STUDY OF ANALYTIC NUMBER THEORY: RIEMANN’S HYPOTHESIS AND PRIME NUMBER THEOREM WITH ADDENDUM ON INTEGER PARTITIONS

 
Citation
Title:
STUDY OF ANALYTIC NUMBER THEORY: RIEMANN’S HYPOTHESIS AND PRIME NUMBER THEOREM WITH ADDENDUM ON INTEGER PARTITIONS
Author:
Editor:
City:
Cambridge, UK
Publisher:
Cambridge International Science Publishing
Year:
2013
Volume:
No. of Volumes:
Edition:
1st
Pages:
108
Series Volume:
Series Editor:
Series Title:
Translator:
Language:
English
URL:
http://www.cisp-publishing.com/acatalog/info_104.html
DOI:
LCCN:
OCLC Number:
License
No License (All right reserved)
ISBN:
1907343988
1907343988 More info
Last Updated:
December 29th, 2013
Abstract

This monograph explores several classical issues of modern mathematics, and discusses both the historical and research aspects. The brief historical part is focused on Riemann’s zeta function and few related problems. The research part starts from direct formulation of simple proofs of both the prime number theorem and Riemann’s hypothesis, two intriguing problems of modern mathematics, which applies the concept of Mertens’s function and is based on Apostol’s and Littlewood’s criterions of equivalence. The second research problem discussed in this book is distribution of prime numbers. Elementary derivation of the explicit formulae for both the prime-counting function and Riemann’s weighted-prime counting function, based on the technique of Mellin’s transform, is presented in some detail. As the third research problem, the binomial regularization of Riemann’s zeta function is introduced. Finally, as the additional research problem, the derivation of the number of partitions of a positive integer into a finite number of positive integer summands, based on Hardy-Ramanujan-Rademacher’s formula, is presented in some detail. The monograph is helpful for the mathematicians working in number theory, combinatorics, mathematical analysis, function theory, and their applications. Also, it can be a higher-level introduction for students starting their work in these branches of mathematics.


Contents

Foreword

1 Brief Historical Overview

1 Riemann’s Zeta Function

2 Mertens’s Conjecture

3 Prime Number Theorem

2 Prime Number Theorem & Riemann’s Hypothesis

3 Distribution of Prime Numbers

4 Binomial Regularization

5 Addendum: Integer Partitions

Bibliography

Index


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