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ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS Developments in Mathematics VOLUME 14 Series Editor: Krishnaswami Alladi, University of Florida, U.S.A. Aims and Scope Developments in Mathematics is a book series publishing (i) Proceedings of conferences dealing with the latest research advances, (ii) Research monographs, and (iii) Contributed volumes focusing on certain areas of special interest. Editors of conference proceedings are urged to include a few survey papers for wider appeal. Research monographs, which could be used as texts or references for graduate level courses, would also be suitable for the series. Contributed volumes are those where various authors either write papers or chapters in an organized volume devoted to a topic of speciaVcurrent interest or importance. A contributed volume could deal with a classical topic that is once again in the limelight owing to new developments. ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS Edited by TAKASHI AOKI Kinki University, Japan SHIGERU KANEMITSU Kinki University, Japan MIKIO NAKAHARA Kinki University, Japan YASUO OHNO Kinki University, Japan a - springer Zeta functions, topology, and quantum physics 1 edited by Takashi Aoki ... [et al.]. p. cm. - (Developments in mathematics ; v. 14) Includes bibliographical references. ISBN 0-387-24972-9 (acid-freep aper) - ISBN 0-387-24981-8 (e-book) 1. Functions, Zeta-Congresses. 2. Mathematical physics-Congresses. 3. Differential geometry-Congresses. I. Aoki, Takashi, 1953- 11 .Series. AMS Subiect Classifications: 1 1M xx. 35Qxx. 34Mxx. 14Gxx. 51 PO5 ISBN- 10 : 0-387-24972-9 ISBN-13 : 978-0387-24972-8 e-ISBN-10: 0-387-24981- 8 e-ISBN-13: 978-0387-24981- 0 Printed on acid-free paper. O 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 11161400 Contents Preface xi Conference schedule xii List of participants xiv Gollnitz-Gordon partitions with weights and parity conditions 1 Krishnaswami Alladi and Alexander Berlcovich 1 Introduction 2 A new weighted partition theorem 3 Series representations 4 A new infinite hierarchy Acknowledgments References Partition Identities for the Multiple Zeta Function David M. Bradley 1 Introduction 2 Definitions 3 Rational Functions 4 Stuffles and Partition Identities References A perturbative theory of the evolution of the center of typhoons Sergey Dobrolchotov, Evgeny Semenov, Brunello Tirozzi 1 Introduction 2 D namics of vortex square-root type singularities and Hugonibt- dslov chains 3 Equation for the smooth and singular part of the solutions Cauchy-Riemann conditions 4 Derivation of the Hugonibt-Maslov chain using complex vari- ables and its integrals Acknowledgments References viii ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS Algebraic Aspects of Multiple Zeta Values 51 Michael E. Hoffman 1 Introduction 51 2 The Shuffle Algebra 54 3 The Harmonic Algebra and Quasi-Symmetric Functions 56 4 Derivations and an Action by Quasi-Symmetric Functions 60 5 Cyclic Derivations 63 6 Finite Multiple Sums and Mod p Results 64 References 71 On the local factor of the zeta function of quadratic orders Masanobu Kaneko Acknowledgments 79 References 79 Sums involving the Hurwitz zeta-function values S. Kanemitsu, A. Schinzel, Y. Tanigawa 1 Introduction and statement of results 2 Proof of results References 89 Crystal Symmetry Viewed as Zeta Symmetry 9 1 Shigeru Kanemitsu, Yoshio Tanigawa, Haruo Tsukada, Masami Yoshimoto 1 Introduction 92 2 Lattice zeta-functions and Epstein zeta-functions 103 3 Abel means and screened Coulomb potential 120 References 128 Sum relations for multiple zeta values 131 Yasuo Ohno 1 Introduction 131 2 Generalizations of the sum formula 133 3 Identities associated with Arakawa-Kaneko zeta functions 140 4 Multiple zeta-star values and restriction on weight, depth, and height 142 Acknowledgment 143 References 143 The Sum Formula for Multiple Zeta Values OKUDA Jun-ichi and UENO Kimio 1 Introduction Contents ix Acknowledgment 2 Shuffle Algebra 3 Multiple Polylogarithms and the formal KZ equation 4 Mellin transforms of polylogarithms and the sum formula for MZVs 5 Knizhnik-Zamolodchikov equation over the configuration space x3 (@) References Zeta functions over zeros of general zeta and L-functions Andre' Voros 1 Generalities 2 The first family {T(s, x)) 3 The second family (2( a,v) ) 4 The third family {3(a,y )) 5 Concrete examples References Hopf Algebras and Transcendental Numbers Michel Waldschmidt 1 Transcendence, exponential polynomials and commutative lin- ear algebraic groups 2 Bicommutative Hopf algebras 3 Hopf algebras and multiple zeta values References 218 Preface This volume contains papers by invited speakers of the symposium "Zeta Functions, Topology and Quantum Physics" held at Kinki Uni- versity in Osaka, Japan, during the period of March 3-6, 2003. The aims of this symposium were to establish mutual understanding and to exchange ideas among researchers working in various fields which have relation to zeta functions and zeta values. We are very happy to add this volume to the series Developments in Mathematics from Springer. In this respect, Professor Krishnaswami Alladi helped us a lot by showing his keen and enthusiastic interest in publishing this volume and by contributing his paper with Alexander Berkovich. We gratefully acknowledge financial support from Kinki University. We would like to thank Professor Megumu Munakata, Vice-Rector of Kinki University, and Professor Nobuki Kawashima, Director of School of Interdisciplinary Studies of Science and Engineering, Kinki Univer- sity, for their interest and support. We also thank John Martindale of Springer for his excellent editorial work. Osaka, October 2004 Takashi Aoki Shigeru Kanemitsu Mikio Nakahara Yasuo Ohno xii ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS Zeta Functions, Topology, and Quantum Physics Kinki University, Osaka, Japan 3 - 6 March 2003 3 March M. Waldschmidt (Paris VI) How to prove relations between polyzeta values using automata H. Tsukada (Kinki Univ.) Crystal symmetry viewed as zeta symmetry (cowork with S. Kanemitsu, Y. Tanigawa and M. Yoshimoto) S. Akiyama (Niigata Univ.) Quasi-crystals and Pisot dual tiling K. Alladi (Florida) Insights into the structure of Rogers-Ramanujan type identities, some from physics A. Voros (Saclay) Zeta functions for the Riemann zeros 4 March Y. Ohno (Kinki Univ.) Sum relations for multiple zeta values M. Hoffman (U. S. Naval Acad.) Algebraic aspects of multiple zeta values B. Tirozzi (Rome) Application of shallow water equation to typhoons J. Okuda (Waseda Univ.) Multiple zeta values and Mellin transforms of multiple polylogarithms (cowork with K. Ueno) D. Broadhurst (The Open Univ.) Polylogarithms in quantum field theory High School Session (Two lectures for younger generation) (i) K. Alladi (Univ. Florida) Prime numbers and primality testing (ii) M. Waldschmidt (Univ. Paris VI) Error correcting codes