Zeta Functions, Topology and Quantum Physics

227 Pages·2005·29.411 MB·English

by  Takashi Aoki

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
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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