This is the first book on the theory of multiple zeta values since its birth around 1994. Readers will find that the shuffle products of multiple zeta values are applied to complicated counting problems in combinatorics, and numerous interesting identities are produced that are ready to be used. This will provide a powerful tool to deal with problems in multiple zeta values, both in evaluations and shuffle relations. The volume will benefit graduate students doing research in number theory.

K Venkatachaliengar, Shaun Cooper, K Venkatachaliengar, and Shaun Cooper

Subjects

Elliptic functions

Abstract

This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter.The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan's work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

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.

Helmut Maier, Laszlo Toth, Michael Th Rassias, Helmut Maier, Laszlo Toth, and Michael Th Rassias

Subjects

Number theory, Riemann hypothesis, Trigonometric sums, and Mathematical analysis

Abstract

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.

The guiding principle in this monograph is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called'Gauss-Manin connection in disguise'.

Bernoulli numbers are often found in mathematical analysis, number theory, combinatorics, and other areas of mathematics. In some monographs on number theory there are separate chapters devoted only to Bernoulli numbers and their properties. Algorithms for calculating Bernoulli numbers are built into all popular mathematical packages: Mathematica, Matlab, Magma, Pari GP, etc. In this paper, we propose an algorithm for calculating Bernoulli numbers, which is faster than the known ones. The essence of our approach is to improve the Harvey multimodular method due to making sparse of large calculated sums, when we express the sums in half the interval in terms of sums in the intervals of length 1/12 or even 1/15 of the length of the summation interval. It is proved in the paper that such a reduction in the summation intervals can be achieved for the vast majority of prime numbers. At the same time, inconvenient prime numbers (and there are them no more than 0.01% for large values of n) can be simply excluded from the list those by modulo of which the current Bernoulli number is calculated. The algorithm of fast calculation proposed by us in the article (acceleration is more than three times as compared with the Harvey algorithm) of Bernoulli numbers modulo prime numbers can also be successfully used to find irregular prime numbers, irregular pairs (p, n), and also when calculating Iwasawa invariants. When calculating the irregular pairs (p, n) and Iwasawa invariants, the algorithm presented in our work is significantly (more than 10 times) more efficient.