Download or read book Value Distribution of L Functions written by Jr̲n Steuding and published by Springer Science & Business Media. This book was released on 2007-06-06 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.
Download or read book Value Distribution Theory written by L. Sario and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this research monograph is to build up a modern value distribution theory for complex analytic mappings between abstract Riemann surfaces. All results presented herein are new in that, apart from the classical background material in the last chapter, there is no over lapping with any existing monograph on merom orphic functions. Broadly speaking the division of the book is as follows: The Introduction and Chapters I to III deal mainly with the theory of mappings of arbitrary Riemann surfaces as developed by the first named author; Chapter IV, due to Nakai, is devoted to meromorphic functions on parabolic surfaces; Chapter V contains Matsumoto's results on Picard sets; Chapter VI, pre dominantly due to the second named author, presents the so-called nonintegrated forms of the main theorems and includes some joint work by both authors. For a complete list of writers whose results have been discussed we refer to the Author Index.
Download or read book Advanced Analytic Number Theory L Functions written by Carlos J. Moreno and published by American Mathematical Soc.. This book was released on 2005 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the pioneering work of Euler, Dirichlet, and Riemann, the analytic properties of L-functions have been used to study the distribution of prime numbers. With the advent of the Langlands Program, L-functions have assumed a greater role in the study of the interplay between Diophantine questions about primes and representation theoretic properties of Galois representations. This book provides a complete introduction to the most significant class of L-functions: the Artin-Hecke L-functions associated to finite-dimensional representations of Weil groups and to automorphic L-functions of principal type on the general linear group. In addition to establishing functional equations, growth estimates, and non-vanishing theorems, a thorough presentation of the explicit formulas of Riemann type in the context of Artin-Hecke and automorphic L-functions is also given. The survey is aimed at mathematicians and graduate students who want to learn about the modern analytic theory of L-functions and their applications in number theory and in the theory of automorphic representations. The requirements for a profitable study of this monograph are a knowledge of basic number theory and the rudiments of abstract harmonic analysis on locally compact abelian groups.
Download or read book Nevanlinna s Theory of Value Distribution written by William Cherry and published by Springer Science & Business Media. This book was released on 2001-04-24 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution as well as a valuable reference for research specialists. Authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its number theoretic digressions These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation.
Download or read book VALUE DISTRIBUTION OF AUTOMORPHIC L FUNCTIONS written by Krzysztof Pawelec and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Significant attention has been given to study various moments of the Riemann zeta function, $\zeta$, its logarithm and their generalizations However, not much is known about the moments of $\frac{\zeta'}{\zeta}$. and the logarithmic derivative of more general L-functions. For $\pi$, a cuspidal automorphic representation of $GL_d( \mathbb{A}_{\mathbb{Q}})$, there is an associated L-function, $L(s, \pi)$. We study the value distribution of its logarithmic derivative on the 1-line, $\frac{L'}{L}(1+it, \pi).$ We are able to prove that for $t \in [T, 2T]$, in some sense, $\frac{L'}{L}(1+it, \pi)$ has ``almost'' normal distribution with mean 0 and variance $\sqrt{\frac{\log(y(T))}{2y(T)}}$. An essential ingredient of the proof is the fact that our function of interest can be approximated by Dirichlet polynomial with coefficients supported on prime powers. We prove similar results for $\frac{L'}{L}(1+it, \pi \times \overline{\pi})$ and $\log(L(1+it, \pi))$.
Download or read book Non vanishing of L Functions and Applications written by Ram M. Murty and published by Birkhäuser. This book was released on 2013-11-09 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph brings together a collection of results on the non-vanishing of L functions. The presentation, though based largely on the original papers, is suitable for independent study. A number of exercises have also been provided to aid in this endeavour. The exercises are of varying difficulty and those which require more effort have been marked with an asterisk. The authors would like to thank the Institut d'Estudis Catalans for their encouragement of this work through the Ferran Sunyer i Balaguer Prize. We would also like to thank the Institute for Advanced Study, Princeton for the excellent conditions which made this work possible, as well as NSERC, NSF and FCAR for funding. Princeton M. Ram Murty August, 1996 V. Kumar Murty Introduction Since the time of Dirichlet and Riemann, the analytic properties of L-functions have been used to establish theorems of a purely arithmetic nature. The distri bution of prime numbers in arithmetic progressions is intimately connected with non-vanishing properties of various L-functions. With the subsequent advent of the Tauberian theory as developed by Wiener and Ikehara, these arithmetical the orems have been shown to be equivalent to the non-vanishing of these L-functions on the line Re(s) = 1. In the 1950's, a new theme was introduced by Birch and Swinnerton-Dyer.
Download or read book Analytic Properties of Automorphic L Functions written by Stephen Gelbart and published by Academic Press. This book was released on 2014-07-14 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analytic Properties of Automorphic L-Functions is a three-chapter text that covers considerable research works on the automorphic L-functions attached by Langlands to reductive algebraic groups. Chapter I focuses on the analysis of Jacquet-Langlands methods and the Einstein series and Langlands’ so-called “Euler products . This chapter explains how local and global zeta-integrals are used to prove the analytic continuation and functional equations of the automorphic L-functions attached to GL(2). Chapter II deals with the developments and refinements of the zeta-inetgrals for GL(n). Chapter III describes the results for the L-functions L (s, ?, r), which are considered in the constant terms of Einstein series for some quasisplit reductive group. This book will be of value to undergraduate and graduate mathematics students.
Download or read book The Lindelof Class of L functions written by Anup Biswanath Dixit and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Meromorphic functions, called L-functions, play a vital role in number theory. In 1989, Selberg defined a class of L-functions that serves as an axiomatic model for L-functions arising from geometry and arithmetic. Even though the Selberg class successfully captures many characteristics common to most L-functions, it fails to be closed under addition. This creates obstructions, in particular, not allowing us to interpolate between L-functions. To overcome this limitation, V. K. Murty defined a general class of L-functions based on their growth rather than functional equation and Euler product. This class, which is called the Lindelof class of L-functions, is endowed with the structure of a ring. In this thesis, we study further properties of this class, specifically, its ring structure and topological structure. We also study the zero distribution and the a-value distribution of elements in this class and prove certain uniqueness results, showing that distinct elements cannot share complex values and L-functions in this class cannot share two distinct values with any other meromorphic function. We also establish the value distribution theory for this class with respect to the universality property, which states that every holomorphic function is approximated infinitely often by vertical shifts of an L-function. In this context, we precisely formulate and give some evidence towards the Linnik-Ibragimov conjecture.
Download or read book Distribution Theorems of L functions written by David Joyner and published by Longman Scientific and Technical. This book was released on 1986 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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
Download or read book Upper Bounds and Moments of L functions written by Vorrapan Chandee and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: L-functions are some of the most studied objects in number theory. Although many crucial properties of L-functions remain mysterious, central conjectures such as the generalized Riemann hypothesis (GRH). This thesis concerns properties of L-functions. In particular, we focus on studying upper bounds and moments of $L$-functions. Assuming GRH, we give effective explicit upper bounds for L-functions on the critical line and apply these bounds to determine what numbers are represented by a given ternary quadratic form. Moreover the best known version of the Lindelof hypothesis from the Riemann hypothesis (RH) is also derived. Another important way of understanding LH is through moments of L-functions. Information about moments sheds light on the distribution of values of \zeta(1/2 + it). We try to understand the joint distribution of quantities like \zeta(1/2 + it) and \zeta(1/2 + it + i). To study these we consider "shifted moments" of the zeta function and obtain good upper and lower estimates for such moments.
Download or read book The Conference On L functions written by Lin Weng and published by World Scientific. This book was released on 2006-12-29 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: This invaluable volume collects papers written by many of the world's top experts on L-functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole.The contributions reflect the latest, most advanced and most important aspects of L-functions. In particular, it contains Hida's lecture notes at the conference and at the Eigenvariety semester in Harvard University and Weng's detailed account of his works on high rank zeta functions and non-abelian L-functions./a
Download or read book Number Theory Plowing And Starring Through High Wave Forms Proceedings Of The 7th China japan Seminar written by Shigeru Kanemitsu and published by World Scientific. This book was released on 2015-02-10 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the successful 7th China-Japan seminar on number theory conducted in Kyushu University, this volume is a compilation of survey and semi-survey type of papers by the participants of the seminar. The topics covered range from traditional analytic number theory to elliptic curves and universality. This volume contains new developments in the field of number theory from recent years and it provides suitable problems for possible new research at a level which is not unattainable. Timely surveys will be beneficial to a new generation of researchers as a source of information and these provide a glimpse at the state-of-the-art affairs in the fields of their research interests.
Download or read book The Behaviour of L functions at the Edge of the Critical Strip and Applications written by Xiannan Li and published by Stanford University. This book was released on 2011 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: A large number of problems in number theory can be reduced to statements about L-functions. In this thesis, we study L-functions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest.
Download or read book A Study of L Functions written by Allysa Lumley and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In analytic number theory, and increasingly in other surprising places, L-functions arise naturally when describing algebraic and geometric phenomena. For example, when attempting to prove the Prime Number Theorem the values of L-functions on the one-line played a crucial role. In this thesis we discuss the theory of L-functions in two different settings. In the classical context we provide results which give estimates for the size of a general L-function on the right edge of the critical strip, that is complex numbers with real part one. We also provide a bound for the number of zeros for the classical Riemann zeta function inside the critical strip commonly referred to as a zero density estimate. In the second setting we study L-functions over the polynomial ring A, which is all polynomials with coefficients in a finite field of size q. As A and the ring of integers have similar structure, A is a natural candidate for analyzing classical number theoretic questions. Additionally, the truth of the Riemann Hypothesis (RH) in A yields deeper unconditional results currently unattainable over the integers. We will focus on the distribution of values of specific L-functions in two different places: On the right edge of the critical strip, that is complex numbers with real part one, and inside of the critical strip, meaning the complex numbers will have real part between one half and one.
Download or read book Value Distribution of Meromorphic Functions written by Anatoliĭ Asirovich Golʹdberg and published by American Mathematical Soc.. This book was released on 2008 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book contains a comprehensive exposition of the Nevanlinna theory of meromorphic functions of one complex variable, with detailed study of deficiencies, value distribution, and asymptotic properties of meromorphic functions." "The main body of the book is a translation of the Russian original published in 1970, which has been one of the most popular sources in this field since then. New references and footnotes related to recent achievements in the topics considered in the original edition have been added and a few corrections made. A new Appendix with a survey of the results obtained after 1970 and extensive bibliography has been written by Alexandre Ermenko and James K. Langley for this English edition." "The only prerequisite for understanding material of this book is an undergraduate course in the theory of functions of one complex variable."--BOOK JACKET.
Download or read book Analytic Number Theory Approximation Theory and Special Functions written by Gradimir V. Milovanović and published by Springer. This book was released on 2014-07-08 with total page 873 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, in honor of Hari M. Srivastava, discusses essential developments in mathematical research in a variety of problems. It contains thirty-five articles, written by eminent scientists from the international mathematical community, including both research and survey works. Subjects covered include analytic number theory, combinatorics, special sequences of numbers and polynomials, analytic inequalities and applications, approximation of functions and quadratures, orthogonality and special and complex functions. The mathematical results and open problems discussed in this book are presented in a simple and self-contained manner. The book contains an overview of old and new results, methods, and theories toward the solution of longstanding problems in a wide scientific field, as well as new results in rapidly progressing areas of research. The book will be useful for researchers and graduate students in the fields of mathematics, physics and other computational and applied sciences.
