Recent Progress On Topics Of Ramanujan Sums And Cotangent Sums Associated With The Riemann Hypothesis
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Recent Progress On Topics Of Ramanujan Sums And Cotangent Sums Associated With The Riemann Hypothesis
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Author : Helmut Maier
language : en
Publisher: World Scientific
Release Date : 2021-12-28
Recent Progress On Topics Of Ramanujan Sums And Cotangent Sums Associated With The Riemann Hypothesis written by Helmut Maier and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-12-28 with Mathematics categories.
In this monograph, we study recent results on some categories of trigonometric/exponential sums along with various of their applications in Mathematical Analysis and Analytic Number Theory. Through the two chapters of this monograph, we wish to highlight the applicability and breadth of techniques of trigonometric/exponential sums in various problems focusing on the interplay of Mathematical Analysis and Analytic Number Theory. We wish to stress the point that the goal is not only to prove the desired results, but also to present a plethora of intermediate Propositions and Corollaries investigating the behaviour of such sums, which can also be applied in completely different problems and settings than the ones treated within this monograph.In the present work we mainly focus on the applications of trigonometric/exponential sums in the study of Ramanujan sums — which constitute a very classical domain of research in Number Theory — as well as the study of certain cotangent sums with a wide range of applications, especially in the study of Dedekind sums and a facet of the research conducted on the Riemann Hypothesis. For example, in our study of the cotangent sums treated within the second chapter, the methods and techniques employed reveal unexpected connections with independent and very interesting problems investigated in the past by R de la Bretèche and G Tenenbaum on trigonometric series, as well as by S Marmi, P Moussa and J-C Yoccoz on Dynamical Systems.Overall, a reader who has mastered fundamentals of Mathematical Analysis, as well as having a working knowledge of Classical and Analytic Number Theory, will be able to gradually follow all the parts of the monograph. Therefore, the present monograph will be of interest to advanced undergraduate and graduate students as well as researchers who wish to be informed on the latest developments on the topics treated.
Analytic Methods In Number Theory When Complex Numbers Count
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Author : Wadim Zudilin
language : en
Publisher: World Scientific
Release Date : 2023-08-22
Analytic Methods In Number Theory When Complex Numbers Count written by Wadim Zudilin and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-22 with Mathematics categories.
There is no surprise that arithmetic properties of integral ('whole') numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other 'better known' numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of π and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible — almost popular — level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.
The Riemann Hypothesis
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Author : Peter B. Borwein
language : en
Publisher: Springer Science & Business Media
Release Date : 2008
The Riemann Hypothesis written by Peter B. Borwein and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors." The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.
Series Associated With The Zeta And Related Functions
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Author : Hari M. Srivastava
language : en
Publisher: Springer Science & Business Media
Release Date : 2001
Series Associated With The Zeta And Related Functions written by Hari M. Srivastava and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s,a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions.
The Riemann Zeta Function
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Author : Anatoly A. Karatsuba
language : en
Publisher: Walter de Gruyter
Release Date : 2011-05-03
The Riemann Zeta Function written by Anatoly A. Karatsuba and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-05-03 with Mathematics categories.
The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany
A Primer Of Analytic Number Theory
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Author : Jeffrey Stopple
language : en
Publisher: Cambridge University Press
Release Date : 2003-06-23
A Primer Of Analytic Number Theory written by Jeffrey Stopple and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-06-23 with Mathematics categories.
An undergraduate-level 2003 introduction whose only prerequisite is a standard calculus course.
Summability Calculus
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Author : Ibrahim M. Alabdulmohsin
language : en
Publisher: Springer
Release Date : 2018-03-07
Summability Calculus written by Ibrahim M. Alabdulmohsin and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-03-07 with Mathematics categories.
This book develops the foundations of "summability calculus", which is a comprehensive theory of fractional finite sums. It fills an important gap in the literature by unifying and extending disparate historical results. It also presents new material that has not been published before. Importantly, it shows how the study of fractional finite sums benefits from and contributes to many areas of mathematics, such as divergent series, numerical integration, approximation theory, asymptotic methods, special functions, series acceleration, Fourier analysis, the calculus of finite differences, and information theory. As such, it appeals to a wide audience of mathematicians whose interests include the study of special functions, summability theory, analytic number theory, series and sequences, approximation theory, asymptotic expansions, or numerical methods. Richly illustrated, it features chapter summaries, and includes numerous examples and exercises. The content is mostly developed from scratch using only undergraduate mathematics, such as calculus and linear algebra.
Analytic Number Theory For Undergraduates
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Author : Heng Huat Chan
language : en
Publisher: World Scientific Publishing Company
Release Date : 2009-04-21
Analytic Number Theory For Undergraduates written by Heng Huat Chan and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-04-21 with Mathematics categories.
This book is written for undergraduates who wish to learn some basic results in analytic number theory. It covers topics such as Bertrand's Postulate, the Prime Number Theorem and Dirichlet's Theorem of primes in arithmetic progression. The materials in this book are based on A Hildebrand's 1991 lectures delivered at the University of Illinois at Urbana-Champaign and the author's course conducted at the National University of Singapore from 2001 to 2008.
Stalking The Riemann Hypothesis
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Author : Daniel Nahum Rockmore
language : en
Publisher: Random House
Release Date : 2011-06-08
Stalking The Riemann Hypothesis written by Daniel Nahum Rockmore and has been published by Random House this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-06-08 with Mathematics categories.
Like a hunter who sees 'a bit of blood' on the trail, that's how Princeton mathematician Peter Sarnak describes the feeling of chasing an idea that seems to have a chance of success. If this is so, then the jungle of abstractions that is mathematics is full of frenzied hunters these days. They are out stalking big game: the resolution of 'The Riemann Hypothesis', seems to be in their sights. The Riemann Hypothesis is about the prime numbers, the fundamental numerical elements. Stated in 1859 by Professor Bernhard Riemann, it proposes a simple law which Riemann believed a 'very likely' explanation for the way in which the primes are distributed among the whole numbers, indivisible stars scattered without end throughout a boundless numerical universe. Just eight years later, at the tender age of thirty-nine Riemann would be dead from tuberculosis, cheated of the opportunity to settle his conjecture. For over a century, the Riemann Hypothesis has stumped the greatest of mathematical minds, but these days frustration has begun to give way to excitement. This unassuming comment is revealing astounding connections among nuclear physics, chaos and number theory, creating a frenzy of intellectual excitement amplified by the recent promise of a one million dollar bounty. The story of the quest to settle the Riemann Hypothesis is one of scientific exploration. It is peopled with solitary hermits and gregarious cheerleaders, cool calculators and wild-eyed visionaries, Nobel Prize-winners and Fields Medalists. To delve into the Riemann Hypothesis is to gain a window into the world of modern mathematics and the nature of mathematics research. Stalking the Riemann Hypothesis will open wide this window so that all may gaze through it in amazement.
The Distribution Of Prime Numbers
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Author : Albert Edward Ingham
language : en
Publisher: Cambridge University Press
Release Date : 1990-09-28
The Distribution Of Prime Numbers written by Albert Edward Ingham and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990-09-28 with Mathematics categories.
Originally published in 1934, this volume presents the theory of the distribution of the prime numbers in the series of natural numbers. Despite being long out of print, it remains unsurpassed as an introduction to the field.