The Pennsylvania State University The Graduate School ON A VARIANCE ASSOCIATED WITH THE DISTRIBUTION OF REAL SEQUENCES IN ARITHMETIC PROGRESSIONS A Dissertation in Mathematics by Pengyong Ding © 2021 Pengyong Ding Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2021 The dissertation of Pengyong Ding was reviewed and approved by the following: Robert Charles Vaughan Professor of Mathematics Dissertation Adviser Committee Chair Wen-Ching Winnie Li Distinguished Professor of Mathematics Ae Ja Yee Professor of Mathematics Mary Kathleen Heid Distinguished Professor of Education Mark Levi Professor of Mathematics Head of the Department ii Abstract This dissertation is composed of two parts. The first part concerns the general result on the following variance associated with the distribution of a real sequence {an} in arithmetic progressions: q V (x, Q) = X X |A(x; q, a) − f(q, a)M(x)|2, q≤Q a=1 where A(x; q, a) represents the sum of {an} in the residue class of a modulo q, and f(q, a) and M(x) approximately reflect the local and global properties of {an} respectively. We will give a brief history of the problem and introduce two powerful methods, and then we will provide the standard initial procedure. The second part is an example on calculating the variance V (x, Q) when an = r3(n), the number of (ordered) representations of n as the sum of three positive cubes: X r3(n) = 1. x1,x2,x3 3 3 3 x1+x2+x3=n We will introduce the properties of the function and show how to calculate the main terms and estimate the error terms. The conclusion will be stated as Theorem 5.1. Finally, several special cases and similar questions will be listed at the end of the dissertation. iii Table of Contents List of Symbols vi Acknowledgments viii Chapter 1 Introduction 1 1.1 An Introduction to Analytic Number Theory . 1 1.2 An Introduction to the Problem . 3 1.3 The History of the Problem . 4 Chapter 2 Two Powerful Methods 8 2.1 The Hardy-Littlewood Circle Method . 8 2.2 The Farey Sequence . 10 Chapter 3 The Standard Initial Procedure 13 3.1 The Treatment of S1 ............................. 14 3.2 Further Arrangement . 16 Chapter 4 An Example: an = r3(n) 17 4.1 Introduction to r3(n) ............................. 17 4.2 Upper and Lower Exponents . 19 Chapter 5 The Variance in the Special Case 21 5.1 Several Lemmata . 21 5.2 The Main Term S3 .............................. 31 5.3 The Error Term S2 − S3 ........................... 33 5.4 The Major Arcs . 35 5.5 The Optimal Choice for R .......................... 38 5.6 The Main Term S6 .............................. 40 5.7 The Calculation of W (X) .......................... 42 iv 5.8 Conclusion: Theorem 5.1 . 48 Chapter 6 Several Notes 50 6.1 Results for Large Q .............................. 50 6.2 Similar Questions . 54 Bibliography 56 v List of Symbols (a, b) The greatest common divisor of integers a and b, p.1 C The set of complex numbers, p.1 <s The real part of a complex number s, p.1 ζ(s) The Riemann-zeta function of a complex number s, p.2 f(x) ∼ g(x) lim f(x)/g(x) = 1, p.2 log x The natural logarithm of a real number x, p.2 N The set of natural numbers, p.3 R The set of real numbers, p.3 a ≡ b (mod c)( a − b)/c is an integer where a, b, c are integers, p.3 f(x) g(x) |f(x)| ≤ Cg(x) where C is an absolute constant, p.3 f(x) = O(g(x)) f(x) g(x), p.4 Λ(n) The von Mangoldt function of a natural number n, p.4 φ(n) The Euler totient function of a natural number n, p.4 exp(x) ex where x is a real number, p.5 a|b b/a is an integer where a and b are integers and a 6= 0, p.6 d(n) The divisor function of a natural number n, p.7 [x] The integer part of a real number x, p.9 e(z) e2πiz where z is a complex number, p.9 Z The set of integers, p.9 vi a - b b/a is not an integer where a and b are integers, p.14 Γ(t) The gamma function of a real number t, p.17 B(a, b) The beta function of real numbers a and b, p.28 ||r|| The distance from a real number r to the nearest integer, p.29 γ The Euler constant, p.32 =s The imaginary part of a complex number s, p.46 f(x) g(x) f(x) g(x) and g(x) f(x), p.50 vii Acknowledgments First, I would like to thank my advisor, Professor Robert C. Vaughan, for his guidance during my Ph.D. career. Next, I would like to thank Professor Wen-Ching W. Li, Ae Ja Yee and M. Kathleen Heid for serving on my committee. I would also like to thank Allyson Borger, the administrative support assistant of the graduate studies program in the Department of Mathematics, for her kindly assistance in the past four years. Finally, I would like to thank my parents for their love and support, both physically and mentally. I would never have made such a success without them. viii Chapter 1 | Introduction This dissertation is composed of two parts. The first part is the general result on a variance associated with the distribution of a real sequence {an} in arithmetic progressions, including the first three chapters: Chapter 1 introduces the problem and gives a brief historical overview, Chapter 2 contains two powerful methods to study the problem, including the Hardy-Littlewood circle method and the Farey sequence, Chapter 3 gives a standard initial procedure to estimate the variance. The second part of the dissertation is a specific example when an = r3(n), including Chapter 4 and onward: Chapter 4 introduces the function r3(n) and shows some preliminary results, Chapter 5 shows further results and the conclusion on the variance, and finally, Chapter 6 gives several notes on the problem, including several special cases for r3(n) as well as similar questions. 1.1 An Introduction to Analytic Number Theory Number Theory is a branch of pure mathematics to the study of integers and integer- valued functions, and analytic number theory is a branch of number theory where we use analytic methods to solve problems about integers, especially the methods where we use mathematical analysis like real analysis and complex analysis. It may be said to begin with Dirichlet’s introduction of L-function in 1837, when he proved the existence of primes in an arithmetic progression, namely for any integers a and q such that (a, q) = 1, there are infinitely many primes of the form a + nq, where n is also an integer. He defined the L-function as follows: ∞ L(s, χ) := X χ(n)n−s, (1.1) n=1 where s ∈ C and <s > 1, and χ is a Dirichlet character modulo q, i.e. a complex-valued function with the following properties: 1 (1) χ(n) = χ(n + q) for all integers n, (2) χ(n) 6= 0 if and only if (n, q) = 1, (3) χ(mn) = χ(m)χ(n) for all integers m and n. And if χ0 is a character such that for all integers n, χ0(n) = 1, then define the Riemann-zeta function: ∞ X −s ζ(s) := L(s, χ0) = n . (1.2) n=1 Since then, many well-known results have been discovered, including: (1) The prime number theorem, namely π(x) ∼ x/ log x, where π(x) denotes the number of prime numbers less than or equal to x. (2) The existence of zeros of the Riemann-zeta function ζ(s). For a fixed character χ, if L(s0, χ) = 0 for a negative real number s0, then s0 is called a trivial zero of L(s, χ). It is proved that there are infinitely many trivial zeros of ζ(s): s = −2n where n is a positive integer. A zero of L(s, χ) is called non-trivial if it is not trivial. It is also proved that if a non-trivial zero s0 of L(s, χ) exists, then it satisfies 0 ≤ <s0 ≤ 1. (3) Goldbach ternary problem, namely any odd number n ≥ 7 is a sum of three primes. However, there are still many unsolved problems and conjectures in analytic number theory, including the following famous problems: (1) Riemann Hypothesis (RH) and generalized Riemann Hypothesis (GRH). RH indicates that the real part of every non-trivial zero of ζ(s) is 1/2, and GRH indicates that for every Dirichlet character χ, the real part of every non-trivial zero of L(s, χ) is 1/2. (2) Waring’s problem: Finding the values of G(k), which is defined to be the least s such that every sufficiently large integer n is the sum of at most s k-th power of positive integers. (3) Goldbach binary problem (conjecture), namely any even number n ≥ 4 is a sum of two primes. In order to prove the results or to study conjectures and other problems, many useful methods have been found. There are some classical methods. For example, the Selberg Sieve and the Bombieri - A. I. Vinogradov Theorem is concerned with the distribution of primes, and the Hardy - Littlewood circle method is commonly used to study the Goldbach problems and the Waring’s problem.