Irregularities in the Distribution of Prime Numbers From the Era of
This volume presents research and expository papers highlighting the vibrant and fascinating study of irregularities in the distribution of primes. Written by an international group of experts, contributions present a self-contained yet unified explo
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gularities in the Distribution of Prime Numbers From the Era of Helmut Maier's Matrix Method and Beyond
Irregularities in the Distribution of Prime Numbers
János Pintz • Michael Th. Rassias Editors
Irregularities in the Distribution of Prime Numbers From the Era of Helmut Maier’s Matrix Method and Beyond
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Editors János Pintz Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest, Hungary
Michael Th. Rassias Institute of Mathematics University of Zürich Zürich, Switzerland Moscow Institute of Physics and Technology Dolgoprudny, Russia Institute for Advanced Study Program in Interdisciplinary Studies Princeton, NJ, USA
ISBN 978-3-319-92776-3 ISBN 978-3-319-92777-0 (eBook) https://doi.org/10.1007/978-3-319-92777-0 Library of Congress Control Number: 2018948355 Mathematics Subject Classification (2010): 11-XX, 05-XX © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Foreword
Although prime numbers have been of interest to mathematicians since ancient times, reasonably precise rules on their distribution were found relatively late. The first known successful guess was accomplished by Gauss who announced in 1849 that he had come to the conclusion that “at around x, the primes occur with density 1/ log x” [10]. He concluded that π(x) could be approximated by
x
Li(x) := 2
x x dt = + +O log t log x log2 x
x log3 x
.
A sharp version of this result, the Prime Number Theorem with error term, could be proven by Hadamard [11] and de le Vallée-Poussin [2] in 1896. Cramér in 1936 interpreted Gauss’ statement in terms of probability theory:
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