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Exploring the Riemann Zeta Function Hugh Montgomery • Ashkan Nikeghbali Michael Th Exploring the Riemann Zeta Function Hugh Montgomery • Ashkan Nikeghbali Michael Th. Rassias Editors Exploring the Riemann Zeta Function 190 years from Riemann’s Birth With a preface by Freeman J. Dyson 123 Editors Hugh Montgomery Ashkan Nikeghbali Department of Mathematics Institut für Mathematik University of Michigan Universität Zürich Ann Arbor, MI, USA Zürich, Switzerland Michael Th. Rassias Institut für Mathematik Universität Zürich Zürich, Switzerland ISBN 978-3-319-59968-7 ISBN 978-3-319-59969-4 (eBook) DOI 10.1007/978-3-319-59969-4 Library of Congress Control Number: 2017947901 Mathematics Subject Classification (2010): 11-XX, 30-XX, 33-XX, 41-XX, 43-XX, 60-XX © Springer International Publishing AG 2017 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 Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface by Freeman J. Dyson: Quasi-Crystals and the Riemann Hypothesis On the occasion of the 190 years from Riemann’s birth, which led to the conception of this book, I am presenting a challenge to young mathematicians to study quasicrystals and use them to prove the Riemann Hypothesis. Among the readers, those who are experts in number theory may consider the challenge frivolous. Those who are not experts may consider it uninteresting or unintelligible. Nevertheless, I am putting it forward for your serious consideration. One should always be guided by the wisdom of the physicist Leo Szilard, who wrote his own Ten Commandments to stand beside those of Moses. Szilard’s second commandment says, “Let your acts be directed toward a worthy goal, but do not ask if they will reach it; they are to be models and examples, not means to an end.” The Riemann Hypothesis is a worthy goal, and it is not for us to ask whether we can reach it. I will give you some hints describing how it might be achieved. Here, I will be giving voice to the mathematician that I was almost 70 years ago before I became a physicist. There were until recently two supreme unsolved problems in the world of pure mathematics, the proof of Fermat’s Last Theorem and the proof of the Riemann Hypothesis. In 1998, my former Princeton colleague Andrew Wiles polished off Fermat’s Last Theorem, and only the Riemann Hypothesis remains. Wiles’ proof of the Fermat Theorem was not just a technical stunt. It required the discovery and exploration of a new field of mathematical ideas, far wider and more consequential than the Fermat Theorem itself. It is likely that any proof of the Riemann Hypothesis will likewise lead to a deeper understanding of many diverse areas of mathematics and perhaps of physics too. The Riemann Hypothesis says that one specific function, the zeta-function that Riemann named, has all its complex zeros upon a certain line. Riemann’s zeta-function and other zeta-functions similar to it appear ubiquitously in number theory, in the theory of dynamical systems, in geometry, in function theory, and in physics. The zeta-function stands at a junction where paths lead in many directions. A proof of the hypothesis will illuminate all the connections. Like every serious student of pure mathematics, when I was young, I had dreams of proving the Riemann Hypothesis. I had some vague ideas that I thought might lead to a proof, but never pursued them vigorously. In recent years, after the discovery v vi Preface by Freeman J. Dyson: Quasi-Crystals and the Riemann Hypothesis of quasicrystals, my ideas became a little less vague. I offer them here for the consideration of any young mathematician who has ambitions to win a Fields Medal. Now, I jump from the Riemann Hypothesis to quasicrystals. Quasicrystals were one of the great unexpected discoveries of recent years. They were discovered in two ways, as real physical objects formed when alloys of aluminum and manganese are solidified rapidly from the molten state and as abstract mathematical structures in Euclidean geometry. They were unexpected, both in physics and in mathematics. I am here discussing only their mathematical properties. Quasicrystals can exist in spaces of one, two, or three dimensions. From the point of view of physics, the three-dimensional quasicrystals are the most interesting, since they inhabit our three-dimensional world and can be studied experimentally. From the point of view of a mathematician, one-dimensional quasicrystals are more interesting than two-dimensional quasicrystals because they exist in far greater variety, and the two-dimensional are more interesting than the three-dimensional for the same reason. The mathematical definition of a quasicrystal is as follows. A quasicrystal is a distribution of discrete point masses whose Fourier transform is a distribution of discrete point frequencies. Or to say it more briefly, a quasicrystal is a pure point distribution that has a pure point spectrum. This definition includes as a special case the ordinary crystals which are exactly periodic point distributions with exactly periodic point spectra. Excluding the ordinary crystals, quasicrystals in three dimensions come in very limited variety, all of them being associated with the icosahedral rotation group. The two-dimensional quasicrystals are more numerous, roughly one distinct type associated with each regular polygon in a plane. The two-dimensional quasicrystal with pentagonal symmetry is the famous Penrose tiling of the plane, discovered by Penrose before the general concept of quasicrystal was invented. Finally, the one-dimensional quasicrystals have a far richer structure since they are not tied to any rotational symmetries. So far as I know, no complete enumeration of one- dimensional quasicrystals exists. It is known that a unique quasi-crystal exists corresponding to every Pisot-Vijayaraghavan number or PV number. A PV number is a real algebraic integer, a root of a polynomial equation with integer coefficients, such that all the other roots have absolute value less than one (see [1]). The set of all PV numbers is infinite and has a remarkable topological structure. The set of all one-dimensional quasicrystals has a structure at least as rich as the set of all PV numbers and probably much richer. We do not know for sure, but it is likely that a huge universe of one-dimensional quasicrystals not associated with PV numbers is waiting to be discovered. Here comes the punch line of my argument, the essential point linking one- dimensional quasicrystals with the Riemann Hypothesis. If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime-power numbers. Andrew Odlyzko (see [2]) has published a beautiful computer calculation of the Fourier transform Preface by Freeman J. Dyson: Quasi-Crystals and the Riemann Hypothesis vii of the zeta-function zeros. The calculation shows precisely the expected structure of the Fourier transform, with a sharp discontinuity at every logarithm of a prime- power number and nowhere else. My proposal is the following. Let us pretend that we do not know that the Riemann Hypothesis is true. Let us tackle the problem from the other end. Let us try to obtain a complete enumeration and classification of all one-dimensional quasicrystals. That is to say, we enumerate and classify all point distributions that have a discrete point spectrum. We shall then find the well-known quasicrystals associated with PV numbers and also a whole slew of other quasicrystals, known and unknown. Among the slew of other quasicrystals, we should be able to identify one corresponding to the Riemann zeta-function and one corresponding to each of the other zeta-functions that resemble the Riemann zeta-function. Suppose that we can prove rigorously that one of the quasicrystals in our enumeration has properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and can wait for the telephone call announcing the award of the Fields Medal. These are of course idle dreams. The problem of classifying one-dimensional quasicrystals is horrendously difficult, probably at least as difficult as the problems that Andrew Wiles took 7 years to fight his way through. But still, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible. The classification of quasicrystals is a worthy goal and might even turn out to be achievable.
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