ANALYTIC PROOF OF THE PRIME NUMBER THEOREM A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science by Maha Mosaad Alghamdi May, 2019 © Copyright All rights reserved Except for previously published materials Thesis written by Maha Mosaad Alghamdi B.S., Dammam University, 2011 M.S., Kent State University, 2019 Approved by Gang Yu , Advisor Andrew Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS TABLE OF CONTENTS ......................................................................................................... III LIST OF FIGURES ............................................................................................................................................. V ACKNOWLEDGMENTS................................................................................................................................. VI CHAPTERS 1. INTRODUCTION ................................................................................................................................. 1 2. SUMMATION FORMULAS............................................................................................................... 6 2.1 Abelian Summation Formula ............................................................................................... 6 2.2 Euler-Maclaurin Summation Formula ............................................................................. 8 3. DIRICHLET SERIES AND EULER PRODUCTS ...................................................................... 10 3.1 The Half-Plane of Absolute Convergence of a Dirichlet Series ............................. 11 3.2 The Function Defined by a Dirichlet Series ................................................................. 12 3.3 Multiplication of the Dirichlet Series ............................................................................. 13 3.4 Euler Product .......................................................................................................................... 16 4. BASIC PROPERTIES OF ζ(s) ....................................................................................................... 17 4.1 Definition and Basic Properties of the Euler Gamma Function .......................... 17 4.2 Stirlig’s Formula ..................................................................................................................... 19 4.3 Analytic Continuation of ζ(s) ............................................................................................ 20 4.4 A Zero-Free Region of ζ(s) ................................................................................................. 30 5. PRIME NUMBER THEOREM ....................................................................................................... 34 5.1 Truncated Form of Perron’s Formula ........................................................................... 36 5.2 Proof of the Prime Number Theorem ............................................................................ 40 5.3 An Explicit Formula of ......................................................................................................... 42 iii BIBLIOGRAPHY ............................................................................................................................................. 46 iv LIST OF FIGURES Figure 5.1. Rectangular contour with vertices (푏 ± 푖푇, −푈 ± 푖푇) oriented counterclockwise ........................................................................................................................... 36 Figure 5.2. Rectangular contour with vertices (푏 ± 푖푇, −푈 ± 푖푇) oriented clockwise ........................................................................................................................................... 37 Figure 5.3. The integral 퐽 around the contour defined by Γ where 퐽 is defined by 1 휁′ 푥푠 퐽 = ∫ (− (푠)) 푑푠 ............................................................................................................... 41 2휋푖 Γ 휁 푠 Figure 5.4. The integral 퐽 around the contour Γ where 퐽 is defined by 1 휁′ 푥푠 퐽 = ∫ (− (푠)) 푑푠 ............................................................................................................... 43 2휋푖 Γ 휁 푠 v ACKNOWLEDGMENTS I owe my thanks and appreciation to my thesis advisor, Dr. Gang Yu, for his patience, advice, guidance, and assistance throughout the time of my thesis research. He was always available and willing to help me and explain any difficulties I faced. I am really proud to have a professor like him who is full of knowledge and experience, which makes someone learn a lot from him on both the scientific and the personal sides. I wish to express my sincere thanks to both Dr. Ulrike Vorhauer and Dr. Morley Davidson for agreeing to be the members of my defense committee. Also, I am thankful for our graduate coordinator, Dr. Artem Zvavitch, for his help and guidance. I would love to thank Kent State University and especially the Department of Mathematical Sciences, for providing me an opportunity to study for a master’s degree in pure mathematics. I also would like to thank Imam Abdulrahman Bin Faisal University for its support and for giving me the opportunity to complete the master’s degree. Thanks as well to officials at the Saudi Arabian Cultural Mission in the United States for their efforts and cooperation while I was studying for my master’s degree and for their support of my family. I would like to thank for the blessing of my life my dear mother for her patience and for letting me find a way forward. Thank you, my dear mother, for sowing the ambition in my heart until what I am now. Thank you, my dear mother, for your support and sincere prayers, which makes me very happy. Thank you, for your patience and waiting for me in the days and years until I return home. Thank you, my brothers and sisters, for your support, encouragement, and motivation, and especially for your trust and belief in me. vi I could not achieve success easily without the support of my beloved husband and my daughters, Malak, Jumanh, Sadeem, my wonderful child Abdullah, and my little angel Ward. A ton of thanks to my husband and children for their sacrifices and patience as I worked to achieve my goal. They supported and helped me unconditionally in my success. They share the most accurate details who rejoice to my joy and sadness for my sorrow. vii CHAPTER 1 INTRODUCTION A prime number is a positive integer that has exactly two positive integer divisors: 1 and itself. In other words, a number is prime if it is greater than 1 and cannot be written as a product of two natural numbers that are both less than it. The most important property of prime numbers is the Fundamental Theorem of Arithmetic (FTA), which describes the relationship between prime numbers and natural numbers. Every natural number n can be uniquely written as 푎1 푎푘 푛 = 푝1 … 푝푘 , where 푎1, … , 푎푘 are natural numbers, and 푝1 < 푝2 < ⋯ < 푝푘 are prime numbers. This theorem shows that prime numbers are the multiplicative building blocks of natural numbers. Based on the FTA, there is an infinite number of prime numbers. Proof of the infinitude of prime numbers based on the FTA has a long history. Euclid was the first to do so in roughly 300 BC. In the 18th century, Leonhard Euler offered a different proof, though it, too, was based on the FTA. Euler proved a factorization formula, which said that for any fixed real number s > 1, the following is true when the product is taken over all prime numbers: −1 ∞ 1 1 ∑ = ∏ (1 − ) (1.1) 푛=1 푛푠 푝 푝푠 1 This formula is essentially the analytic equivalence of the FTA. Euler’s formula represents an important discovery because it does much more than merely the infinitude of prime numbers. The formula also relates prime numbers to the more familiar natural numbers. While a simple formula does not exist to distinguish prime numbers from composite numbers, the quantitative distribution of prime numbers within the natural numbers becomes extremely important. For a positive real number 푥, let 휋(푥) = ∑ 1, 푝≤푥 where 푝 runs over prime numbers up to 푥. Thus, 휋(푥) serves as the “counting function” of prime numbers. While the unboundedness of 휋(푥) had long been known, a natural question is: Does 휋(푥) have an asymptotic formula as 푥 → ∞? The Prime Number Theorem (PNT), which serves as the focus of this thesis, gives an affirmative answer to the question posed. In 1798, based on the existing numerical evidence, Adrien-Marie Legendre made the following very accurate conjecture: 푥 휋(푥)~ as 푥 → ∞. (1.2) 푙표푔 푥−1.08366 In 1849, Johann Carl Friedrich Gauss wrote about his own findings in a letter to German astronomer Johann Franz Encke, saying in 1792 or 1793, when he was still a boy, that he had noticed that the average density of prime numbers up to 푥 should be 1 . In other words, log 푥 푥 휋(푥)~ as 푥 → ∞. (1.3) log 푥 2 As 푥 → ∞, Equation 1.2 and Equation 1.3 are equivalent; however, Equation 1.3 is the weakest form of the PNT and often is referred to as the PNT without an error term. For the next few years, the PNT attracted the attention of mathematicians but remained unproved. In 1852 or so, Pafnuty Lvovich Chebyshev identified the existence of two positive constants: 퐶1 and 퐶2, such that 푥 푥 퐶 < 휋(푥) < 퐶 (푥 > 2) 1 log 푥 2 log 푥 Chebyshev also introduced two functions, which now are referred to as Chebyshev’s functions: 휃(푥) = ∑ log 푝 푝≤푥 and 휓(푥) = ∑ Λ(푛), 푛≤푥 where Λ(푛) is the von Mangoldt function and is defined, for every