Vinogradovs theorem and its generalization on primes in arithmetic progression

VINOGRADOV’S THEOREM AND ITS GENERALIZATION ON PRIMES IN ARITHMETIC PROGRESSION WONG WEI PIN (B.Sc.(Hons.) NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2009 i Acknowledgements First and foremost, it is my great honor to work under Assistant Professor Chin Chee Whye again, for he has been more than just a supervisor to me but as well as a supportive friend; never in my life I have met another person who is so knowledgeable but yet is extremely humble at the same time. Apart from the inspiring ideas and endless support that Prof. Chin has given me, I would like to express my sincere thanks and heartfelt appreciation for his patient and selfless sharing of his knowledge on algebraic number theory, which has tremendously enlighten me. Also, I would like to thank him for entertaining all my impromptu visits to his office for consultation and entrusting me to be the grader for his Galois Theory module. I would like to express my profound gratitude to Prof. Régis de la Bretèche, who was my supervisor of my scientific internship at the Mathematics Institute of Jussieu, Paris, for having equipped me with a solid foundation on understanding Vinogradov’s theorem. Many thanks to all the professors in the Mathematics department who have taught me before. Also, special thanks to Professor Chan Heng Huat and Dr. Toh Pee Choon for patiently attending my seminar series on this research as well as giving me constructive suggestions to improve my thesis. I would also like to take this opportunity to thank the administrative staff of the Department of Mathematics for all their kindness in offering administrative assistant to me throughout my Master’s study in NUS. Special mention goes to Ms. Shanthi D/O D Devadas, Mdm. Tay Lee Lang and Mdm. Lum Yi Lei for always entertaining my request with a smile on their face. Last but not least, to my family and my fellow peers, Siong Thye, Jia Le, Jian Xing and Tao Xi, thanks for all the laughter and support you have given me throughout my Master’s study. It will be a memorable chapter of my life. Wong Wei Pin Spring 2009 Contents Acknowledgements i Summary iii Notation vi 1 Analytic tools 1 1.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Prime Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 The Ternary Goldbach Problem 22 2.1 The Minor Arcs m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 The Major Arcs M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Vinogradov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Generalized Ternary Goldbach Problems 36 3.1 Ternary Goldbach Problem in Number Fields . . . . . . . . . . . . . . . . 36 3.2 The Minor Arcs m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 The Major Arcs M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Bibliography 63 ii iii Summary Christian Goldbach first made his famous conjecture in 1742 that every even number larger than 2 is a sum of two prime numbers. Although Goldbach’s Conjecture still remains unsolved today, great progress has been achieved by many mathematical giants such as Hardy, Littlewood, Vinogradov, Estermann, Chen Jinrun and Heath-Brown, in proving weaker variants of the conjecture. One such variant is the Weak Goldbach’s Conjecture (also known as the ternary Goldbach problem), which says that every odd number larger than 5 is a sum of three prime numbers. The central idea in proving these variants is to find a good estimate of the number of representation of an integer as a sum of primes, using revolutionary and accurate counting methods. One such method is the Hardy-Littlewood circle method, first invented by Hardy and Ramanujan and then further developed and applied by Hardy and Littlewood in solving the Waring’s problem. In fact, the circle method has far reaching applications in other additive number theory problems, such as Birch’s theorem, Roth’s theorem and the ternary Goldbach problem. In 1923, Hardy and Littlewood made a remarkable progress on the ternary Goldbach problem by showing that every sufficiently large odd number is a sum of three prime numbers, with the assumption of a zero free domain for the Dirichlet L-functions (which is true if one assume the Generalized Riemann Hypothesis). The ultimate breakthrough for the ternary Goldbach problem was done by Vinogradov in 1937. With his ingenious estimation of the exponential sum on prime numbers as well as his aptly application of the Siegel-Walfisz Theorem, Vinogradov managed to remove the assumption in Hardy and Littlewood’s proof and thus proved unconditionally the Vinogradov’s theorem: every sufficiently large odd number is a sum of three prime numbers. Subsequently, Chinese mathematicians Chen and Wang showed that the condition for being sufficiently large is to be larger than 1043000 , but this astronomical number is still far to be reached by numerical verification with computer programs in order to prove the Weak Goldbach’s Conjecture completely. ∗∗∗ iv The aim of this thesis is to first study a simplified proof of the Vinogradov’s theorem given by Vaughan and to generalize the Vinogradov’s theorem to the quadratic fields. This generalization will in turn give the motivation to formulate the Vinogradov’s theorem for primes in arithmetic progression: Let x1 , x2 , x3 and y be integers such that 1 < y and (xi , y) = 1 for i = 1, 2, 3. Then for all sufficiently large odd integer N ≡ x1 + x2 + x3 mod y, there exist primes pi ≡ xi mod y for i = 1, 2, 3, such that N = p1 + p2 + p3 . This was first conditionally proven in 1926 by Rademacher with the assumption of the Generalized Riemann Hypothesis (see [9]). By borrowing ideas from the proof of Vinogradov’s theorem, Ayoub improved Rademacher’s argument to give an unconditional proof of this theorem in 1953 (see [10]). However, the proof of this theorem presented in this thesis is mainly original. Once this theorem is established, the Vinogradov’s theorem for quadratic fields will follow immediately as a corollary. ∗∗∗ This thesis is organized in three chapters. Chapter 1 concentrates on developing the crucial analytical tools that will serve us in the later chapters. These include an important inequality for exponential sums, some typical arithmetic functions, Ramanujan’s sums, Dirichlet’s series, infinite products and Euler products. The chapter winds up with a short exposition on two important results of modern prime number theory: the prime number theorem and the Siegel-Walfisz Theorem. The entire Chapter 2 is dedicated to study Vaughan’s proof of Vinogradov’s theorem. A general outline of the Hardy-Littlewood circle method in Vinogradov’s theorem will be presented first, followed by the definition of the major arcs and the minor arcs. Then the chapter proceeds on to find the asymptotic estimate of integrations over these two arcs. Compiling all these estimations, the last section of this chapter will prove the Vinogradov’s theorem by analyzing the behavior of the singular series S(N ). At the beginning of Chapter 3, some possible generalizations of Goldbach’s Conjecture to number fields will be discussed and eventually we will focus our interest on the ternary Goldbach problem on quadratic fields. After formulating our conjecture on quadratic fields, we will explain why the conjecture is a direct consequence of the v Vinogradov’s theorem for primes in arithmetic progression and move on to prove this theorem. The underlying idea of the proof follows exactly the one presented in Chapter 2, i.e. we will apply the Hardy-Littlewood circle method with the similar treatment of integrations over the major arcs and the minor arcs. However, great effort is put in to handle the twisted Ramanujan sum ηx,y that occurs in the estimation over the major arcs. Once this is overcome, we conclude by proving Vinogradov’s theorem for primes in arithmetic progression as well as Vinogradov’s theorem for quadratic fields. vi Notation 1. The letters a, b, d, j, k, , m, n, q, r, s always stand for integers. The letter p is always reserved for prime numbers. 2. We abbreviate e2πiα as e(α). 3. If f (x), g(x) ≥ 0, then f (x) g(x) means there exists an absolute constant C > 0 such that |f (x)| ≤ Cg(x). f g means g f. y means the constant C depends on y. 4. α is the largest integer that is less than or equal to α. 5. {α} := α − α . 6. f (n) = O(g(n)) means lim f (n) n→∞ g(n) < ∞. It is by default that the limit is taken when n tends to ∞ unless stated otherwise. 7. f (n) = o(g(n)) means lim f (n) n→∞ g(n) = 0. It is by default that the limit is taken when n tends to ∞ unless stated otherwise. 8. The notation (n, m) refers to gcd(n, m). 9. [a, b] := {x ∈ ❘ : a ≤ x ≤ b}. 10. always refers to the product over all prime numbers, unless stated otherwise. p 11. log is the natural logarithm function. 12. ❩n is the set of classes of residues modulo n and ❩∗n is the set of multiplicative invertible elements in ❩n . 13. The notation pr ||n means that pr is the highest power of p dividing n. Chapter 1 Analytic tools In this chapter, we develop some crucial analytical tools that will serve us in proving Vinogradov’s theorem in Chapter 2 and the generalized Vinogradov’s theorem in Chapter 3. The most important of these tools is an inequality for exponential sums. We shall also introduce some typical arithmetic functions, Ramanujan sums, infinite products and Euler products. In the last section of this chapter, we state some important results of modern prime number theory, which will be needed as well in the later chapters. 1.1 Inequalities Lemma 1.1.1. (Dirichlet) Let α be a real number. Then for any real number X ≥ 1, there exist integers a and q, such that (a, q) = 1, 1 ≤ q ≤ X and α− 1 a 1 ≤ ≤ 2. q qX q Proof. It suffices to prove the first inequality, and without the condition (a, q) = 1. We first let m = X and βq = αq − αq ∈ [0, 1) for q = 1, 2, . . . , m. Next, we consider r−1 r the m + 1 disjoint intervals Br = [ m+1 , m+1 ), for r = 1, 2, . . . , m + 1. If there exists 1 m a q such that βq ∈ [0, m+1 ) (resp. [ m+1 , 1)), then we have α − (resp. α− αq +1 q < 1 q(m+1) ). αq q < < 1 (u−v)X , < 1 qX Otherwise, by the pigeonhole principle, there exists u > v and some r ∈ {2, 3, . . . , m} such that βu , βv ∈ Br , which implies α − 1 (u−v)(m+1) 1 q(m+1) αu − αv u−v < with u − v < X. The lemma then follows with a = αu − αv 1 CHAPTER 1. ANALYTIC TOOLS 2 and q = u − v. Definition 1.1.2. Given any real number α, we denote the distance from α to the nearest integer as : ||α|| := min |α − m|. m∈❩ Remark: By the definition of ||α||, we have || − α|| = ||α|| ∈ [0, 1/2] and α = n ± ||α|| for some integer n. Proposition 1.1.3. For all real numbers α, β, the following triangle inequalities hold: ||α|| − ||β|| ≤ ||α + β|| ≤ ||α|| + ||β||. Proof. The first inequality in the assertion is an immediate consequence of the second, so it suffices to prove the second inequality. Let m, n be integers such that α = m ± ||α|| and β = n±||β||. Without loss of generality, let ||α|| ≥ ||β||, so that ||α||−||β|| ∈ [0, 1/2] and hence ||α|| − ||β|| = ||α|| − ||β||. With this, we obtain easily that ||α + β|| = m + n ± ||α|| ± ||β|| = ||α|| ± ||β|| ≤ ||α|| + ||β||. Lemma 1.1.4. For every real number α, we have | sin(πα)| = sin(π||α||) ≥ 2||α||. Proof. We have ||α|| ∈ [0, 21 ] and α = n ± ||α|| for some integer n. Hence | sin(πα)| = | sin(πn ± π||α||)| = sin(π||α||) ≥ 2||α||, as the function sin(πx) is concave when x ∈ [0, 12 ]. CHAPTER 1. ANALYTIC TOOLS 3 Lemma 1.1.5. For every real number α and all integers N1 < N2 , we have N2 e(αn) ≤ min(N2 − N1 , ||α||−1 ). n=N1 +1 Proof. It is obvious that N2 N2 e(αn) ≤ 1 = N2 − N1 . n=N1 +1 n=N1 +1 For α ∈ / ❩, we have ||α|| > 0 and e(α) = 1. Since the sum is a geometric series, we have N2 e(αn) = n=N1 +1 ≤ = = = ≤ e(α(N2 + 1)) − e(α(N1 + 1)) e(α) − 1 2 |e(α/2) − e(−α/2)| 2 |2i sin(πα)| 1 | sin(πα)| 1 sin(π||α||) 1 1 ≤ . 2||α|| ||α|| Lemma 1.1.6. Let α, X, Y be real numbers, X ≥ 1, Y ≥ 1 and let q and a be integers such that |α − aq | ≤ q −2 with (a, q) = 1 and q ≥ 1. Then min n≤X XY + X + q log(2Xq). q XY , ||αn||−1 n Proof. The case of q = 1 for the assertion is trivial because min n≤X XY , ||αn||−1 n ≤ XY n≤X 1 n XY log X ≤ (XY + X + 1) log(2X). So we assume q > 1 for the rest of this proof. By rewriting the summing index as residues CHAPTER 1. ANALYTIC TOOLS 4 modulo q, we obtain the following: S := XY , ||αn||−1 n min n≤X q ≤ min r=1 0≤j≤ X q XY , ||α(qj + r)||−1 . qj + r We now focus on finding a bound for most of the terms ||α(qj+r)||−1 . Define yj := αjq 2 for j = 0, . . . , X q and θ := q 2 α − qa ∈ [−1, 1], which then give the identity αjq 2 + {αjq 2 } αq 2 r ar arq + 2 + − 2 q q q q 2 yj + ar {αjq } rθ = + + 2, q q q α(qj + r) = We show that for a fixed j ∈ {0, . . . , X q j = 0, . . . , X . q }, the inequality ||α(qj + r)|| ≥ 1 (yj + ar) 2 q (1) fails to hold only for at most 7 exceptional values of r in {1, . . . , q}. By writing xr := yj +ar q and := {αjq 2 } q + rθ q2 < 2q , it suffices to show that the inequality 1 ||xr + || ≥ ||xr || 2 fails to hold for at most 7 exceptional values of r if q ≥ 8 (so that | | < 1 4 and thus || || = | |). For those r’s where ||xr || ≥ 2| |; which is equivalent to ||xr || − | | ≥ 21 ||xr ||, we have the desired inequality: 1 ||xr + || ≥ ||xr || − || || = ||xr || − | | ≥ ||xr ||. 2 Thus, the r’s for which (1) might fail correspond to ||xr || < 2| | < 4q , which implies these r’s can only correspond to the case yj + ar ≡ 0, ±1, ±2, ±3 mod q. So we conclude that for q > 1, j ∈ {0, . . . , X q }, the inequality (1) holds for all r ∈ {1, . . . , q} with at most 7 exceptions. Notice that the 7 exceptions include also the unique r ∈ {1, . . . , q} for which yj + ar ≡ 0 mod q. CHAPTER 1. ANALYTIC TOOLS 5 In fact, in the case of j = 0, we need furthermore that the inequality ||αr|| = ||α(qj + r)|| ≥ holds for all r ∈ {1, . . . , || ≤ || rθ q2 1 2q . q 2 }. Indeed, for 1 ≤ r ≤ For q > 1, we have || ar q || ≥ for r ∈ {1, . . . , q 2 1 (yj + ar) 1 ar = >0 2 q 2 q 1 q q 2, 1 2q | ≤ we get | rθ q2 and thus ≥ 2|| rθ ||, because (a, q) = 1 and thus q2 ar q ∈ /❩ }. So we obtain ar rθ + 2 q q rθ ar − 2 ≥ q q 1 ar ≥ > 0. 2 q ||α(qj + r)|| = by Proposition 1.1.3 Now, we return to find an upper bound for S. For j = 0 or j = 0 and r ≥ 2q , we have q(j + 1) ≤ 2(qj + r). This implies that for those r’s for which (1) does not hold, we can bound the summing terms by q S≤ min r=1 0≤j≤ X q 2XY q(j+1) . XY , ||α(qj + r)||−1 qj + r q ≤ 2 0≤j≤ X q ≤2 ≤4 ≤ r=1 yj +ar≡0 mod q X +1 q X +1 q q−1 s=1 q/2 s=1 Thus s q (yj + ar) q −1 + 14 XY q −1 +7 0≤j≤ X q X j=0 1 j+1 q XY + 14 2 log(2X) s q X XY + 1 q log(2q) + log(2X) q q XY + q + X log(2Xq). q 2XY q(j + 1) by (1) CHAPTER 1. ANALYTIC TOOLS 1.2 6 Arithmetic Functions Definition 1.2.1. I. An arithmetic function is a complex-valued function whose domain is the set of all positive integers. II. An arithmetic function f (n) is multiplicative if f (mn) = f (m)f (n), whenever (m, n) = 1. In this case, it is easy to see that if f is not identically zero, then f (1) = 1. III. An arithmetic function f (n) is completely multiplicative if f (mn) = f (m)f (n), for all positive integers m, n. Theorem 1.2.2. Let f be a multiplicative function. If lim f (pk ) = 0 pk →∞ as pk runs through the sequence of all prime powers (i.e. both p and k vary such that pk tends to ∞), then lim f (n) = 0. n→∞ Proof. Since lim f (pk ) = 0, there are only finitely many prime powers pk such that pk →∞ |f (pk )| ≥ 1. We denote |f (pk )|. A := |f (pk )|≥1 Now, given any > 0, there exists only finitely many prime powers pk for |f (pk )| > A+ , and thus for n ∈ ◆ large enough, the prime factorization of n must contain at least a prime power pk with |f (pk )| ≤ A+ . Hence, n can be written as r r+s pki i n= i=1 r+s+t pki i i=r+1 pki i , i=r+s+1 CHAPTER 1. ANALYTIC TOOLS 7 where pi are pairwise distinct prime numbers such that 1 ≤|f (pki i )| A+ for i = 1, . . . , r ≤|f (pki i )| ≤ 1 |f (pki i )| ≤ for i = r + 1, . . . , r + s for i = r + s + 1, . . . , r + s + t, t ≥ 1. A+ Using the fact that f is multiplicative, we have for all n ∈ ◆ large enough, r r+s r+s+t |f (pki i )| |f (n)| = i=1 |f (pki i )| i=r+1 |f (pki i )| ≤ A · 1 · i=r+s+1 A+ < . We present here some arithmetic functions that appears frequently in the study of analytic number theory. Definition 1.2.3. The Möbius function is defined by:    1    µ(n) := 0      (−1)r if n = 1, if n is not square-free, i.e. divisible by the square of a prime , if n is the product of r distinct primes. It is easy to check that the arithmetic function µ(n) is multiplicative and µ3 = µ. Proposition 1.2.4. For any natural number n ∈ ◆>0 , we have the following identity: µ(d) = δ(n) =    1 if n = 1,   0 otherwise. d|n Proof. The assertion is trivial for n = 1. For n > 1, we write n in its unique prime decomposition : r pki i , n= i=1 where pi are pairwise distinct primes and r ≥ 1. Since µ vanishes at non square-free integers, we have µ(d) = d|n µ(d) = d|p1 ···pr µ(d)+ d|p2 ···pr µ(p1 d) = d|p2 ···pr µ(d)− d|p2 ···pr d|p2 ···pr µ(d) = 0. CHAPTER 1. ANALYTIC TOOLS 8 Definition 1.2.5. The von Mangoldt’s function is defined as: Λ(n) :=    log p if n = pk , k ≥ 1,   0 otherwise . The arithmetic function Λ(n) is not multiplicative. Proposition 1.2.6. For any natural number n ∈ ◆>0 , we have the following identity: Λ(d) = log n. d|n Proof. The assertion is trivial for n = 1. For n > 1, we write n in its unique prime decomposition : r pki i , n= i=1 where pi are pairwise distinct primes and r ≥ 1. Since Λ is non zero only at prime powers, we have r ki r Λ(pji ) = Λ(d) = d|n i=1 j=1 r log pki i = log n. ki log pi = i=1 i=1 Definition 1.2.7. The divisor function d(n) counts the number of positive divisors of positive integer n. If we write n in its unique prime factorization: n = pk11 · · · pkr r , where p1 , . . . , pr are distinct primes and k1 , . . . , kr are nonnegative integers, then it is straight forward to deduce that d(n) = (k1 + 1) · · · (kr + 1). With this formula, we see that d(n) is multiplicative. In general, for any positive integer m and n, d(mn) ≤ d(m)d(n). This inequality follows from the inequality (α + β + 1) ≤ (α + 1)(β + 1). CHAPTER 1. ANALYTIC TOOLS 9 Theorem 1.2.8. For real number Z ≥ 2, we have d(k)2 Z log3 Z. k≤Z Proof. For X ≥ 1, we have m≤X 1≤ 1= d(m) = d≤X m≤X d|m m≤X d|m d≤X X d X log(2X). Now, if Z ≥ 2, we have d(n)2 = n≤Z n≤Z d(k) k≤Z d(mk) ≤ 1= d(n) k|n k≤Z Z log k 2Z k m≤ Z k d(k) k≤Z Z log Z k≤Z d(m) m≤ Z k d(k) . k But k≤Z d(k) = k k≤Z 1 k 1= d|k d≤Z k≤Z d|k 1 = k d≤Z 1 d m≤ Z d 1 m log2 Z. This completes the proof. Definition 1.2.9. The Euler φ-function is defined as: φ(n) := Card{1 ≤ a ≤ n : (a, n) = 1}, which is also the order of the multiplicative group ❩∗n . As a consequence of Chinese remainder theorem, φ is multiplicative and has the explicit formula: 1− φ(n) = n p|n 1 p Theorem 1.2.10. For any δ > 0, we always have n1−δ = 0. n→∞ φ(n) lim . CHAPTER 1. ANALYTIC TOOLS 10 Proof. Since φ is multiplicative, by Theorem 1.2.2, it is sufficient to prove pk(1−δ) = 0. pk →∞ φ(pk ) lim This is easily obtained, as the formula of φ gives pk(1−δ) 2 pk(1−δ) ≤ kδ −→ 0. = p−1 k φ(p ) p pk →∞ pk ( p ) Definition 1.2.11. Given any nonzero natural number n ∈ ◆>0 , there are φ(n) classes of residues relatively prime to n. Any set of φ(n) residues, one from each class, is called a complete set of residues relatively prime to n. If there is no ambiguity, we denote any complete set of residues relatively prime to n as ❩× n , which is different from the notation of the multiplicative group ❩∗n . Definition 1.2.12. Let q and n be integers with q ≥ 1. The exponential sum cq (n) := e a∈❩× q an q is called the Ramanujan sum. Remark: It is straight forward that the sum is independent of the chosen complete set of residues relatively prime to q but some authors prefer to define the function by summing over the set {a ∈ ◆ : 1 ≤ a ≤ q, (a, q) = 1}. This function will appear frequently in the proofs of Vinogradov’s theorem and generalized Vinogradov’s theorem. Hence, an explicit formula of Ramanujan sum will be very useful. Theorem 1.2.13. For a fixed integer n, the Ramanujan sum cq (n) is a multiplicative function of q, i.e., if (a, b) = 1, then cab (n) = ca (n)cb (n). × × Proof. If (a, b) = 1, then {as + br : r ∈ ❩× a , s ∈ ❩b } = ❩ab . In fact, if there exist × r1 , r2 ∈ ❩× a and s1 , s2 ∈ ❩b such that as1 + br1 ≡ as2 + br2 mod ab, CHAPTER 1. ANALYTIC TOOLS then 11    as1 + br1 ≡ as2 + br2 mod a,   as1 + br1 ≡ as2 + br2 mod b, which implies    r1 ≡ b−1 br1 ≡ b−1 br2 ≡ r2 mod a,   s1 ≡ a−1 as1 ≡ a−1 as2 ≡ s2 mod b, where b−1 is the inverse of b modulo a and a−1 is the inverse of a modulo b. Also, (as + br, ab) = (as + br, a) · (as + br, b) = (br, a) · (as, b) = 1. This proves the claim and thus rn sn + = a b e ca (n)cb (n) = r∈❩ × a s∈❩ × b e rb+sa∈❩ × ab (rb + sa)n ab = cab (n). Lemma 1.2.14. The Ramanujan sum can be expressed in the form cq (n) = µ d|(q,n) q d. d Proof. Since the geometric series d e =1 n d =    d if d|n,   0 if d n, it follows that q cq (n) = e k=1 (k,q)=1 q = e k=1 kn q kn q µ(d) d|(k,q) by Proposition 1.2.4 (2) CHAPTER 1. ANALYTIC TOOLS q d µ(d) = dn q e =1 d|q q d µ(d) = n e q d =1 d|q q µ d = d|q = d n d e =1 q d d µ d|q d|n by (2) q d d µ = 12 d|(q,n) Theorem 1.2.15. The Ramanujan sum has an explicit formula: µ cq (n) = q (q,n) φ φ(q) . q (q,n) Proof. We define q := q . (q, n) Then the formula of φ gives 1− q φ(q) = φ(q ) p|q q p |q 1 p 1 1− p 1− = (q, n) p|q pq 1 p 1− = (q, n) p|(q,n) pq 1 p . CHAPTER 1. ANALYTIC TOOLS 13 Then cq (n) = µ q d d µ q (q, n) · (q, n) d d|(q,n) = d|(q,n) µ(q c) = c|(q,n) by Lemma 1.2.14 (q, n) c µ(q )µ(c) = (q, n) c|(q,n) (q ,c)=1 = µ(q )(q, n) c|(q,n) (q ,c)=1 1 c because µ(q c) = 0 if (q , c) > 1 µ(c) c 1− = µ(q )(q, n) p|(q,n) pq = d 1 p µ(q )φ(q) . φ(q ) We remark that |cq (n)| ≤ min{φ(q), n} and cq (n) = µ(q) if (q, n) = 1. 1.3 Dirichlet Series In this section, we give a brief introduction of Dirichlet series and state some of their important properties without providing proofs. Readers who want to learn more about this topic are advised to refer to [4]. Definition 1.3.1. A Dirichlet series is a series of the form ∞ n=1 Cn , ns with Cn , s ∈ ❈. Proposition 1.3.2. (Uniqueness of Dirichlet series) The function f defined by: ∞ f (s) := n=1 Cn ns is identically zero on the domain of convergence of the Dirichlet series if and only if all CHAPTER 1. ANALYTIC TOOLS 14 the coefficients Cn = 0. Example 1.3.3. The Riemann zeta function ζ, its derivative ζ and the derivative of log ζ are the standard Dirichlet series that have been frequently studied in analytic number theory. They have the explicit formula ∞ ζ(s) := m=1 1 , ms ∞ −ζ (s) = m=1 log m , ms ζ − (s) = ζ ∞ m=1 Λ(m) ms (s) > 1 and they all converges absolutely on this domain. The proofs of these for identities can be found in [5] G. Tenenbaum and [6] G. H. Hardy and E. M. Wright. 1.4 Infinite Products Definition 1.4.1. I. Let (αk )k∈◆>0 be a sequence of complex numbers. The nth partial product of this sequence is the number n α1 · · · αn = αk . k=1 II. If the sequence of nth partial product converges to a limit α when n tends to infinity, ∞ αk converges and we denote we say that the infinite product k=1 ∞ n αk := lim n→∞ k=1 αk = α. k=1 III. If the sequence of partial products does not converge when n tends to infinity, then we say that the infinite product diverges. ∞ Theorem 1.4.2. Let ak ≥ 0 for all k ∈ ◆>0 . The infinite product k=1 ∞ ak converges. if and only if the infinite series n Proof. Let sn := k=1 n (1+ak ). Since ak ≥ 0, we have the strict inequality ak and pn := k=1 (1 + ak ) converges k=1 sn < pn . Also, since the inequality 1 + x ≤ ex CHAPTER 1. ANALYTIC TOOLS 15 is true for all x ∈ ❘, we have n 0≤ n k=1 n n eak = exp (1 + ak ) ≤ ak < k=1 k=1 ak , k=1 i.e. 0 ≤ sn < pn ≤ esn . Since both sequences {sn }n∈◆>0 and {pn }n∈◆>0 are monotone increasing, they converges if and only if they are bounded. Thus the inequality above implies that the sequence {sn }n∈◆>0 converges if and only if the sequence {pn }n∈◆>0 converges. Notice that pn ≥ 1 for all n and hence its limit, if it exists, is non zero. ∞ (1 + ak ) is said to converge absolutely if Definition 1.4.3. The infinite product k=1 ∞ (1 + |ak |) converges. the infinite product k=1 ∞ (1+ak ) converges absolutely, then it converges Theorem 1.4.4. If the infinite product k=1 and the limit is independent of the order of which the product is taken, i.e. for any permutation σ of ◆>0 , we have ∞ ∞ (1 + aσ(k) ). (1 + ak ) = k=1 k=1 Furthermore, the limit is zero if and only if 1 + ak = 0 for some k. ∞ ∞ Proof. By Theorem 1.4.2, ak converges (1 + ak ) converges absolutely if and only if k=1 k=1 absolutely. Thus, let ∞ |ak | . C := exp k=1 For any 0 < < 21 , there exists N0 such that |ak | < . k>N0 Let σ be a permutation of ◆>0 . Then for any integer N ≥ N0 , there exists an integer M ≥ N such that {1, . . . , N } ⊂ {σ(1), . . . , σ(M )}. CHAPTER 1. ANALYTIC TOOLS 16 Then we have M N (1 + aσ(k) ) − k=1 (1 + ak ) = k=1   N (1 + ak ) − 1 , (1 + ak )  k=1 k∈E(M,N ) where E(M, N ) := {σ(1), . . . , σ(M )} \ {1, . . . , N }. Since E(M, N ) ⊂ ◆>N0 , thus (1 + |ak |) − 1 (1 + ak ) − 1 ≤ k∈E(M,N ) k∈E(M,N ) e|ak | − 1 ≤ k∈E(M,N )   |ak | − 1 ≤ exp  k∈E(M,N )   ≤ exp  |ak | − 1 k>N0 ≤e −1≤2 . This gives M N (1 + aσ(k) ) − k=1 N N (1 + ak ) ≤ k=1 |ak | (1 + ak ) · 2 ≤ exp · 2 ≤ 2C . k=1 k=1 Now, if σ is the identity function, the inequality above is satisfied for any M ≥ N > N0 N (1 + ak ) and hence the sequence of partial product k=1 converges. Furthermore, for any M ≥ N = N0 , we have N0 M (1 + ak ) − k=1 k=1 k∈◆>0 N0 M (1 + ak ) ≤ is Cauchy and thus (1 + aσ(k) ) − N0 (1 + ak ) ≤ 2 k=1 k=1 which implies N0 M (1 + ak ) ≥ (1 − 2 ) k=1 (1 + ak ) . k=1 By taking the limit when M tends to infinity, we obtain ∞ N0 (1 + ak ) ≥ (1 − 2 ) k=1 (1 + ak ) . k=1 (1 + ak ) k=1 CHAPTER 1. ANALYTIC TOOLS 17 Hence, the limit of the infinite product is not zero if none of the 1 + ak = 0 and the converse is trivially true. To complete the proof, let L be the limit of the sequence of N partial product (1 + ak ) k=1 that and by increasing N0 if necessary, we can assume k∈◆>0 N0 (1 + ak ) − L ≤ . k=1 Then for any arbitrarily permutation σ, let N = N0 and M ≥ N as defined before, then for any K ≥ M , we have K N0 K (1 + aσ(k) ) − L ≤ k=1 (1 + aσ(k) ) − k=1 N0 (1 + ak ) − L (1 + ak ) + k=1 k=1 ≤ 2C + , which yields the assertion that the limit of the infinite product is independent of the order of which the product is taken. Definition 1.4.5. An Euler product is an infinite product over the prime numbers. Theorem 1.4.6. Let f (n) be a multiplicative function that is not identically zero. If the ∞ f (n) converges absolutely, then series n=1 ∞ ∞ 1 + f (p) + f (p2 ) + . . . = f (n) = n=1 p f (pk ) . 1+ p k=1 Furthermore, the limit of the infinite product is independent of the ordering of the prime index. ∞ f (n) converges absolutely, then the series Proof. If n=1 an :=     ∞ f (pk ) if n = p is prime, k=1    0 otherwise, CHAPTER 1. ANALYTIC TOOLS 18 is well-defined. Also, we have ∞ ∞ ∞ p n=1 ∞ |f (pk )| ≤ f (pk ) ≤ |an | = p k=1 k=1 |f (n)| < ∞. n=1 ∞ ∞ p n=1 f (pk ) 1+ (1 + an ) = Thus, the infinite product converges absolutely and k=1 by Theorem 1.4.4, it converges and the limit is independent of the ordering of the prime index. We proceed to complete the proof by establishing the equality. Let > 0, then there exists an integer N0 such that |f (n)| < . n≥N0 For every positive integer n, let P (n) denote the greatest prime factor of n. Since the ∞ |f (pk )| converges for any finite set I of prime numbers, therefore by Fubini’s series p∈I k=0 theorem and the fact that f is multiplicative, we have, for any N > N0 , ∞ ∞ k 1+ p≤N f (p ) f (pk ) = p≤N k=1 = k=0 f (n) P (n)≤N and so ∞ ∞ f (n) − n=1 1+ p≤N ∞ k f (p ) f (n) − = n=1 k=1 ≤ f (n) P (n)≤N |f (n)| ≤ n>N |f (n)| < . n>N0 By taking the limit when N tends to infinity, we obtain the desired equality. CHAPTER 1. ANALYTIC TOOLS 1.5 19 Prime Number Theory Theorem 1.5.1. (Prime Number Theorem) There exists a constant C > 0, such that for any positive real number X, we have the asymptotic formula : X π(X) := 1= n=2 p≤X = 1 + O(X exp(−C log n X +O log X X log2 X Remark: Notice that the second asymptotic formula log X)) . X log X can be obtained directly by X X 1 1 dt ∼ . The proof of this theorem involves mainly log n 2 log t n=2 complex analysis but yet most of steps are extremely technical. It can be shown that integration by part of the first asymptotic formula above is equivalent to X ψ(X) := Λ(n) = X + O(X exp(−C log X)). n=1 This is where the Prime Number Theorem is related to the analytic properties of Riemann zeta function, as the study of ψ(X) involves the estimation of an integration associated ∞ ζ Λ(n) with the function − (s) = . Apart from being technical, the estimation reζ ns n=1 quires one to find a zero free region of the Riemann zeta function near (s) = 1, such that the domain of integration is free of poles of the function ζ ζ (s). The proof of the theorem above can serve as a research topic but this is not the aim of this thesis. We invite the readers to refer to (1) in Chapter 1 of [1] Estermann or Chapters 7 and 18 of [7] Davenport for a complete proof of the refined Prime Number Theorem stated above. Readers who are contented with just the asymptotic behavior π(X) ∼ X log X can refer to Chapter XXII of [6] G. H. Hardy and E. M. Wright for a simpler proof that does not involve the Riemann zeta function. The same chapter contains also a short proof the Tchebycheff’s theorem, namely, π(X) = O( logXX ), which in fact, as far as this thesis is concerned, is sufficient for our application as we need only the upper bound π(X) X log X . We cannot resist from presenting to the readers a more precise and beautiful theorem of the prime numbers. CHAPTER 1. ANALYTIC TOOLS 20 Theorem 1.5.2. (Siegel-Walfisz Theorem) Given B > 0, there exists a constant CB > 0 such that for any positive real number X, any integers 1 ≤ q ≤ logB X and (a, q) = 1, the following asymptotic formula holds: log p = p≤X p≡a mod q X + O(X exp(−CB φ(q) log X)), where the big-O constant depends only on B and is independent of q and X. Remark: This theorem is an intermediate result in the proof of Dirichlet’s Theorem for primes in arithmetic progressions, as the asymptotic formula above implies that 1= p≤X p≡a mod q 1 φ(q) X n=2 1 + O(X exp(−CB log n log X)), uniformly for all q ≤ logB X. By using the orthogonality of characters, one has 1 φ(q) φ(q) χi (a)χi (n) =    1 if n ≡ a mod q ,   0 otherwise i=1 where χi ’s are Dirichlet’s character modulo q. With this identity, one sees that  log p = p≤X p≡a mod q p≤X 1 log p  φ(q)  φ(q) χi (a)χi (p) = i=1 1 φ(q) φ(q) χi (a) i=1 χi (p) log p. p≤X Again the proof of Siegel-Walfisz Theorem will recourse to finding the asymptotic formula for X ψ(X, χ) := χ(n)Λ(n), p=1 which can be obtained through integration involving the Dirichlet L-function L(s, χ) := ∞ χ(n) . Of course one can achieve this by generalizing the proof of the Prime Number ns n=1 Theorem mentioned above; the main difficulty in the proof of Siegel-Walfisz Theorem is to obtain an asymptotic formula that is uniform on q ≤ logB X. This difficulty is translated into the search of a single zero free region near (s) = 1 for all Dirichlet’s L-functions L(s, χ) with Dirichlet’s character χ modulo q ≤ logB X. This difficulty is CHAPTER 1. ANALYTIC TOOLS 21 resolved by Siegel’s Theorem, a theorem that gives a lower bound of the value of L(1, χ) for real primitive character χ. Again, the proof of Siegel’s Theorem is purely analytical and is too lengthy to be included in this thesis. For a complete proof of Siegel-Walfisz Theorem, readers can refer to (40) in Chapter 1 and theorem 53 of [1] Estermann or (4) in Chapters 22 of [7] Davenport. Chapter 2 The Ternary Goldbach Problem The ternary Goldbach problem conjectures that every odd integer greater than 5 can be written as a sum of three prime numbers. In this chapter, we are going to prove Vinogradov’s theorem: there exists an absolute constant N0 , such that for all odd integer N ≥ N0 , N can be expressed as a sum of three prime numbers. This was first proved by Vinogradov in 1937. In fact, Vinogradov proved furthermore that the number of representations of an odd integer N as a sum of three prime numbers increases with N . More precisely, for odd integer N ≥ N0 , we have (log p1 )(log p2 )(log p3 ) N 2, p1 +p2 +p3 =N which implies the number of representations 1≥ p1 +p2 +p3 =N 1 log3 N (log p1 )(log p2 )(log p3 ) p1 +p2 +p3 =N N2 . log3 N We attack this problem using the Hardy-Littlewood method, which is also known as the circle method. We consider the following period 1 function defined for α ∈ ❘ : f (α) = (log p) e(αp), p≤N 22 CHAPTER 2. THE TERNARY GOLDBACH PROBLEM 23 and thus 3N f 3 (α) = R(m, N ) e(αm), (log p1 )(log p2 )(log p3 ). where R(m, N ) = p1 +p2 +p3 =m pi ≤N m=1 By denoting R(N ) := R(N, N ) = I f 3 (α)e(−N α)dα, where I is an interval of length 1, we transform the ternary Goldbach problem into the search for an asymptotic estimate for the integral. We fix from now on a positive constant B > 10 and let P := logB N , where N is a fixed positive integer that is large enough in order for certain inequalities to hold in the proof. We first divide the domain of integration into two parts: the major arcs and the minor arcs. The major arcs are defined by M= M(q, a), where M(q, a) := α∈ q≤P (a,q)=1 P P ,1 + N N : α− a P ≤ q N . This integration domain will give a major contribution to the integral, because it contains the α’s that are close to some rational number with a “small” denominator, and thus the chances of having a huge cancellation between the e(αp) terms in the sum which defines f (α) is small. The reason for choosing the bound P = logB N for the “small” denominator is due to the hypothesis needed in Siegel-Walfisz Theorem, which we will apply in the estimation of the integral over the major arcs M. On the other hand, the minor arcs P P ,1 + \M N N m= do not contribute to the asymptotic approximation of the integral due to the huge cancellation between the e(αp) terms for the α’s close to a rational number with a “big” denominator. The reason that we consider the sum (log p1 )(log p2 )(log p3 ) p1 +p2 +p3 =N instead of the sum 1, p1 +p2 +p3 =N CHAPTER 2. THE TERNARY GOLDBACH PROBLEM 24 is because this will simplify a great deal most of the estimations in the proof. After all, the actual motivation is not to estimate the number of representations of an odd integer as the sum of three primes, but to show the existence of such representations. In the following sections, we begin by finding a bound for the integral mf 3 (α)e(−N α)dα over the minor arcs m and after that we deal with the case of major arcs M. 2.1 The Minor Arcs m Before we begin to bound the integral over the minor arcs, we need the following lemma concerning the Dirichlet series. Lemma 2.1.1. (Dirichlet’s convolution identity) For any two arithmetic functions ∞ ∞ h(m) g(m) and L(h, s) = be two Dirichlet’s series and let g, h, let L(g, s) = s m ms m=1 m=1 (g ∗ h)(m) := g(a)h(b). Then L(g ∗ h, s) converges absolutely for all s for which both ab=m L(g, s) and L(h, s) converge absolutely, and one has L(g, s)L(h, s) = L(g ∗ h, s). Proof. For any positive integer k, a direct calculation gives m≤k (g ∗ h)(m) ≤ ms m≤k2 1 ms    |g(a)h(b)| =  g(a)   as h(b)  bs ab=m a,b≤k a≤k b≤k and thus L(g ∗ h, s) converges absolutely whenever L(g, s) and L(h, s) both converge absolutely. Under this hypothesis, the same calculation shows    g(a)   as a≤k  b≤k h(b)  = bs m≤k2 1 ms g(a)h(b) − ab=m kk We obtain the lemma by making k tend to infinity, observing that the last term tends ∞ 1 to zero by the convergence of |g(a)h(b)|. ms m=1 ab=m CHAPTER 2. THE TERNARY GOLDBACH PROBLEM 25 Theorem 2.1.2. For any real number α such that there exists integers a, q with 1 ≤ q ≤ N and α − a q ≤ 1 , q2 we have N √ + q (log4 N ) f (α) Proof. We first fix a positive real number X ∈ [1, 4 Nq + N 5 √ . N ] and establish the following iden- tity: Λ(b)e(αb) = S1 − S2 − S3 (3) X[...]... {sn }n∈◆>0 converges if and only if the sequence {pn }n∈◆>0 converges Notice that pn ≥ 1 for all n and hence its limit, if it exists, is non zero ∞ (1 + ak ) is said to converge absolutely if Definition 1.4.3 The infinite product k=1 ∞ (1 + |ak |) converges the infinite product k=1 ∞ (1+ak ) converges absolutely, then it converges Theorem 1.4.4 If the infinite product k=1 and the limit is independent... to infinity, ∞ αk converges and we denote we say that the infinite product k=1 ∞ n αk := lim n→∞ k=1 αk = α k=1 III If the sequence of partial products does not converge when n tends to infinity, then we say that the infinite product diverges ∞ Theorem 1.4.2 Let ak ≥ 0 for all k ∈ ◆>0 The infinite product k=1 ∞ ak converges if and only if the infinite series n Proof Let sn := k=1 n (1+ak ) Since ak... fix from now on a positive constant B > 10 and let P := logB N , where N is a fixed positive integer that is large enough in order for certain inequalities to hold in the proof We first divide the domain of integration into two parts: the major arcs and the minor arcs The major arcs are defined by M= M(q, a), where M(q, a) := α∈ q≤P (a,q)=1 P P ,1 + N N : α− a P ≤ q N This integration domain will give... major contribution to the integral, because it contains the α’s that are close to some rational number with a “small” denominator, and thus the chances of having a huge cancellation between the e(αp) terms in the sum which defines f (α) is small The reason for choosing the bound P = logB N for the “small” denominator is due to the hypothesis needed in Siegel-Walfisz Theorem, which we will apply in the... estimations in the proof After all, the actual motivation is not to estimate the number of representations of an odd integer as the sum of three primes, but to show the existence of such representations In the following sections, we begin by finding a bound for the integral mf 3 (α)e(−N α)dα over the minor arcs m and after that we deal with the case of major arcs M 2.1 The Minor Arcs m Before we begin to... where the big-O constant depends only on B and is independent of q and X Remark: This theorem is an intermediate result in the proof of Dirichlet’s Theorem for primes in arithmetic progressions, as the asymptotic formula above implies that 1= p≤X p≡a mod q 1 φ(q) X n=2 1 + O(X exp(−CB log n log X)), uniformly for all q ≤ logB X By using the orthogonality of characters, one has 1 φ(q) φ(q) χi (a)χi (n) =... have the strict inequality ak and pn := k=1 (1 + ak ) converges k=1 sn < pn Also, since the inequality 1 + x ≤ ex CHAPTER 1 ANALYTIC TOOLS 15 is true for all x ∈ ❘, we have n 0≤ n k=1 n n eak = exp (1 + ak ) ≤ ak < k=1 k=1 ak , k=1 i.e 0 ≤ sn < pn ≤ esn Since both sequences {sn }n∈◆>0 and {pn }n∈◆>0 are monotone increasing, they converges if and only if they are bounded Thus the inequality above... distinct primes It is easy to check that the arithmetic function µ(n) is multiplicative and µ3 = µ Proposition 1.2.4 For any natural number n ∈ ◆>0 , we have the following identity: µ(d) = δ(n) =    1 if n = 1,   0 otherwise d|n Proof The assertion is trivial for n = 1 For n > 1, we write n in its unique prime decomposition : r pki i , n= i=1 where pi are pairwise distinct primes and r ≥ 1 Since... the integral over the minor arcs, we need the following lemma concerning the Dirichlet series Lemma 2.1.1 (Dirichlet’s convolution identity) For any two arithmetic functions ∞ ∞ h(m) g(m) and L(h, s) = be two Dirichlet’s series and let g, h, let L(g, s) = s m ms m=1 m=1 (g ∗ h)(m) := g(a)h(b) Then L(g ∗ h, s) converges absolutely for all s for which both ab=m L(g, s) and L(h, s) converge absolutely, and. .. ◆>0 , we have the following identity: Λ(d) = log n d|n Proof The assertion is trivial for n = 1 For n > 1, we write n in its unique prime decomposition : r pki i , n= i=1 where pi are pairwise distinct primes and r ≥ 1 Since Λ is non zero only at prime powers, we have r ki r Λ(pji ) = Λ(d) = d|n i=1 j=1 r log pki i = log n ki log pi = i=1 i=1 Definition 1.2.7 The divisor function d(n) counts the number ... our conjecture on quadratic fields, we will explain why the conjecture is a direct consequence of the v Vinogradov’s theorem for primes in arithmetic progression and move on to prove this theorem. .. the big-O constant depends only on B and is independent of q and X Remark: This theorem is an intermediate result in the proof of Dirichlet’s Theorem for primes in arithmetic progressions, as the... Vinogradov’s theorem to the quadratic fields This generalization will in turn give the motivation to formulate the Vinogradov’s theorem for primes in arithmetic progression: Let x1 , x2 , x3 and