The Riemann Hypothesis and the Distribution of Prime Numbers

This book provides an accessible and comprehensive introduction to the Riemann Hypothesis,one of the most important open questions in math today. It covers the definition of the zeta function, complex numbers, the functional equation of the zeta function, the Riemann Hypothesis, and its connections to zeta's zeros and prime numbers.

Format: Hardback
Length: 219 pages
Publication date: 01 June 2021
Publisher: Nova Science Publishers Inc

This book is a comprehensive and introductory exploration of the Riemann Hypothesis, one of the most significant and challenging open questions in mathematics today. Written in an accessible and detailed format, it makes the theory easy to understand and grasp. The book is comprehensive in its coverage, explaining and proving all the mathematical concepts surrounding and leading to the formulation of the hypothesis.

Chapter 1 begins by defining the zeta function and delving into its properties when the argument is a real number. It then identifies the convergence domain of the series and establishes Euler's product formula. Chapter 2 introduces complex numbers and the complex analytic tools necessary to understand the zeta function in the complex plane. Chapter 3 extends the domain of the zeta function for the first time by introducing the eta function. Through proofs by Sondow, it is demonstrated that the zeta function can be defined for any complex number with a positive real part. Chapter 4 derives the functional equation of the zeta function, providing a method to extend its definition to the entire complex plane. Chapter 5 introduces the Riemann Hypothesis for the first time, linking the zeros of the zeta and eta functions and leading 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's explicit formula and explains its use in rewriting the prime counting function in terms of the Riemann prime counting function. It provides a detailed numerical example on how to apply Riemann's formula. Chapter 7 derives the von Mangoldt formula via the residue theorem and elucidates some of its important properties.

In conclusion, this book is an invaluable resource for anyone interested in delving into the world of mathematics and understanding the Riemann Hypothesis. Its comprehensive and accessible approach makes it an ideal introduction to the topic, while its rigorous proofs and detailed explanations provide a deeper understanding for advanced students and researchers.

Weight: 437g
ISBN-13: 9781536194227