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

Author : Helmut Maier
Publisher : World Scientific
ISBN 13 : 9811246904
Total Pages : 165 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Recent Progress On Topics Of Ramanujan Sums And Cotangent Sums Associated With The Riemann Hypothesis by : Helmut Maier

Download or read book Recent Progress On Topics Of Ramanujan Sums And Cotangent Sums Associated With The Riemann Hypothesis written by Helmut Maier and published by World Scientific. This book was released on 2021-12-28 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 And Combinatorial Number Theory: The Legacy Of Ramanujan - Contributions In Honor Of Bruce C. Berndt

Author : George E Andrews
Publisher : World Scientific
ISBN 13 : 9811277389
Total Pages : 704 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Analytic And Combinatorial Number Theory: The Legacy Of Ramanujan - Contributions In Honor Of Bruce C. Berndt by : George E Andrews

Download or read book Analytic And Combinatorial Number Theory: The Legacy Of Ramanujan - Contributions In Honor Of Bruce C. Berndt written by George E Andrews and published by World Scientific. This book was released on 2024-08-19 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume reflects the contributions stemming from the conference Analytic and Combinatorial Number Theory: The Legacy of Ramanujan which took place at the University of Illinois at Urbana-Champaign on June 6-9, 2019. The conference included 26 plenary talks, 71 contributed talks, and 170 participants. As was the case for the conference, this book is in honor of Bruce C Berndt and in celebration of his mathematics and his 80th birthday.Along with a number of papers previously appearing in Special Issues of the International Journal of Number Theory, the book collects together a few more papers, a biography of Bruce by Atul Dixit and Ae Ja Yee, a preface by George Andrews, a gallery of photos from the conference, a number of speeches from the conference banquet, the conference poster, a list of Bruce's publications at the time this volume was created, and a list of the talks from the conference.

Analytic Methods In Number Theory: When Complex Numbers Count

Author : Wadim Zudilin
Publisher : World Scientific
ISBN 13 : 9811279330
Total Pages : 192 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Analytic Methods In Number Theory: When Complex Numbers Count by : Wadim Zudilin

Download or read book Analytic Methods In Number Theory: When Complex Numbers Count written by Wadim Zudilin and published by World Scientific. This book was released on 2023-08-22 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author : Peter B. Borwein
Publisher : Springer Science & Business Media
ISBN 13 : 0387721258
Total Pages : 543 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis The Riemann Hypothesis by : Peter B. Borwein

Download or read book The Riemann Hypothesis written by Peter B. Borwein and published by Springer Science & Business Media. This book was released on 2008 with total page 543 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Riemann Zeta-Function

Author : Anatoly A. Karatsuba
Publisher : Walter de Gruyter
ISBN 13 : 3110886146
Total Pages : 409 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis The Riemann Zeta-Function by : Anatoly A. Karatsuba

Download or read book The Riemann Zeta-Function written by Anatoly A. Karatsuba and published by Walter de Gruyter. This book was released on 2011-05-03 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author : Jeffrey Stopple
Publisher : Cambridge University Press
ISBN 13 : 9780521012539
Total Pages : 404 pages
Book Rating : 4.0/5 (125 download)

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Book Synopsis A Primer of Analytic Number Theory by : Jeffrey Stopple

Download or read book A Primer of Analytic Number Theory written by Jeffrey Stopple and published by Cambridge University Press. This book was released on 2003-06-23 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: An undergraduate-level 2003 introduction whose only prerequisite is a standard calculus course.

The Distribution of Prime Numbers

Author : Albert Edward Ingham
Publisher : Cambridge University Press
ISBN 13 : 9780521397896
Total Pages : 140 pages
Book Rating : 4.3/5 (978 download)

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Book Synopsis The Distribution of Prime Numbers by : Albert Edward Ingham

Download or read book The Distribution of Prime Numbers written by Albert Edward Ingham and published by Cambridge University Press. This book was released on 1990-09-28 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

CRC Standard Mathematical Tables and Formulae, 32nd Edition

Author : Daniel Zwillinger
Publisher : CRC Press
ISBN 13 : 1439835500
Total Pages : 792 pages
Book Rating : 4.4/5 (398 download)

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Book Synopsis CRC Standard Mathematical Tables and Formulae, 32nd Edition by : Daniel Zwillinger

Download or read book CRC Standard Mathematical Tables and Formulae, 32nd Edition written by Daniel Zwillinger and published by CRC Press. This book was released on 2011-06-22 with total page 792 pages. Available in PDF, EPUB and Kindle. Book excerpt: With over 6,000 entries, CRC Standard Mathematical Tables and Formulae, 32nd Edition continues to provide essential formulas, tables, figures, and descriptions, including many diagrams, group tables, and integrals not available online. This new edition incorporates important topics that are unfamiliar to some readers, such as visual proofs and sequences, and illustrates how mathematical information is interpreted. Material is presented in a multisectional format, with each section containing a valuable collection of fundamental tabular and expository reference material. New to the 32nd Edition A new chapter on Mathematical Formulae from the Sciences that contains the most important formulae from a variety of fields, including acoustics, astrophysics, epidemiology, finance, statistical mechanics, and thermodynamics New material on contingency tables, estimators, process capability, runs test, and sample sizes New material on cellular automata, knot theory, music, quaternions, and rational trigonometry Updated and more streamlined tables Retaining the successful format of previous editions, this comprehensive handbook remains an invaluable reference for professionals and students in mathematical and scientific fields.

数论导引

Author :
Publisher :
ISBN 13 : 9787115156112
Total Pages : 435 pages
Book Rating : 4.1/5 (561 download)

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Book Synopsis 数论导引 by :

Download or read book 数论导引 written by and published by . This book was released on 2007 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: 本书内容包括素数、无理数、同余、费马定理、连分数、不定方程、二次域、算术函数、分化等。

The Riemann Zeta-Function

Author : Aleksandar Ivic
Publisher : Courier Corporation
ISBN 13 : 0486140040
Total Pages : 548 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis The Riemann Zeta-Function by : Aleksandar Ivic

Download or read book The Riemann Zeta-Function written by Aleksandar Ivic and published by Courier Corporation. This book was released on 2012-07-12 with total page 548 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text covers exponential integrals and sums, 4th power moment, zero-free region, mean value estimates over short intervals, higher power moments, omega results, zeros on the critical line, zero-density estimates, and more. 1985 edition.

Analytic Combinatorics in Several Variables

Author : Robin Pemantle
Publisher : Cambridge University Press
ISBN 13 : 1107031575
Total Pages : 395 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Analytic Combinatorics in Several Variables by : Robin Pemantle

Download or read book Analytic Combinatorics in Several Variables written by Robin Pemantle and published by Cambridge University Press. This book was released on 2013-05-31 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed at graduate students and researchers in enumerative combinatorics, this book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective.

Computing the Continuous Discretely

Author : Matthias Beck
Publisher : Springer
ISBN 13 : 1493929690
Total Pages : 295 pages
Book Rating : 4.4/5 (939 download)

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Book Synopsis Computing the Continuous Discretely by : Matthias Beck

Download or read book Computing the Continuous Discretely written by Matthias Beck and published by Springer. This book was released on 2015-11-14 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device. The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more. With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume? Reviews of the first edition: “You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.” — MAA Reviews “The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate rial, exercises, open problems and an extensive bibliography.” — Zentralblatt MATH “This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.” — Mathematical Reviews “Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.” — CHOICE

Summability Calculus

Author : Ibrahim M. Alabdulmohsin
Publisher : Springer
ISBN 13 : 3319746480
Total Pages : 168 pages
Book Rating : 4.3/5 (197 download)

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Book Synopsis Summability Calculus by : Ibrahim M. Alabdulmohsin

Download or read book Summability Calculus written by Ibrahim M. Alabdulmohsin and published by Springer. This book was released on 2018-03-07 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Introduction to Real Analysis

Author : William F. Trench
Publisher : Prentice Hall
ISBN 13 : 9780130457868
Total Pages : 0 pages
Book Rating : 4.4/5 (578 download)

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Book Synopsis Introduction to Real Analysis by : William F. Trench

Download or read book Introduction to Real Analysis written by William F. Trench and published by Prentice Hall. This book was released on 2003 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Using an extremely clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. The real number system. Differential calculus of functions of one variable. Riemann integral functions of one variable. Integral calculus of real-valued functions. Metric Spaces. For those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.

An Introduction to the Theory of the Riemann Zeta-Function

Author : S. J. Patterson
Publisher : Cambridge University Press
ISBN 13 : 9780521499057
Total Pages : 176 pages
Book Rating : 4.4/5 (99 download)

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Book Synopsis An Introduction to the Theory of the Riemann Zeta-Function by : S. J. Patterson

Download or read book An Introduction to the Theory of the Riemann Zeta-Function written by S. J. Patterson and published by Cambridge University Press. This book was released on 1995-02-02 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the analytic techniques used in the investigation of zeta functions through the example of the Riemann zeta function. It emphasizes central ideas of broad application, avoiding technical results and the customary function-theoretic appro

Analytic Number Theory For Undergraduates

Author : Heng Huat Chan
Publisher : World Scientific Publishing Company
ISBN 13 : 9814365270
Total Pages : 125 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Analytic Number Theory For Undergraduates by : Heng Huat Chan

Download or read book Analytic Number Theory For Undergraduates written by Heng Huat Chan and published by World Scientific Publishing Company. This book was released on 2009-04-21 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Discrete Mathematics and Applications

Author : Andrei M. Raigorodskii
Publisher : Springer Nature
ISBN 13 : 3030558576
Total Pages : 499 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Discrete Mathematics and Applications by : Andrei M. Raigorodskii

Download or read book Discrete Mathematics and Applications written by Andrei M. Raigorodskii and published by Springer Nature. This book was released on 2020-11-21 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: Advances in discrete mathematics are presented in this book with applications in theoretical mathematics and interdisciplinary research. Each chapter presents new methods and techniques by leading experts. Unifying interdisciplinary applications, problems, and approaches of discrete mathematics, this book connects topics in graph theory, combinatorics, number theory, cryptography, dynamical systems, finance, optimization, and game theory. Graduate students and researchers in optimization, mathematics, computer science, economics, and physics will find the wide range of interdisciplinary topics, methods, and applications covered in this book engaging and useful.