A Survey of Algebraic Exponential Sums and Some Applications
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On Moments of Twisted L-Functions
ARIZONA WINTER SCHOOL 2016: ANALYTIC THEORY OF TRACE FUNCTIONS PHILIPPE MICHEL, EPF LAUSANNE 1. Functions on congruence rings As was emphasized by C.F. Gauss in the Disquisitiones Arithmeticae (and probably much earlier), the concept of congruences (a.k.a arithmetic progressions of some modulus q, a.k.a the ring Z=qZ) is central in number theory. It is therefore fundamental to understand spaces of functions over the ring Z=qZ. One of the first classes of such functions to have been studied systematically (and one of the most interesting even today) is the group of Dirichlet's characters of modulus q, (Z\=qZ)×, the group homomorphisms χ :(Z=qZ)× ! C×: Indeed these functions enter crucially in Dirichlet's proof that there are infinitely many primes in any admissible arithmetic progression of modulus q [7]. The studying of integral solutions of diophantine equations is another important source of func- tions on congruence rings: these arise as a manifestation of some general local-global principles or more directly when using the circle method. For instance, while investigating the set of representa- tions of an integer d by some integral, positive, definite, quadratic form in four variables q(x; y; z; t), ie. the integral solutions to the equation (1.1) d = q(x; y; z; t); x; y; z; t 2 Z; Kloosterman came up with the very important function modulo q, the so-called "Kloosterman sums", 1 X m (mod q) 7! Kl (m; q) := p e (mx + y); 2 q q xy=1 (mod q) where eq(·) denote the additive character of (Z=qZ; +) 2πix e (x) := exp( ): q q By evaluating the fourth moment of this function over Z=qZ, Kloosterman established the following non-trivial bound 2=3−1=2+o(1) (1.2) Kl2(m; q) q and deduced from it an asymptotic formula for the number of solutions to the equations (1.1). -
Exponential Sums and the Distribution of Prime Numbers
CORE Metadata, citation and similar papers at core.ac.uk Provided by Helsingin yliopiston digitaalinen arkisto Exponential Sums and the Distribution of Prime Numbers Jori Merikoski HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI Tiedekunta/Osasto Fakultet/Sektion Faculty Laitos Institution Department Faculty of Science Department of Mathematics and Statistics Tekijä Författare Author Jori Merikoski Työn nimi Arbetets titel Title Exponential Sums and the Distribution of Prime Numbers Oppiaine Läroämne Subject Mathematics Työn laji Arbetets art Level Aika Datum Month and year Sivumäärä Sidoantal Number of pages Master's thesis February 2016 102 p. Tiivistelmä Referat Abstract We study growth estimates for the Riemann zeta function on the critical strip and their implications to the distribution of prime numbers. In particular, we use the growth estimates to prove the Hoheisel-Ingham Theorem, which gives an upper bound for the dierence between consecutive prime numbers. We also investigate the distribution of prime pairs, in connection which we oer original ideas. The Riemann zeta function is dened as s in the half-plane Re We extend ζ(s) := n∞=1 n− s > 1. it to a meromorphic function on the whole plane with a simple pole at s = 1, and show that it P satises the functional equation. We discuss two methods, van der Corput's and Vinogradov's, to give upper bounds for the growth of the zeta function on the critical strip 0 Re s 1. Both of ≤ ≤ these are based on the observation that ζ(s) is well approximated on the critical strip by a nite exponential sum T s T Van der Corput's method uses the Poisson n=1 n− = n=1 exp s log n . -
Uniform Distribution of Sequences
Alt 1-4 'A I #so 4, I s4.1 L" - _.u or- ''Ifi r '. ,- I 'may,, i ! :' m UNIFORM DISTRIHUTION OF SEQUENCES L. KUIPERS Southern Illinois University H. NIEDERREITER Southern Illinois University A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS New York London Sydney Toronto Copyright © 1974, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language with- out the written permission of the publisher. Library of Congress Cataloging in Publication Data: Kuipers, Lauwerens. Uniform distribution of sequences. (Pure and applied mathematics) "A Wiley-Interscience publication." Bibliography: 1. Sequences (Mathematics) 1. Niederreiter, H., joint author. II. Title. QA292.K84 515'.242 73-20497 ISBN 0-471-51045-9 Printed in the United States of America 10-987654321 To Francina and Gerlinde REFACCE The theory of uniform distribution modulo one (Gleichverteilung modulo Eins, equirepartition modulo un) is concerned, at least in its classical setting, with the distribution of fractional parts of real numbers in the unit interval (0, 1). The development of this theory started with Hermann Weyl's celebrated paper of 1916 titled: "Uber die Gleichverteilung von Zahlen mod. Eins." Weyl's work was primarily intended as a refinement of Kronecker's approxi- mation theorem, and, therefore, in its initial stage, the theory was deeply rooted in diophantine approximations. During the last decades the theory has unfolded beyond that framework. Today, the subject presents itself as a meeting ground for topics as diverse as number theory, probability theory, functional analysis, topological algebra, and so on. -
Exponential Sums and Applications in Number Theory and Analysis
Faculty of Sciences Department of Mathematics Exponential sums and applications in number theory and analysis Frederik Broucke Promotor: Prof. dr. J. Vindas Master's thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Mathematics Academic year 2017{2018 ii Voorwoord Het oorspronkelijke idee voor deze thesis was om het bewijs van het ternaire vermoeden van Goldbach van Helfgott [16] te bestuderen. Al snel werd mij duidelijk dat dit een monumentale opdracht zou zijn, gezien de omvang van het bewijs (ruim 300 bladzij- den). Daarom besloot ik om in de plaats de basisprincipes van de Hardy-Littlewood- of cirkelmethode te bestuderen, de techniek die de ruggengraat vormt van het bewijs van Helfgott, en die een zeer belangrijke plaats inneemt in de additieve getaltheorie in het algemeen. Hiervoor heb ik gedurende het eerste semester enkele hoofdstukken van het boek \The Hardy-Littlewood method" van R.C. Vaughan [37] gelezen. Dit is waarschijnlijk de moeilijkste wiskundige tekst die ik tot nu toe gelezen heb; de weinige tussenstappen, het gebrek aan details, en zinnen als \one easily sees that" waren vaak frustrerend en demotiverend. Toch heb ik doorgezet, en achteraf gezien ben ik echt wel blij dat ik dat gedaan heb. Niet alleen heb ik enorm veel bijgeleerd over het onderwerp, ik heb ook het gevoel dat ik beter of vlotter ben geworden in het lezen van (moeilijke) wiskundige teksten in het algemeen. Na het lezen van dit boek gaf mijn promotor, professor Vindas, me de opdracht om de idee¨en en technieken van de cirkelmethode toe te passen in de studie van de functie van Riemann, een \pathologische" continue functie die een heel onregelmatig puntsgewijs gedrag vertoont. -
On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017
On Some Variants of the Gauss Circle Problem by David Lowry-Duda B.S. in Applied Mathematics, Georgia Institute of Technology, Atlanta, GA, 2011 B.S. in International Affairs and Modern Languages, Georgia Institute of Technology, Atlanta, GA, 2011 M.Sc. in Mathematics, Brown University, Providence, RI, 2015 A dissertation submitted in partial fulfillment of the arXiv:1704.02376v2 [math.NT] 2 May 2017 requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2017 c Copyright 2017 by David Lowry-Duda Abstract of “On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017 The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly accurate. This seemingly simple problem has prompted new ideas in many areas of number theory and mathematics, and it is now recognized as one instance of a general phenomenon. In this work, we describe two variants of the Gauss Circle problem that exhibit similar characteristics. The first variant concerns sums of Fourier coefficients of GL(2) cusp forms. These sums behave very similarly to the error term in the Gauss Circle problem. Normalized correctly, it is conjectured that the two satisfy essentially the same asymptotics. We introduce new Dirichlet series with coefficients that are squares of partial sums of Fourier coefficients of cusp forms. We study the meromorphic properties of these Dirichlet series and use these series to give new perspectives on the mean square of the size of sums of these Fourier coefficients. -
Estimating the Growth Rate of the Zeta-Function Using the Technique of Exponent Pairs
Estimating the Growth Rate of the Zeta-function Using the Technique of Exponent Pairs Shreejit Bandyopadhyay July 1, 2014 Abstract The Riemann-zeta function is of prime interest, not only in number theory, but also in many other areas of mathematics. One of the greatest challenges in the study of the zeta function is the understanding of its behaviour in the critical strip 0 < Re(s) < 1, where s = σ+it is a complex number. In this note, we first obtain a crude bound for the growth of the zeta function in the critical strip directly from its approximate functional equation and then show how the estimation of its growth reduces to a problem of evaluating an exponential sum. We then apply the technique of exponent pairs to get a much improved bound on the growth rate of the zeta function in the critical strip. 1 Introduction The Riemann zeta function ζ(s) with s = σ+it a complex number can be defined P 1 Q 1 −1 in either of the two following ways - ζ(s) = n ns or ζ(s) = p(1− ps ) , where in the first expression we sum over all natural numbers while in the second, we take the product over all primes. That these two expressions are equivalent fol- lows directly from the expression of n as a product of primes and the fundamen- tal theorem of arithmetic. It's easy to check that both the Dirichlet series, i.e., the first sum and the infinite product converge uniformly in any finite region with σ > 1. -
New Equidistribution Estimates of Zhang Type
NEW EQUIDISTRIBUTION ESTIMATES OF ZHANG TYPE D.H.J. POLYMATH Abstract. We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese 1 7 Remainder Theorem conditions, obtaining an exponent of distribution 2 ` 300 . Contents 1. Introduction 1 2. Preliminaries 8 3. Applying the Heath-Brown identity 14 4. One-dimensional exponential sums 23 5. Type I and Type II estimates 38 6. Trace functions and multidimensional exponential sum estimates 59 7. The Type III estimate 82 8. An improved Type I estimate 94 References 105 1. Introduction In May 2013, Y. Zhang [52] proved the existence of infinitely many pairs of primes with bounded gaps. In particular, he showed that there exists at least one h ¥ 2 such that the set tp prime | p ` h is primeu is infinite. (In fact, he showed this for some even h between 2 and 7 ˆ 107, although the precise value of h could not be extracted from his method.) Zhang's work started from the method of Goldston, Pintz and Yıldırım [23], who had earlier proved the bounded gap property, conditionally on distribution estimates arXiv:1402.0811v3 [math.NT] 3 Sep 2014 concerning primes in arithmetic progressions to large moduli, i.e., beyond the reach of the Bombieri{Vinogradov theorem. Based on work of Fouvry and Iwaniec [11, 12, 13, 14] and Bombieri, Friedlander and Iwaniec [3, 4, 5], distribution estimates going beyond the Bombieri{Vinogradov range for arithmetic functions such as the von Mangoldt function were already known. However, they involved restrictions concerning the residue classes which were incompatible with the method of Goldston, Pintz and Yıldırım. -
Exponential Sums and the Distribution of Prime Numbers
Exponential Sums and the Distribution of Prime Numbers Jori Merikoski HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI Tiedekunta/Osasto Fakultet/Sektion Faculty Laitos Institution Department Faculty of Science Department of Mathematics and Statistics Tekijä Författare Author Jori Merikoski Työn nimi Arbetets titel Title Exponential Sums and the Distribution of Prime Numbers Oppiaine Läroämne Subject Mathematics Työn laji Arbetets art Level Aika Datum Month and year Sivumäärä Sidoantal Number of pages Master's thesis February 2016 102 p. Tiivistelmä Referat Abstract We study growth estimates for the Riemann zeta function on the critical strip and their implications to the distribution of prime numbers. In particular, we use the growth estimates to prove the Hoheisel-Ingham Theorem, which gives an upper bound for the dierence between consecutive prime numbers. We also investigate the distribution of prime pairs, in connection which we oer original ideas. The Riemann zeta function is dened as s in the half-plane Re We extend ζ(s) := n∞=1 n− s > 1. it to a meromorphic function on the whole plane with a simple pole at s = 1, and show that it P satises the functional equation. We discuss two methods, van der Corput's and Vinogradov's, to give upper bounds for the growth of the zeta function on the critical strip 0 Re s 1. Both of ≤ ≤ these are based on the observation that ζ(s) is well approximated on the critical strip by a nite exponential sum T s T Van der Corput's method uses the Poisson n=1 n− = n=1 exp s log n . -
18.785 Notes
Contents 1 Introduction 4 1.1 What is an automorphic form? . 4 1.2 A rough definition of automorphic forms on Lie groups . 5 1.3 Specializing to G = SL(2; R)....................... 5 1.4 Goals for the course . 7 1.5 Recommended Reading . 7 2 Automorphic forms from elliptic functions 8 2.1 Elliptic Functions . 8 2.2 Constructing elliptic functions . 9 2.3 Examples of Automorphic Forms: Eisenstein Series . 14 2.4 The Fourier expansion of G2k ...................... 17 2.5 The j-function and elliptic curves . 19 3 The geometry of the upper half plane 19 3.1 The topological space ΓnH ........................ 20 3.2 Discrete subgroups of SL(2; R) ..................... 22 3.3 Arithmetic subgroups of SL(2; Q).................... 23 3.4 Linear fractional transformations . 24 3.5 Example: the structure of SL(2; Z)................... 27 3.6 Fundamental domains . 28 3.7 ΓnH∗ as a topological space . 31 3.8 ΓnH∗ as a Riemann surface . 34 3.9 A few basics about compact Riemann surfaces . 35 3.10 The genus of X(Γ) . 37 4 Automorphic Forms for Fuchsian Groups 40 4.1 A general definition of classical automorphic forms . 40 4.2 Dimensions of spaces of modular forms . 42 4.3 The Riemann-Roch theorem . 43 4.4 Proof of dimension formulas . 44 4.5 Modular forms as sections of line bundles . 46 4.6 Poincar´eSeries . 48 4.7 Fourier coefficients of Poincar´eseries . 50 4.8 The Hilbert space of cusp forms . 54 4.9 Basic estimates for Kloosterman sums . 56 4.10 The size of Fourier coefficients for general cusp forms . -
Number Theory Books Pdf Download
Number theory books pdf download Continue We apologise for any inconvenience caused. Your IP address was automatically blocked from accessing the Project Gutenberg website, www.gutenberg.org. This is due to the fact that the geoIP database shows that your address is in Germany. Diagnostic information: Blocked at germany.shtml Your IP address: 88.198.48.21 Referee Url (available): Browser: Mozilla/5.0 (Windows NT 6.1) AppleWebKit/537.36 (KHTML, as Gecko) Chrome/41.0.2228.0 Safari/537.36 Date: Thursday, 15-October-2020 19:55:54 GMT Why did this block happen? A court in Germany ruled that access to some items from the Gutenberg Project collection was blocked from Germany. The Gutenberg Project believes that the Court does not have jurisdiction over this matter, but until the matter is resolved, it will comply. For more information about the German court case, and the reason for blocking the entire Germany rather than individual items, visit the PGLAF information page about the German lawsuit. For more information on the legal advice the project Gutenberg has received on international issues, visit the PGLAF International Copyright Guide for project Gutenberg This page in German automated translation (via Google Translate): translate.google.com how can I get unlocked? All IP addresses in Germany are blocked. This unit will remain in place until the legal guidance changes. If your IP address is incorrect, use the Maxmind GeoIP demo to verify the status of your IP address. Project Gutenberg updates its list of IP addresses about monthly. Sometimes a website incorrectly applies a block from a previous visitor. -
Lectures on Mean Values of the Riemann Zeta Function
Lectures on Mean Values of The Riemann Zeta Function By A.Ivic Tata Institute of Fundamental Research Bombay 1991 Lectures on Mean Values of The Riemann Zeta Function By A. Ivic Published for the Tata Institute of Fundamental Research SPRINGER-VERLAG Berlin Heidelberg New York Tokyo Author A. Ivic S.Kovacevica, 40 Stan 41 Yu-11000, Beograd YUGOSLAVIA © Tata Institute of Fundamental Research, 1991 ISBN 3-350-54748-7-Springer-Verlag, Berlin. Heidelberg. New York. Tokyo ISBN 0-387-54748-7-Springer-Verlag, New York, Heidelberg. Berlin. Tokyo No part of this book may e reproduced in any form by print, microfilm or any other mans with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by Anamika Trading Company Navneet Bhavan, Bhavani Shankar Road Dadar, Bombay 400 028 and Published by H.Goetze, Springer-Verlag, Heidelberg, Germany PRINTED IN INDIA Preface These lectures were given at the Tata Institute of Fundamental Research in the summer of 1990. The specialized topic of mean values of the Riemann zeta-function permitted me to go into considerable depth. The central theme were the second and the fourth moment on the critical line, and recent results concerning these topic are extensively treated. In a sense this work is a continuation of my monograph [1], since except for the introductory Chapter 1, it starts where [1] ends. Most of the results in this text are unconditional, that is, they do not depend on unproved hypothesis like Riemann’s (all complex zeros of ζ(s) have real parts equal to 1 ) or Lindelof’s¨ (ζ( 1 + it) tǫ). -
Kloosterman Sums
On SL2(Z) and SL3(Z) Kloosterman sums Olga Balkanova Erasmus Mundus Master ALGANT University of Bordeaux 1 Supervisors Guillaume Ricotta and Jean-Marc Couveignes 2011 Acknowledgements I wish to thank my advisor Dr. Guillaume Ricotta for designing this project for me and making my work interesting. I am grateful for his encouragements and inspiring interest, for his knowledge and clear explanations, for our discussions and emails. It is really valuable experience to work under the guidance of advisor Guillaume. I thank my co-advisor Prof. Jean-Marc Couveignes for the help with computational part of my thesis, useful advices and fruitful ideas. I am also grateful to my russian Professor Maxim Vsemirnov for in- troducing me to the world of analytic number theory and providing interesting problems. Many thanks to my family, whose love always helps me to overcome all the difficulties. Finally, I thank my ALGANT friends Andrii, Katya, Liu, Martin, Novi, Nikola for all shared moments. Contents Contents ii List of Figures iii 1 Classical Kloosterman sums3 1.1 SL(2) modular forms . .3 1.2 Construction of SL(2) Poincar´eseries . .6 1.3 Fourier expansion of Poincar´eseries . .9 1.4 Some properties of Kloosterman sums . 13 1.5 Distribution of Kloosterman angles . 15 1.6 Numerical computation of Poincar´eseries . 21 1.6.1 Poincar´eseries and fundamental domain reduction algorithm 21 1.6.2 Absolute error estimate . 23 2 SL(3) Kloosterman sums 27 2.1 Generalized upper-half space and Iwasawa decomposition . 27 2.2 Automorphic forms and Fourier expansion . 31 2.3 SL(3) Poincar´eseries .