as the circle method, is the basis for many problems in additive number theory. Through refinements of the Hardy-Littlewood method, the value, g(k) is almost completely settled. In particular, the integer n = 2 k [(3/2) k ] is less than 3 k and thus can only be represented by k-th powers of and 2. The best representation includes [(3/2) k ] k-th powers of 2 and 2 k powers of. Thus g(k) [(3/2) k ] + 2 k 2. It is likely that this always holds with equality. In fact Mahler (957) [65] showed this holds for all but finitely many k. No exceptions are currently known, and unfortunately there is no known bound for which there are no exceptions. In 942 Linnik [56] proved only finitely many integers require 8 or more cubes. That is if n is sufficiently large, it can be represented as a sum of 7 cubes. Since g(k) is usually determined by the requirements of a few specific integers (for instance 23 requires 9 cubes), it is more interesting to study the case for sufficiently large integers. Let G(k) be the minimum s such that all sufficiently large integers are representable by s k-th powers. Linnik s result can be restated as G(3) 7. Clearly G(k) g(k). Since G(k) is much smaller than g(k) when k is large, its evaluation is more difficult. Currently, the only known values of G(k) are G(2) = 4 and G(4) = 6, the latter result due to Davenport in 939 [25]..2 Goldbach s Problem Another historically famous additive problem is Goldbach s conjecture. In 742 Christian Goldbach conjectured in a letter to Euler that every even integer is a sum of 2 primes and every integer is a sum of 3 primes. Goldbach included as a prime number and thus Goldbach s conjectures have become the statements every even integer greater than 2 is the sum of two primes and every odd integer greater than 5 is the sum of 3 primes. Hardy and 2

10 Littlewood [35] showed all sufficiently large odd integers are representable as a sum of three primes under the assumption of the generalized Riemann hypothesis. In 937 Vinogradov removed the dependence of the generalized Riemann hypothesis and gave an unconditional proof of Hardy and Littlewood s result. In particular Vinogradov [89] proved the following result: For a positive integer odd n, let R(n) denote the number of representations of n as the sum of three primes. Then where R(n) = S 0 (n) = p n n 2 2(log n) 3 S 0(n) + O(n 2 (log n) 4 ) ( (p ) 2 ) p n ( ) + (p ) 3. In particular, every sufficiently large odd integer is a sum of three primes. It should be remarked that the removal of the dependence on GRH comes at the cost of a large implied constant. Hardy and Littlewood s proof (GRH dependent) establishes the ternary Goldbach conjecture for all n 0 50, where as Vinogradov s method requires n The bound, 0 50, is beyond checking the remaining cases by computer. However, recently Deshouillers et al. [46] proved that if GRH is true then the ternary Goldbach conjecture holds for all odd n 7. Most recently, Harold Helfgott claims to have settled the problem unconditionally [39] [38]. Work on the binary Goldbach problem has been less conclusive. Following Vinogradov s method, Chudakov [22], van der Corput [82], and Estermann [29] each showed that almost all even integers n x are sums of two primes. More precisely, let E(x) denote the number of integers n x such that n is not the sum of two primes. They showed that for any A > 0 we have E(x) = O(x(log x) A ). 3

11 Some improvements have been made on this result. Montgomery and Vaughan [67] proved the existence of an absolute constant δ > 0 such that E(x) = O(x δ ). Chen and Pan [20] computed a numerical value for the constant. They showed that the result of Montgomery and Vaughan gives the constant δ = 0.0. This result has been improved by a number of authors and in 2004 Pintz established the bound with δ = /3 [69]. Although the binary Goldbach conjecture is out of reach, there have been some results just short of the conjecture. We denote P r as a number having at most r prime factors counting multiplicity. Such a number is called an almost prime of order r. In 973, Chen proved a result that is just short of the binary Goldbach conjecture. His result is stated below. [9] For an even integer n, let r(n) denote the number of representations of n in the form n = p + P 2, where p is a prime and P 2 is an almost prime of order 2. There exists an absolute constant n 0 such that if n n 0, then r(n) > 0.67 p>2 ( (p ) 2 ) p>2 p n ( ) p n p 2 (log n) 2. In particular, every sufficiently large even integer n can be represented in the form n = p+p 2..3 Waring-Goldbach Problems In the last century several variations to Waring s problem have been considered. Vinogradov s work on Goldbach s conjecture provides a model to adapt Hardy and Littlewood s method to problems involving sums of primes. We will begin by describing the necessary congruence conditions for Waring s problem over primes. Suppose k is a positive integer and p is a prime, and denote θ = θ(k, p) to be the power of p that exactly divides k i.e. p θ k 4

12 but p θ+ k. Now, define γ = γ(k, p) by θ + 2, when p = 2 and θ > 0, γ(k, p) = θ +, otherwise. Lastly, define K(k) = p γ. (p ) k The Diophantine equation in consideration is everywhere locally solvable if it is solvable modulo K(k). Now, we let H(k) denote the minimal s such that every sufficiently large n congruent to s modulo K(k) can be represented as a sum of s kth powers of primes. Following the earlier work of Vinogradov [89] [90], Hua studied a variety of problems involving primes in his book. Indeed, in [44] he proved H(k) 2 k + and when k is large H(k) 4k(log k + log log k + O()). 2 The approach to the Waring-Goldbach problem is to proceed with the Hardy-Littlewood method as in the classic Waring s problem and removing the primality condition on some estimates to obtain an upper bound. There have been some improvements to H(k) for small values of k. In particular Kawada and Wooley proved H(4) 4, and H(5) 2 [50]. Also, Thanigasalam proved the estimates H(6) 33, H(7) 47, H(8) 63, H(9) 83, and H(0) 07 [8]..4 Mixed Power Problems The Hardy-Littlewood method can be adapted to attack many different types of additive problems. In addition to Waring-Golbach problems, there has been some attention given to mixed power problems. Hardy and Littlewood formulated several conjectures on asymptotic 5

13 formulas concerning different representations of integers. In particular, they conjectured an asymptotic formula for the number of representations of a sufficiently large integer n in the form n = p + x 2 + y 2 where p is a prime and x and y are integers. Their conjecture was confirmed by Hooley [4] under the conditions of GRH and later unconditionally by Linnik [58]. Another famous conjecture of Hardy and Littlewood is concerning an asymptotic formula for the number of representations of a sufficiently large integer, n, as a sum of a prime and a square. For non-square n, let R(n) be the number of representations of n as n = m 2 + p. The conjectured asymptotic formula is given by R(n) = n log n p>2 ( ( n p ) ) p (.) where ( n p ) represents the Legendre symbol. This result appears to be out of reach with current methods, but a number of authors have considered this problem. In 936 Davenport and Heilbronn [26] gave a proof that the asymptotic formula (.) holds for all but O(N(log N) A ) integers n N and some positive constant A. Meich [66] provided the first improvement to Davenport and Heilbronn. In particular, let E k (N) denote the number of integers n N not representable in the form n = p + x k where p is a prime and x an integer. Meich proved E 2 (N) N(log N) A for any fixed A > 0. The next improvement came from Polyakov [7]. He showed that R(n) = n log n p>2 ( ( n p ) )( + O ( ) exp ( log/2 n p log 3 log n )) for all but O(N exp ( c log N)) integers n N for some constant c. Polyakov s improvement was followed by a result of Vinogradov. Vinogradov [88] showed that the number of exceptions is at most O(N θ ) for some constant θ <. Brünner, Perelli, and Pintz [8] 6

14 independently proved the same result as Vinogradov, but with a different approach. Vinogradov used the original approach of Hardy and Littlewood. where Brünner, Perelli, and Pintz used a method developed by Montgomery and Vaughan. In addition to the study of the classic conjecture, a number of authors have considered the more challenging generalization of representing a positive integer n in the form n = m k + p. In the study of this generalization, Davenport and Heilbronn s 936 paper included a result on the general problem. They proved that almost all positive integers n can be represented in the form n = m k + p. The most recent result concerning this problem is due to A. Zaccagnini in 992. Zaccagnini [92] proved that the number of exceptions is bounded by E k (X) X δ for some δ = δ(k) > 0. One final variation of this conjecture has been studied throughout the years. This variation concerns representing an integer n in the form n = p + q k where p and q are primes. This problem was first considered by W. Schwarz in 96. Schwarz built on the earlier work of Prachar to establish a number of results for various prime diophantine equations. Plaksin [70] improved this result in 990. Plaksin showed that the number of even integers, n N, not representable in the form n = p + q k is N γ for some γ < and γ < 37k 3 log k for large k. Liu and Chiu [2] considered variations of the Hardy-Littlewood conjecture which included coefficients. They considered representations of integers in the forms n = a p + a 2 m k and n = a p + a 2 q k, where p and q are primes and m is an integer. If we let E (N), and E 2 (N) denote the number of exceptions n N of these problems respectively, Liu and Chiu proved E (N) N θ and E 2 (N) N θ 2 for some θ, θ 2 <. Although this conjecture is presently unobtainable, we prove the following result inspired by this conjecture. 7

15 Theorem. Every sufficiently large n is representable in the form n = [m c ] + p k where m is an integer and p is a prime provided c θ. Here, θ = k ( + µ λ ) ( λ 2 (k )µ 2 ) µ ( λ 2 ) 2(µ 2 + )( + µ λ ) + 2( λ 2 )( + µ λ ) + ( λ 2 ) where (µ, λ ) and (µ 2, λ 2 ) are exponent pairs. In particular, if k = 2, c We remark that the result for k = 2 is not optimal. Here we obtain a representation of all sufficiently large integers, but in this case [m c ] is far from a square. We also have a generalization in that we include a power of p as well. In addition to these conjectures, many other variations of mixed power problems have been considered. A special case concerns the representation of n as a sum of s consecutive powers starting with a square. This question was answered by Roth [79] in 95 for sufficiently large n with s = 50. This result was later improved by Thanigasalam [8], Vaughan [83], Brüdern [3], and Ford [30]. The current best result belongs to Ford with s = 5. In general, we consider representations as non-consecutive mixed powers. That is suppose that k, k 2,... k s are natural numbers with 2 k k 2 k s. Then we consider representations of integers n as n = x k + xk xks s where each x i is a positive integer. Heuristic applications of the Hardy-Littlewood method suggest that all sufficiently large n have such a representation provided that k + k k s > 2, (.2) and a representation exists for almost all n provided k + k k s >. (.3) 8

16 Table.: Representations of almost all integers Davenport and Heilbronn [27] x 2 + x2 2 + yk Davenport and Heilbronn[28] x 2 + y 3 + y3 2 Roth [78] x 2 + y 3 + z 4 Vaughan[84] x 2 + y 3 + z 5 Hooley [43] x 2 + y 3 + z 6 + w k Davenport [24] x 3 + x3 2 + x3 3 + x3 4 Brüdern [0] x 3 + x3 2 + x3 + y4 Brüdern [0] and Lu [64] x 3 + x 2 + x 3 + y5 Brüdern and Wooley [7] x 3 + x 2 + x y6 Kawada and Wooley [49] x 3 + y 4 + y4 2 + y4 3 + y4 4 Vaughan [87] x 4 + x4 2 + x4 3 + x4 4 + x4 5 + x4 6 Kawada and Wooley [49] x 4 + x4 2 + x4 3 + x4 4 + x4 5 + yk (k odd) Of course, the condition (.3) is necessary for such a representation of all integers. The study of such problems has stimulated progress on the methods used in the classical Waring s problem. Problems involving smaller powers tend to be of greater interest than those of higher powers, and a great deal is known about these cases. In particular, the problem of representing a natural number n in the form n = x 2 + y 3 + z 5 for x, y and z in the naturals has been of particular interest. R.C. Vaughan [84] first showed that almost all integers have such a form. Later Brüdern improved Vaughan s result by showing there are at most O(N /42+ε ) exceptions up to N [2]. He later improved this result to at most O(N /30+ε ) exceptions [4]. This is the best result that can be expected with the current tools available. Many other variations have been studied as well. Here we summarize the results concerning mixed power problems. In particular, for all ternary problems subject to (.3), an almost all type of result has been proved. As with Waring s problem, it is natural to consider mixed power problems with prime variables. That is, we also consider representations of integers as a sum of mixed powers of 9

17 Table.2: Representations of all sufficiently large integers Gauss [33] x 2 + x2 2 + x2 3 Hooley [42] x 2 + x2 2 + y3 + y3 2 + y3 3 Hooley [43] x 2 + x2 2 + assorted powers Vaughan [86] x 2 + y3 + y3 2 + y3 3 + y3 4 + y3 5 Brüdern and Wooley [5] x 2 + y3 + y3 2 + y3 3 + y3 4 + y3 5 + z4 Ford [3] x 2 + x3 2 + x x5 4 Linnik [57] x 3 + x3 2 + x3 3 + x3 4 + x3 5 + x3 6 + x3 7 Brüdern [] x 3 + x3 2 + x3 3 + x3 4 + x3 5 + x3 6 = Y 4 + y4 2 Brüdern [] x 3 + x3 2 + x3 3 + x3 4 + x3 5 + y4 + y4 2 + y4 3 Kawada and Wooley [49] x 3 + x3 2 + x3 3 + y4 + y4 2 + y4 3 + y4 4 + y4 5 + y4 6 Brüdern and Wooley [6] x 3 + x3 2 + y4 + y4 2 + y4 3 + y4 4 + y4 5 + y4 6 + y4 7 Kawada and Wooley [49] x 3 + y4 + y4 2 + y4 3 + y4 4 + y4 5 + y4 6 + y4 7 + y4 8 + y4 9 Vaughan [87] x 4 + x x4 2 Kawada and Wooley [49] x 4 + x x4 0 + yk (k odd) primes. In particular, the problem of representing an integer n in the form n = p 2 + p p p 4 k (.4) for prime numbers p j, j =, 2, 3, 4. We should mention that, when the p j s are replaced by natural numbers, the equation (.4) has received a lot of attentions (c.f. [80], [8], [23] for example). Let E(k, N) be the number of even integers n N that can not be represented by (.4). In 953, Prachar [72] showed that E(4, N) = O(N(log N) 30/47+ε ). This was later improved by a number of authors (c.f. [5], [6], [7], [73], [74]). The current best result is 47 E(4, N) N 680 +ε, (.5) which belongs to Bauer [8]. For general k, Lu and Shan proved in [63] that E(k, N) = O(N(log N) c ) for some c > 0. This was recently improved to E(k, N) N 3k 2 k 2 +ε (.6) by Liu [62]. Here we give improvements to (.5) and (.6). Theorem 2. We have 67 E(4, N) N ε, (.7) 0

18 and, for k 5, 47 E(k, N) N s +ε, (.8) where s = [ k+ 2 ]. The improvements (.7) and (.8) are obtained by improving the minor arc estimates in [8] and [62], respectively. Additionally, we consider the ternary problem of representing an integer n in the form n = m 2 + p 3 + q 5 (.9) where m is squarefree and p and q are primes. We will establish the following theorem: Theorem 3. Let E(n) be the number of positive integers not exceeding n which cannot be written in the form (.9). Then we have E(n) n 45 +ε, where ε is any fixed positive number. We remark that the number 45 in the exponent can be improved to 30 if, in (.9), m is relaxed to ordinary integers. Concerning representing integers in the form of (.9), the most challenging problems are probably to improve Brüdern s result to O(N 30 δ ) for any δ > 0, or show that almost all odd integers are representable by (.9) with m a prime as well..5 Thin Sets Another application of exponential sums is the study of primes in thin sets. That is a set of primes, S, is said to be thin if = o(π(x)). p x p S

19 In particular, the set P c = {p : p prime and p = [n c ] for some integer n} is a well known thin set of primes known as Piatetski-Shapiro primes. Let π c (x) denote the number of primes, p x, in this form. It is a straightforward result from the prime number theorem to show that π c (x) = x ( ) x c log x + O c log 2 x (.0) holds for 0 < c. This result is expected to hold for c in the range < c < 2 as well. The problem of representing primes by linear polynomials is settled by Dirichlet s theorem on primes in arithmetic progressions, but it is not known if there is a quadratic polynomial which takes on infinitely many primes. Representing primes in the form [n c ] for < c < 2 can be regarded as representing primes as a polynomial of degree c. Piatetski-Shapiro proved that this result holds when 0 < c < 2/ = [68], and such primes are called Piatetski-Shapiro primes in his honor. A number of authors have made improvements to the range of c for which this asymptotic formula holds. Kolesnik [5] was the first to provide a improvement. He proved that (.0) holds for 0 < c < 0/9 =... Graham (unpublished) and Leitmann [55] improved this result to 0 < c < 69/62 = using the method of exponent pairs. Heath-Brown [37] gave the improvement 0 < c < 755/662 = using a discrete Weyl shift and exponent pairs along with his decomposition of the von Mangoldt function. Kolesnik [52] provided another improvement on Heath-Brown s result. He gave the range 0 < c < 39/34 = using multiple exponential sums. Liu and Rivat [59] used the double large sieve technique of Iwaniec and Fouvry to obtain the range 0 < c < 5/3 = The current best result is due to Rivat and Sargos [76]. They gave the range 0 < c < 287/2426 = In addition to the studying the range of c for which (.0) holds, some authors have considered the range for which π c (x) x c log x. (.) 2

20 J. Rivat [75] was the first to consider this problem. In his doctoral thesis, he proved that there exists an absolute constant ρ 0 > 0 such that x π c (x) ρ 0 c log x for each fixed c such that < c < 7/6 = Subsequent improvements came independently from Baker, Harmon, and Rivat [3] and Jia [48] using the double large sieve of Iwaniec and Fouvry. Baker et al proved (.) holds for < c < 20/7 = This was improved by Jia [47] to the range < c < 3/ = The next improvement was due to Kumchev [53] who used a refined version of the double large sieve of Iwaniec and Fouvry to obtain the range < c < 45/38 = The current best result is due to Rivat and Wu who proved that (.) holds when < c < 243/205 [77]. The introduction of Piatetski-Shapiro primes (and other thin sets of primes) has led to hybrid variations in Waring-Goldbach type problems. Balog and Friedlander proved a variation of Vinogradov s three prime theorem involving Piatetski-Shapiro primes. They showed that every sufficiently large odd integer can be expressed as a sum of three primes of the form [n c ] for c < 2/20 [4]. Their result was later improved by Kumchev to include c in the range < c < 53/50. Balog and Friedlander s proof doesn t directly use the Hardy-Littlewood method. The proof of this result comes from reducing the problem to the standard three prime problem and showing the error term is admissible. Bounding the error term is done by directly estimating exponential sums. The construction of a thin set of primes has led to many other variations in mixed power and Waring-Goldbach problems. In view of Balog and Friedlander s work on the hybrid Vinogradov problem, there has been some attention given to using Piatetski-Shapiro primes in other Waring-Goldbach problems. In particular, W. G. Zhai proved a variation of the Waring-Goldbach problem for k = 2 [93]. He gave an asymptotic formula for the number of representations of a large integer n as a sum of five Piatetski-Shapiro prime squares 3

21 provided < c < 44/43. His proof was carried out similarly to Balog and Friedlander s with a slight modification to account for the squares. We also consider a variation of this problem. We examine the representation of integers n as a sum of three prime squares and two Piatetski-Shapiro prime squares. We prove the following theorem: Theorem 4. Let R 2,γ (n) denote the number of solutions of the equation n = p 2 + p p p 2 γ + q 2 γ where p, p 2, p 3 are primes and p γ, q γ are Piatetski-Shapiro primes of type c = /γ. Then provided 7/8 < γ. Here where R 2,γ (n) Γ5 (3/2) Γ(5/2) S(n) n 3/2 (log n) 5 S(n) = q q= a= (a,q)= S(q, a) = S 5 (q, a) ϕ 5 (q) e q m= (m,q)= ( an e ( am 2 ). q Moreover, S(n) is absolutely convergent and S(n) δ > 0 for all n 5 (mod 24) q ), The proof of this theorem uses a combination of the Hardy-Littlewood method and exponential sum estimation. We use a pruning technique to shrink the size of the major arcs in order to obtain an acceptable error for replacing two of the primes with Piatetski- Shapiro primes. 4

22 CHAPTER 2 Preliminaries In this chapter, we provide some of the machinery that will be used to prove the results in later chapters. In particular, we will introduce the basic idea of the Hardy-Littlewood method, and include the proofs of the classic Weyl and Hua inequalities. Later in this chapter, we discuss some more recent contributions to the study of exponential sums such as the exponent pair and double large sieve methods. 2. The Hardy-Littlewood Method In the study of additive problems, the Hardy-Littlewood method is a powerful method of attack. Here we outline its basic ideas. Hardy and Littlewood developed this method to determine values of G(k), but their method is easily adapted to more general additive problems. Suppose that for each j =, 2,... s, A j is a set of non-negative integers. For a fixed large integer n, let f j (α) = f j (α, n) = a j A j [0,n] e(a j α), j =, 2,... s and It is clear that s F (α) = f j (α). j= F (α) = e((a + + a s )α) = R(m, n)e(mα) a A [0,n] a s A s [0,n] m sn where R(m, n) is the number of representations of m as m = a + a a s with a j A j [0, n]. In particular, we define R(n) = R(n, n), the number of representations of 5

23 n as n = a + + a s. Now from the orthogonality relation, if h = 0, e(αh)dα = 0 0 if h Z\{0} we have that R(n) = 0 F (α)e( nα)dα. Now if A j is a well-distributed subset of non-negative integers, then f j (α) is expected to be large when α is close to a rational number with relatively small denominator and small otherwise. In view of this observation, it is natural to dissect the unit interval into two parts; the major arcs and minor arcs denoted M and m respectively. The major arcs are made up of subintervals centered at reduced rational numbers a/q with q small, and the minor arcs are the complement of the major arcs in the unit interval. In most cases, we expect the integral over the major arcs to give us the main term and the integral over the minor arcs gives a negligible contribution. Various analytic tools are used to estimate these integrals. 2.2 Weyl s Inequality Bounds for exponential sums are extremely useful. We will develop some of the tools necessary for estimating exponential sums. In Waring s problem (and many other additive problems), we are interested in representing a natural number n as a sum of k-th powers. Here f(α) has the form f(α) = e(αn k ), n N so we will focus our attention on sums of the form S = e(g(n)) n N 6

24 where g(n) = αn k + is a polynomial of degree k. The case for k = is a sum over a linear polynomial which is a geometric series. We have e(αn) e(αn) e(α) e(α) n N Since sin πα 2 α, we have for α Z e(αn) min n N ( N, sin παn. sin πα ). (2.) 2 α We now give an estimate for a sum involving a polynomial of any degree k 2. We begin with e(g(n)) n N 2 = m N = n N e(g(m) g(n)) n N N n h= n e(g(h + n) g(n)). Now if g(n) is a polynomial of degree k, then g(h + n) g(n) is a polynomial of degree k (for h 0). By repeating this differencing process we arrive at a sum involving a polynomial of degree one for which we have a non-trivial bound. Thus by induction, we obtain Proposition. If g(x) = αx k + is a polynomial of degree k with k, then n N e(g(n)) 2N ( N k N<h,...,h k <N min (N, αk!h... h k ) ) 2 k Proof: For k = we interpret this result as that given by (2.). Now suppose this result holds for some k. Let g(x) = αx k+ + be a polynomial of degree k +. Then we have g(x + h) g(x) = α(k + )hx k + is a polynomial of degree k. Thus, n N e(g(n)) 2 h <N n N 0<h+n N 7 e(α(k + )hn k + )

25 2N h <N ( N k N<h,...,h k <N ) 2 k min (N, α(k + )!h h k h ) by the induction hypothesis. From an application of Hölder s inequality, we have n N e(g(n)) 2 4N 2 ( N k This gives the result for k +. N<h,...,h k <N To finish Weyl s inequality, we require the following result. ) 2 k min (N, α(k + )!h h k h k ). Proposition 2. Suppose that X, Y, α are real numbers with X, Y and that α a/q q 2 with (a, q) =. Then Proof: Let x X ( min (XY x, αx ) XY q + Y + q ) log (2Xq). XY S = x X min (XY x, αx ). We have S q 0 j X/q r= ( ) XY min, α(qj + r). qj + r For each j, let y j = [αjq 2 ] and write θ = q 2 (α a/q). Then When j = 0 and r q/2, α(qj + r) = y j + ar q + {αjq2 } q + θrq 2. α(qj + r) ar/q 2q 2 ar/q. Otherwise for each j, there are at most O() values of r for which α(qj + r) 2 (y j + 8

26 ar)/q fails to hold. Moreover qj + r q(j + ). Thus, S r q/2 XY q ar/q + 0 j X/q ( XY q + Y + 0 j X/q ( XY q(j + ) + j + + (Xq + ) q XY ) log (2Xq) h q/2 q r= q y j +ar q h (y j + ar)/q ) which is the desired result. Estimating exponential sums is critical in applications of the Hardy-Littlewood method. Here, we combine the previous propositions into an estimate for exponential sums. Lemma (Weyl s Inequality). Suppose that (a, q) =, α a/q q 2, g(x) = αx k + α x k + + α k, and S g (N) = e(g(n)). n N Then S g (N) N +ε (q + N + qn k ) 2 k. Proof: From Proposition we have that n N e(g(n)) 2N ( N k N<h,...,h k <N When h h k = 0, each term is bounded by N, so we have n N ( e(g(n)) N N k (N k + h,..., h k <N min (N, αk!h... h k ) ) 2 k. ) 2 k min (N, αk!h h k )). 9

27 Now let h = k!h h k, then we have n N k!n e(g(n)) N (N k ) 2 k k (N k + N ε min (N, αh )) h= k!n N (N k ) 2 k k (N k + N ε min (N k h, αh )). h= Now from Proposition 2 this is bounded by ( N N k (N k + N ε (N k ( q + N + q ) 2 k N k ))) N +ε ( N + q + qn k ) 2 k which completes the proof. 2.3 Exponential Sums over Prime Variables We remark that although we allowed a general polynomial in Weyl s inequality, the result only relies on the leading term. In addition to the standard Weyl inequality, it is necessary to consider an estimate for sums over a prime variable. Such sums are necessary in additive problems involving primes. Vinogradov introduced these sums in the late 930s and provided an unconditional estimate in the case k =. This estimate is the essential innovation in his celebrated proof of the ternary Goldbach problem. Let g(α) = p P e(αp k ) where the sum is over primes. Vinogradov s result essentially states that for k = if a and q are integers satisfying q, (a, q) =, and α a/q q 2 then g(α) q ε P (q + P 2/5 + qp ) /2. 20

28 In addition, Vinogradov gave estimates for the case k 2 and used them to provide the first unconditional results on the Waring-Goldbach problem. For k 2 the sharpest estimates using Vinogradov s approach were provided by Harman [36]. He showed g(α) P +ε (q + P /2 + qp k ) 4 k. A number of authors have worked to improve the estimates over the years. Here we provide the current best result due to A. Kumchev [54]. Lemma 2. Let k N and α R and suppose that there exists a Z and q N satisfying with Q P. Then for any fixed ε > 0, q Q, (a, q) =, α a q < Q qp k g(α) Q /2 P /20+ε + q ε P (log P ) c q /2 ( + P k α a/q ) /2, where c > 0 is an absolute constant and where the constant implied in the depends on at most k and ε. When dealing with sums over prime variables, it is often convenient to weight the sums by Λ(n) where Λ is the von Mangoldt function. Thus, we are required to estimate a sum in the form Λ(n)f(n) for an appropriate function f(n). To bound a sum in this form, we use a decomposition of Λ(n) known as Vaughan s identity. We begin with an identity of ζ(s). We have the following lemma. Lemma 3. Let µ(n) be the Möbius function and define M X (s) = M(s) = n X µ(n) n s and N(s) = Λ(n) n s. n X 2

29 Then we have ζ ( ) ζ ζ (s) = ζ (s)m(s) + ζ(s)m(s)n(s) + (s) + N(s) ( ζ(s)m(s)) N(s) (2.2) ζ From this identity, we select the coefficients of n s on each side of (2.2) to obtain Vaughan s identity. For any arithmetic function f, we have where X<n x S = m X µ(m) S 2 = S 3 = m,n X m,n>x X<mn x Λ(n)f(n) = S S 2 S 3, X<mn x µ(m)λ(n) (log n)f(mn), X<mnr x Λ(m)c(n)f(mn) f(mnr), with c(n) = d n d X µ(d). The sums S and S 2 are estimated by considering a sum in the form S I = m a m f(mn), (2.3) n The sum S 3 can be treated in the form S II = m a m b n f(mn). (2.4) n These sums are known as type I and type II sums respectively. Various methods are applied to bound these sums with the quality of estimate dependent upon the function f, as well as the ranges of m and n. 22

30 2.4 Hua and van der Corput Inequalities Estimating exponential sums directly is essential, but it is also useful to estimate integrals of exponential sums. Here we give a result on such estimates. In this section, we define f(α) = e(αm k ). m N Lemma 4 (Hua s Inequality). Suppose that j k. Then 0 f(α) 2j dα N 2j j+ε. Proof: We proceed by induction on j. The case j = is immediate from Parseval s identity. Now suppose the lemma holds for j k. By a repeated differencing process described above, we have f(α) 2j (2N) 2j j h e(αh h j p j (x; h,..., h j )) h j x I j h i N where p j (x; h,..., h j ) is a polynomial in x of degree k j with integer coefficients. Thus, f(α) 2j (2N) 2j j h c h e(αh) (2.5) where c h is the number of solutions to the equation h h j p j (x; h,..., h j ) = h with h i < N and x I j. We have c 0 N j and c h N ε (h 0). Now, considering f(α) 2j = f(α) 2j f( α) 2j we also have that f(α) 2j = h b h e( αh) (2.6) 23

31 where b h is the number of solutions to the equation x k + + x k 2 j y k y k 2 j = h with x i, y i N. Thus and, by the inductive hypothesis, b h = f(0) 2j = N 2j h b 0 = 0 f(α) 2j dα N 2j j+ε. Now by (2.5), (2.6), and Parseval s identity, we have 0 f(α) 2j+ dα (2N) 2j j h c h b h. Moreover, c h b h c 0 b 0 + N ε b h h 0 h N j N 2j j+ε + N ε N 2j which proves the lemma. Lemma 5 (Weyl-van der Corput Inequality). Suppose ξ(n) is a complex valued function such that ξ(n) = 0 if n I = (a, b]. If Q is a positive integer then ξ(n) n I Proof: We have 2 ( ) I + Q Q q Q ( q ) ξ(n)ξ(n + q). Q n,n+q I Q ξ(n) = n I Q ξ(n + q) = n n q= Q ξ(n + q). q= 24

32 Note that the inner sum is empty unless a Q < n b. By Cauchy s inequality Q 2 ξ(n) n 2 ( I + Q) Q 2 ξ(n + q) n k= Q Q = ( I + Q) ξ(n + k)ξ(n + l). k= l= n Now we collect the terms with l k = q to obtain the result. In applications of the Weyl-van der Corput inequality we take e(f(n)) if n I, ξ(n) = 0 otherwise It is often necessary to apply a pruning technique to the major arcs. This is essentially a result (in slightly different forms) that has been used in several previous papers (e.g., Lemma 2.6 of [9]). First, we give a definition of the major arcs, M(K), M(K) = Here we provide the pruning lemma. K q q= a= (a,q)= [ a q K qn, a q + K ]. qn Lemma 6. For a large n and < K n suppose G(α) is a function satisfying G(α) q (+θ ) ( + n α a/q ) (+θ 2), α M(2K) where θ, θ 2 > 0 are certain constants. The function Φ(α) = h H η(h)e(αh) satisfies Φ(α) 0, η(h) 0 for all α, h and log(h) log(n). Then M(2K)\M(K) G(α)Φ(α) dα n +ɛ ( η(0)(k θ + K θ 2 ) + ( 25 h H h 0 η(h) ) ) (K θ + K θ 2 )

33 Proof: We have M(2K)\M(K) G(α)Φ(α) dα q K q a= K/qn K/qn h H η(h) q K Here we bound the integral trivially to obtain h H η(h) q K q (+θ ) ( + n β ) +θ 2 q e(ah/q)q (+θ ) a= h H K/qn η(h)e(ah/q)e(hβ)dβ K/qn q e(ah/q)q (+θ) ( + K/q) (+θ 2) K qn. a= Noting that the sum over a is equal to q if q divides h and 0 otherwise, we have Thus, we have Hence h H η(h) q K q h n ( h H η(h) q K q h n +ε ( η(0)k θ + q (+θ ) ( + K/q) (+θ 2) K qn. (q (+θ ) K + q θ 2 θ K θ 2 ). h H h 0 η(h)k θ ). ( + n β ) +θ 2 e(hβ)dβ. Here, we also give a lemma for the special case with G(α). We omit the proof as it can be carried out similarly. Lemma 7. Suppose the function Φ(α) = h H η(h)e(αh) satisfies Φ(α) 0, η(h) 0 for all α, h and H N γ for some fixed γ > 0. Then, for any Y N 2 L c, we have M(Y ) ( Φ(α) dα N +ɛ Y η(0)y + 26 h H h 0 ) η(h).

34 2.5 Exponent Pair Method Weyl s inequality is a useful bound for many applications of exponential sums, but some problems require more sophisticated estimation. Additionally we often require an estimate for an exponential sum where the exponent is not a polynomial. Here, we appeal to van der Corput s method of exponent pairs. Suppose F is differentiable on [N, 2N]. We expect an estimate of the form S = N n 2N e(f (n)) (A/N) k N l where AN F (n) AN. We call such a pair (k, l) an exponent pair. Clearly if (k, l) and (µ, λ) are exponent pairs, then so is any linear combination (tk + ( t)µ, tl + ( t)λ) for t [0, ] because S = S t S t (A/N) tk+( t)µ N tl+( t)λ. Trivially, we have that (0, ) is an exponent pair. The power of the theory of exponent pairs is the ability to obtain new pairs from known pairs. Thus, we can estimate an exponential sum with this technique and choose the pair that yields the best bound. There are two processes to determine new pairs from old pairs. They are known as the A process and B process. ( ˆ A process: If (k, l) is an exponent pair, so is k 2(k + ), k + l + 2(k + ) ˆ B process: If (k, l) is an exponent pair, so is (l /2, k + /2). ). Here we include a list of commonly used exponent pairs. These are: ( 2, 2 ) = B(0, ); ( 6, 2 3 ) = A( 2, 2 ); ( 2 7, 4 7 ) = BA( 6, 2 3 ); ( 30, 6 30 ) = BA2 ( 6, ); ( 40, ) = BA2 ( 2 7, ); ( 3, 6 3 ) = BAB( 30, 6 30 ). For a complete treatment of exponent pair theory, we refer the interested reader to chapter 2 of [45]. Exponent pairs are useful for a variety of exponential sum estimates. In particular, we frequently use exponent pairs to bound the error term of the Fourier expansion of 27

35 ψ(t) = t [t] /2. Here we provide the expansion for ψ(t) that will be used in later chapters. For t Z and any J, where with ψ(t) = 0< h J ( { }) 2πih e(ht) + O min,, (2.7) J t { } min, = J t h= b h e(ht), { log 2J b h min, J }. J h 2 In addition to the exponent pair technique, van der Corput s method can be used to obtain other useful estimates for exponential sums. We include one such estimate which we will use in the later chapters. Lemma 8. If > 0 and F (t) or ( F (t) ) for N < t 2N then e(f (n)) N /2 + /2. n N 2.6 Double Large Sieve Method We turn our attention to another method of exponential sum estimation known as the double large sieve. The basis of this method was introduced in a work of Fouvry and Iwaniec on exponential sums with monomials [32]. In their paper, they consider sums of the type e(f(m,..., m j )), m m j where the m i range over integers from an interval and f(m,..., m j ) = xm α mα j j monomial. In the treatment of such sums, they regard them as a special case of bilinear forms B φψ (X, Y ) = r φ r ψ s e(x r y s ), where X = (x r ), Y = (y s ) are finite sequences of real numbers with x r X, y s Y and φ r, ψ s C. Their method is based off the following inequality of [9]. 28 s is a

36 Lemma 9. with and B ψ (Y, X) defined similarly. B φψ (X, Y ) 2 20( + XY )B φ (X, Y )B ψ (Y, X) B φ (X, Y ) = x r x r2 Y φ r φ r2 In the application of this lemma, we must determine the length of the summations involved in the definition of B φ (X, Y ). In our applications, we typically have X = (x m ) = mα M and Y = (y n) = nβ N where m M and n N. Thus, it is natural to consider the spacing of the monomial term m α n β. We provide Fouvry and Iwaniec s result in the following lemma. Lemma 0. Let αβ 0, > 0, M and N. Let A (M, N; ) be the number of quadruples (m, m 2, n, n 2 ) such that ( m m 2 ) α with m, m 2 M and n, n 2 N. We have ( n n 2 ) β <, A (M, N; ) MN log 2MN + M 2 N 2. This lemma provides a method to estimate a common form of exponential sum. Occasionally, more sophisticated spacing estimates are needed. In particular, if we apply a Weyl shift, we introduce a term of the form t α (m, q) = (m + q) α (m q) α. Note that for m M and q Q, t α (m, q) M α Q = T. Thus, we require an estimate for the number of quadruples (m, m 2, q, q 2 ) such that t α (m, q ) t α (m 2, q 2 ) < T. (2.8) Let B(M, Q, ) denote the number of quadruples (m, m 2, q, q 2 ) subject to (2.8). An upper bound for B(M, Q, ) is provided in the following lemma due to Fouvry and Iwaniec. 29

37 Lemma. If Q M 2/3, we have B(M, Q, ) (MQ + M 2 Q 2 + M 2 Q 6 )(log 2M) 2 where the constant implied in depends on α only. In their paper, Fouvry and Iwaniec applied Lemma 9, Lemma 0, and Lemma in combination with other analytic methods (such as Weyl-van der Corput inequality) to obtain a number of bounds for exponential sums with monomials. Here we give one of their results as well as the proof for completeness. Lemma 2. Let α, α, α 2 be real constants such that α and αα α 2 0. Let M, M, M 2, x and ϕ m, ψ m m 2 be complex numbers with ϕ m, ψ m m 2. We then have, S ϕψ (M, M, M 2 ) = m M (x mα m α mα 2 2 ϕ m ψ m m 2 e m M m 2 M 2 ( x /4 M /2 (M M 2 ) 3/4 + M 7/0 M M 2 M α M α M α 2 2 ) M(M M 2 ) 3/4 + x /4 M /0 M M 2 )(log 2MM M 2 ) 2. Proof: By the Weyl-van der Corput inequality we have S ϕψ 2 Q MM M 2 (MM M 2 + S(Q 0 ) log 2Q), for any Q 3 M and some Q 0 Q, where and S(Q 0 ) = ( q ) ϕ m+q ϕ Q m q e (x t(m, ) q)mα mα 2 2 M α M α m m 2 M α 2 2 q Q 0 Thus, by lemma 9 we have m t(m, q) = (m + q) α (m q) α Q 0 M α. S(Q 0 ) (A BxQ 0 M ) /2, (2.9) 30

38 where A is the number of quadruples (m, m 2, m m 2 ) such that ( ) α ( ) α2 m m2 (xq 0 ) M, m m 2 and B is the number of quadruples (m, m, q, q) such that t(m, q) t( m, q) x M α. Now from lemmas 0 and we have A M M 2 log 2M M 2 + (xq 0 ) MM 2 M 2 2, (2.0) and B Q 0 M( + x M 2 )(log 2M) 4, (2.) provided 3Q 0 M 3/5. Now combining (2.9), (2.0), and (2.), we have S(Q 0 ) Q 0 (xm M 2 ) /2 ( + x Q 0 MM M 2 ) /2 ( + x M 2 ) /2 (log 2MM M 2 ) 5/2. Notice that the worst value for Q 0 is Q so we set Q = 3 M 3/5 to conclude the proof. This lemma is one of many results from Fouvry and Iwaniec paper. For a complete list of their results, we refer the reader to their work, [32]. 3

39 CHAPTER 3 Representations of n = p 2 + p2 2 + p2 3 + p2 γ + q 2 γ 3. Introduction In this chapter, suppose N 5 (mod 24) is a large integer. We consider the representations of a large integer, N, in the form N = p 2 + p2 2 + p2 3 + p2 γ + q 2 γ, where p, p 2, p 3 are ordinary prime numbers and p γ, q γ are Piatetski-Shapiro primes of type γ. The first part of the chapter is dedicated to using the Hardy-Littlewood method to reduce the problem to an estimation on small major arcs. From there, the problem reduces to an exponential sum estimate over a prime variable. The remainder of the chapter is dedicated to that estimate. We begin by noting the number of representations is given by R γ(n) = 0 ( ) 3 ( e(αp 2 ) p N /2 p γ N /2 p γ=[n c ] e(αp 2 γ)) 2 e( Nα)dα. (3.) For convenience we will consider a weighted number of representations instead of R γ(n). We define g(α) = Λ(m)e(αm 2 ), m N /2 f γ (α) = m γ Λ(m)e(αm 2 ( ) [(m + ) γ ] [m γ ] ), γ m N /2 and let R γ (N) := 0 g 3 (α)f 2 γ (α)e( Nα)dα. (3.2) In this section we will prove the following theorem. 32

40 Theorem 5. For R γ (N) defined above and γ > 7/8, we have where S(N) is the singular series, defined by R γ (N) = Γ5 (3/2) Γ(5/2) S(N)N 3/2( + o() ), S(N) = q q= a= (a,q)= S 5 (q, a) ϕ 5 (q) e ( an Moreover, S(N) is absolutely convergent and S(N) c > 0 for all N 5 (mod 24). q ). From this theorem, we have the following corollary. Corollary. Every sufficiently large integer N can be represented in the form N = p 2 + p p2 3 + p2 γ + q 2 γ provided γ > 7/8. Note that the asymptotic formula Rγ(N) = Γ5 (3/2) Γ(5/2) S(N) N 2 +γ ( + o()) (log N) 5 follows from the result of Theorem 5. The goal is to use the existing work on the Waring- Goldbach problem of representing an integer as a sum of five prime squares. Recall that if R 2 (N) is the weighted number of representations of N as a sum of five prime squares, we have R 2 (N) Γ5 (3/2) Γ(5/2) S(N)N 3/2 (log N) 5, (3.3) where the singular series is given in the above theorem. In this framework, we wish to show that by replacing two of the primes with Piatetski-Shapiro primes, we obtain an analogous result provided γ is not too small. That is, for R γ (N) given by (3.2), we wish to show R 2 (N) R γ (N) N 3/2 ε. However, the presence of e(αm 2 ) term prevents a suitable estimate for g(α) f γ (α) for large major arcs. To overcome this obstacle, we prune the major arcs to be as small as 33

41 possible. This works best when there are two or fewer Piatetski-Shapiro prime squares among the five terms. We could replace more of the terms by Piatetski-Shapiro prime squares, but the range for γ will be much more restrictive. For α in small major arcs, the term e(αm 2 ) can be well handled. We proceed by using the Hardy-Littlewood method to split R γ (N) into two integrals respectively over major and minor arcs. Suppose N is sufficiently large, and define P = N /4 and Q = N ε. (3.4) We define the major arcs by M = M(P ), where and the minor arcs M(K) = K q q= a= (a,q)= [ a q K qn, a q + K ] qn m = U \M, where U = [P/N, + P/N]. Thus, we rewrite R γ (N) as R γ (N) = R M (N) + R m (N), where R M (N) and R m (N) represent integrals over the major and minor arcs respectively. 3.2 The Minor Arcs We bound the contribution from the minor arcs by ( ) /2 ( /2 R m (N) max g(α) g(α) 4 dα f γ (α) dα) 4. (3.5) α m 0 0 For the two integrals on the right side of (3.5), we will bound them by Hua s inequality and the next lemma respectively. Lemma 3. 0 f γ (α) 4 dα N 2 γ+ε. (3.6) 34

42 Proof: Note that this integral is bounded by the number of integer solutions to m 2 m 2 2 = m 2 3 m 2 4 with each m i weighted by a factor m γ i and m i N /2 for i =, 2, 3, 4. We consider the solutions in two cases. Case : m = m 2. Note that this implies that m 3 = m 4. Now, we have N /2 choices for m and N /2 choices for m 3 both weighted by m γ i N ( γ)/2 for i =, 3. Thus, we have at most (N /2 N ( γ)/2 ) 2 = N 2 γ weighted solutions. Case 2: m m 2. In this case, we have N /2 choices for m and m 2 weighted by m γ i N ( γ)/2 for i =, 2. Thus we have at most (N /2 N ( γ)/2 ) 2 = N 2 γ choices for m and m 2. In addition we have at most N ε choices for m 3 and m 4. Thus the number of solutions is at most O(N 2 γ+ε ), which completes the proof. Also, we use the estimate on g(α) of Ghosh. We state the result in the following lemma. Lemma 4. (Ghosh [34]) Let α R and suppose there exists a Z and q N such that (a, q) = and α a/q < q 2. Then for any positive ε, we have g(α) N /2+ε (q + N /4 + qn ) /4 (3.7) where the constant implied in the depends on at most ε. 35

43 Applying Hua s inequality to the integral over g(α) and combining (3.6), and (3.7) into (3.5), we have Thus, we have which gives, by (3.4) R m (N) N /2+ε (P + N /4 ) /4 N 2 γ 2 N /2. R m (N) N 2 γ/2+ε (P /4 + N /6 ), R m (N) N 2 /6 γ/2+ε. We clearly have R m (N) N 3/2 ε provided γ > 7/8 (and we can take any ε > 0 satisfying ε < 4 (γ 7/8)). 3.3 The Major Arcs Now, our goal is to prune the major arcs to M(Q). We use the pruning technique described in the preliminaries section. First, we require a major arc estimate for g(α). We apply the bound (3.9) in lemma 5. First, we have M(P )\M(Q) g(α) 3 f γ (α) 2 dα log N max g(α) 3 f γ (α) 2 dα. (3.8) Q K P M(2K)\M(K) Here we apply a theorem of Kumchev to the function g(α). We include the theorem here as a lemma. Lemma 5. (Kumchev [?]) Let k N and α R, and suppose that there exists a Z and q N satisfying q Q, (a, q) =, qα a QN with Q N /2. Then, for any fixed ε > 0, g(α) Q /2 N /40+ε + qε N /2 (log N) c, (3.9) (q + N qα a ) /2 where c > 0 is an absolute constant and where the constant implied by the depends on ε only. 36