Zeta Functions, Topology And Quantum Physics

by Aoki, Takashi; Kanemitsu, Shigeru; Nakahara, Mikio; Ohno, Yasuo

• ISBN13: 9780387249728
• ISBN10: 0387249729
• Format: Hardcover
• Copyright: 2005-08-15
• Publisher: Springer Verlag

Summary

This volume focuses on various aspects of zeta functions: multiple zeta values, Ohno's relations, the Riemann hypothesis, L-functions, polylogarithms, and their interplay with other disciplines. Eleven articles on recent advances are written by outstanding experts in the above-mentioned fields. Each article starts with an introductory survey leading to the exciting new research developments accomplished by the contributors. This book will become the major standard reference on the recent advances on zeta functions.

Table of Contents

Preface	xi
Conference schedule	xii
List of participants	xiv
Göllnitz-Gordon partitions with weights and parity conditions	1	(18)
Krishnaswami Alladi and Alexander Berkovich
1 Introduction	1	(3)
2 A new weighted partition theorem	4	(5)
3 Series representations	9	(2)
4 A new infinite hierarchy	11	(5)
Acknowledgments	16	(1)
References	16	(3)
Partition Identities for the Multiple Zeta Function	19	(12)
David M. Bradley
1 Introduction	19	(1)
2 Definitions	20	(2)
3 Rational Functions	22	(2)
4 Stuffles and Partition Identities	24	(5)
References	29	(2)
A perturbative theory of the evolution of the center of typhoons	31	(20)
Sergey Dobrokhotov, Evgeny Semenov, Brunello Tirozzi
1 Introduction	31	(3)
2 Dynamics of vortex square-root type singularities and Hugoniót-Myaslov chains	34	(6)
3 Equation for the smooth and singular part of the solutions Cauchy-Riemann conditions	40	(2)
4 Derivation of the Hugoniót-Maslov chain using complex variables and its integrals	42	(6)
Acknowledgments	48	(1)
References	48	(3)
Algebraic Aspects of Multiple Zeta Values	51	(24)
Michael E. Hoffman
1 Introduction	51	(3)
2 The Shuffle Algebra	54	(2)
3 The Harmonic Algebra and Quasi-Symmetric Functions	56	(4)
4 Derivations and an Action by Quasi-Symmetric Functions	60	(3)
5 Cyclic Derivations	63	(1)
6 Finite Multiple Sums and Mod p Results	64	(7)
References	71	(4)
On the local factor of the zeta function of quadratic orders	75	(6)
Masanobu Kaneko
Acknowledgments	79	(1)
References	79	(2)
Sums involving the Hurwitz zeta-function values	81	(10)
S. Kanemitsu, A. Schinzel, Y. Tanigawa
1 Introduction and statement of results	81	(5)
2 Proof of results	86	(3)
References	89	(2)
Crystal Symmetry Viewed as Zeta Symmetry	91	(40)
Shigeru Kanemitsu, Yoshio Tanigawa, Haruo Tsukada, Masami Yoshimoto
1 Introduction	92	(11)
2 Lattice zeta-functions and Epstein zeta-functions	103	(17)
3 Abel means and screened Coulomb potential	120	(8)
References	128	(3)
Sum relations for multiple zeta values	131	(14)
Yasuo Ohno
1 Introduction	131	(2)
2 Generalizations of the sum formula	133	(7)
3 Identities associated with Arakawa-Kaneko zeta functions	140	(2)
4 Multiple zeta-star values and restriction on weight, depth, and height	142	(1)
Acknowledgment	143	(1)
References	143	(2)
The Sum Formula for Multiple Zeta Values	145	(26)
OKUDA Jun-ichi and UENO Kimio
1 Introduction	145	(2)
Acknowledgment	147	(1)
2 Shuffle Algebra	147	(2)
3 Multiple Polylogarithms and the formal KZ equation	149	(7)
4 Mellin transforms of polylogarithms and the sum formula for MZVs	156	(6)
5 Knizhnik-Zamolodchikov equation over the configuration space X3(C)	162	(8)
References	170	(1)
Zeta functions over zeros of general zeta and L-functions	171	(26)
André Voros
1 Generalities	171	(5)
2 The first family {F(s, x)}	176	(6)
3 The second family {Z(σv)}	182	(4)
4 The third family {3(σy)}	186	(2)
5 Concrete examples	188	(6)
References	194	(3)
Hopf Algebras and Transcendental Numbers	197
Michel Waldschmidt
1 Transcendence, exponential polynomials and commutative linear algebraic groups	198	(9)
2 Bicommutative Hopf algebras	207	(3)
3 Hopf algebras and multiple zeta values	210	(8)
References	218