The Riemann Hypothesis and the Distribution of Prime Numbers


Series: Mathematics Research Developments

BISAC: MAT022000

***Rated as One of the Best New Prime Numbers books for 2023 by BookAuthority!***

This book is an introductory and comprehensive presentation of the Riemann Hypothesis, one of the most important open questions in math today. It is introductory because it is written in an accessible and detailed format that makes it easy to read and understand. And it is comprehensive because it explains and proves all the mathematical ideas surrounding and leading to the formulation of the hypothesis.

Chapter 1 begins by defining the zeta function and exploring some of its properties when the argument is a real number. It proceeds to identify the series’ domain of convergence and proves Euler’s product formula. Chapter 2 introduces complex numbers and the complex analytic tools necessary to understand the zeta function in complex plane. Chapter 3 extends the domain of the zeta function for the first time by introducing the eta function. Presenting proofs by Sondow, it is shown that zeta can be defined for any complex number whose real part is positive. Next, the functional equation of the zeta function is derived in Chapter 4. This provides a method to extend the definition of zeta to the entirety of the complex plane. Chapter 5 is where the Riemann Hypothesis is properly introduced for the first time. It relates the zeros of the zeta and eta functions which leads to a simple formulation of the hypothesis. Chapters 6 and 7 connect the topics of zeta’s zeros and the distribution of prime numbers. Chapter 6 introduces Riemann explicit formula and explains the use of Mobius transform to rewrite the prime counting function in terms of the Riemann prime counting one and it provides a detailed numerical example on how to use the Riemann’s formula. Chapter 7 derives the von Mangoldt formula via the residue theorem and elucidates some of its important properties. Certain necessary mathematical tools, such as Fourier analysis and theta and gamma functional equations, are included in the appendices to make the chapters more concise and focused.

Table of Contents


Chapter 1. The Zeta Function in the World of Real Numbers

Chapter 2. A Guided Tour in the World of Complex Numbers

Chapter 3. The Eta Series and the First Extension of Zeta

Chapter 4. The Functional Equation and the Second Extension of Zeta

Chapter 5. The Roots of the Zeta Function and the Riemann Hypothesis

Chapter 6. The Zeta Function and Counting the Prime Numbers

Chapter 7. Von Mangoldt Formula to the Rescue

Appendix A – Fourier Transform

Appendix B – The Functional Equation of the Jacobi Theta Function

Appendix C – The Functional Equations of the Gamma Function (Legendre’s Duplication and Euler’s Reflection)

Appendix D




“Dr. Arwashan provides a clear and concise account of all the undergraduate-level mathematical topics relevant to an understanding of the Riemann Hypothesis, with careful attention to issues that commonly cause confusion — the multiple values of a logarithm in the complex plane, for example. This is a valuable addition to the literature on the Hypothesis.” – John Derbyshire, writer, critic, commentator, columnist, and magazine journalist

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