Value-Distribution of L-Functions
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Jörn Steuding
Value-Distribution of L-Functions
1877
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1877
Jörn Steuding
Value-Distribution of L-Functions
ABC
Author Jörn Steuding Department of Mathematics Chair of Complex Analysis University of Würzburg Am Hubland 97074 Würzburg Germany e-mail: [email protected]
Library of Congress Control Number: 2007921875 Mathematics Subject Classification (2000): 11M06, 11M26, 11M41, 30D35, 30E10, 60B05, 60F05 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-26526-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26526-9 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-44822-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author and SPi using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
SPIN: 11504160
VA41/3100/SPi
543210
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Riemann Zeta-Function and the Distribution of Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bohr’s Probabilistic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Voronin’s Universality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dirichlet L-Functions and Joint Universality . . . . . . . . . . . . . . . . 1.5 L-Functions Associated with Newforms . . . . . . . . . . . . . . . . . . . . . 1.6 The Linnik–Ibragimov Conjecture . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 9 12 19 24 28
2
Dirichlet Series and Polynomial Euler Products . . . . . . . . . . . . 2.1 General Theory of Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Class of Dirichlet Series: The Main Actors . . . . . . . . . . . . . . . . 2.3 Estimates for the Dirichlet Series Coefficients . . . . . . . . . . . . . . . 2.4 The Mean-
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