Table Of ContentNumber Theoretic Methods
Developments in Mathematics
VOLUMES
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 focussing 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
special/current interest or importance. A contributed volume could deal with a classical
topic which is once again in the limelight owing to new developments.
Number Theoretic Methods
Future Trends
Edited by
Shigeru Kanemitsu
Graduate School ofA dvanced Technology,
University of Kinki, /izuka, Japan
and
ChaohuaJia
Institute of Mathematics,
Academia Sinica, Beijing, China
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-5239-4 ISBN 978-1-4757-3675-5 (eBook)
DOI 10.1007/978-1-4757-3675-5
Printed on acidjree paper
All Rights Reserved
© 2002 Springer Science+B usiness Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 1st edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Contents
Preface vii
A limiting form of the q-Dixon 4<p3 summation and related
partition identities 1
Krishnaswami Alladi and Alexander Berkovich
Arithmetical properties of solutions of linear second order
q-difference equations 15
Masaaki Amou and Tapani Matala-aho
Ramanujan's Contributions to Eisenstein Series, Especially in
His Lost Notebook 31
Bruce C. Berndt and Ae Ja Yee
New Applications of a Result of Galochkin on Linear Independence 55
Peter Bundschuh
Partitions modulo prime powers and binomial coefficients 67
Tianxiri Cai
Infinite sums, diophantine equations and Fermat's last theorem 73
Henri Darmon and Claude Levesque
On the nature of the "explicit formulas" in analytic number theory
- a simple example 97
Christopher Deninger
Product representations by rationals 119
P.D. T.A. Elliott
On the distribution of aP modulo 1 151
Chaohua Jia
Ramanujan's formula and modular forms 159
Shigeru Kanemitsu, Yoshio Tanigawa and Masami Yoshimoto
Waldspurger's formula and central critical values of £-functions
of newforms in weight aspect 213
Winfried Kohnen and Jyoti Sengupta
vi
Primitive roots: a survey 219
Shuguang Li and Carl Pomerance
Zeta-Functions Defined by Two Polynomials 233
K ohji Matsumoto and Lin Weng
Some Aspects on Interactions between Algebraic Number Theory
and Analytic Number Theory 263
K atsuya Miyake
On G-functions and Pade approximations 301
M akota Nagata
A penultimate step toward cubic theta-Weyl sums 311
Yoshinobu Nakai
Some Results in view of Nevanlinna Theory 339
Junjiro Noguchi
A Historical Comment about the GVT in Short Interval 351
Pan Chengbiao ·
Convexity and Intersection of Random Spaces 369
Mariya Shcherbina and Brunella Tirozzi
Generalized hypergeometric series and the symmetries of 3-j
and 6-j coefficients 381
K. Srinivasa Rao, H.D. Doebner and P. Nattermann
Stability and New Non-Abelian Zeta Functions 405
Lin Weng
A hybrid mean value of £-functions and general quadratic
Gauss sums 421
Zhang Wenpeng
index 435
vii
Preface
This volume contains the proceedings of the very successful second China-Japan
Seminar held in lizuka, Fukuoka, Japan, during March 12-16, 2001 under the support
of the Japan Society for the Promotion of Science (JSPS) and the National Science
Foundation of China (NSFC), and some invited papers of eminent number-theorists
who visited Japan during 1999-2001 at the occasion of the Conference at the Research
Institute of Mathematical Sciences (RIMS), Kyoto University.
The proceedings of the 1st China-Japan Seminar held in September 1999 in Beijing
has been published recently {2002) by Kluwer as DEVM 6 which also contains some
invited papers. The topics of that volume are, however, restricted to analytic number
theory and many papers in this field are assembled.
In this volume, we return to the lines of the previous one "Number Theory and
its Applications", published as DEVM 2 by Kluwer in 1999 and uphold the spirit of
presenting various topics in number theory and related areas with possible applica
tions, in a unified manner, and this time in nearly a book form with a well-prepared
index. We accomplish this task by collecting highly informative and readable survey
papers (including half-survey type papers), giving overlooking surveys of the hith
erto obtained results in up-to-the-hour form with insight into the new developments,
which are then analytically continued to a collection of high standard research papers
which are concerned with rather diversed areas and will give good insight into new
researches in the new century.
Survey papers range over Transcendence Theory (Nagata, on G-functions), Prob
abilistic Number Theory (Elliott, on the product representation by rationals, Li and
Pomerance, on "primitive roots" modulo n), History of Number Theory (Miyake,
on the history of interaction between algebraic number theory and analytic num
ber theory and Pan, on the Goldbach-Vinogradov theorem in short intervals), Ra
manujan's Work and Automorphic Forms (Berndt and Yee ,on Ramanujan's contri
butions of Eisenstein series, and Kanemitsu-Tanigawa-Yoshimoto, on Ramanujan's
formulas and modular forms, both are half-survey papers), Analytic Number Theory
(Jia, on the distribution of ap, p ranging primes) and Diophantine Equations (Dar
mon and Levesque) while research papers range over Arithmetic Algebraic Geometry
(Deninger, on explicit formulas in analytic number theory and Wen, on non-abelian
zeta-functions), Transcendence Theory (Amou and Matala-aho, on the arithmetic na
ture of solutions of q-difference equations, Bundschuh, on applications of Galochikin's
result, and Noguchi, on applications of Nevanlinna theory), Analytic Number Theory
(Cai, on partitions modulo prime powers, Matsumoto and Weng, on the zeta-function
of polynomials, Nakai, on Weyl sums as finite theta series, and Zhang, on hybrid mean
values of £-functions), Partition Theory (Alladi and Berkovich), Automorphic Froms
(Kohnen, on Waldspurger's formulas and central critical values of £-functions, Berndt
and Yee, on Ramanujan's contributions of Eisenstein series, Kanemitsu-Tanigawa
Yoshimoto, on Ramanujan's formulas and modular forms) and two papers of authors
who have physical background (Srinivasa Rao, on quantum angular momentum, and
Shcherbina and Tirozzi, on neural network)
Now some passages about the seminar itself are in order. The length of time spent
for preparation notwithstanding, the seminar itself elapsed away in an instant as
with all other pleasant events. Our present feeling is best expressed in the following
Haiku: "Omoshiroute yagate kanashiki hanabikana", which in English would read
"Fireworks, momentary flowering in the darkness, followed by utter vacuousness".
Vlll
We received several (e-mail) messages to the effect that it was a successful and
memorable one as well as oral communications expressing their high-spiritedness in
attending the seminar. We hope everyone felt in the same way and that this was a
pleasant occasion and you got new ideas or expansion of your research fields.
Hereby we wish to express our hearty thanks to the great help given to us by
Dean of University of Kinki, Kyushu School of Engineering, Professor K. Kikukawa
whose presidential address given half in Chinese was very impressive. Although the
second editor expressed his thanks to his ex-students for their devoted help before and
during the seminar, we feel we are to thank them again for their unusually efficient
help: Drs. H. Kumagai and M. Yoshimoto, Ms. K. Mashimo and Ms. K. Wada.
Without their utmost painstaking effort, we could not have made it. We wish to
thank Dr. Yoshimoto also for his devoted help in editing many files of papers in this
volume.
The last but not the least thanks are due to Ms. Chiaki Yokoyama of the JSPS for
her timely and most appropriate adv ices, and above all her patience before, during and
after the seminar. Also we thank all the participants for their active contribution to
the seminar and the authors of papers in this volume for contributing their important
works before the deadline. Finally we wish to thank all anonymous referees for their
thorough scrutinizing of the papers and for their important comments which improved
some of the papers greatly. As usual we should express our thanks to Kluwer Academic
Publishers for publishing this volume, and to the editor John Martindale and his
secretary Angela Quilici for their devoted help.
Let's hope to meet again in another conference in 5 years (which is the length of
time needed before the first editor will get energetized again) in Japan. Many of us
will meet at the next China-Japan Seminar to be held in Xi'an in September, 2003.
July 2002
Shigeru Kanemitsu and Chaohua Jia (Editors)
ix
List of participants
Professor Shigeki Akiyama (Niigata University)
Professor Krishnaswami Alladi (University of Florida)
Professor Masaaki Amou (Gunma University)
Professor Tsuneo Arakawa (Rikkyo University)
Professor Tianxin Cai (Zhejiang University)
Professor Dr. Christopher Deninger (Universitat Mi.inster)
Professor Shigeki Egami (Toyama University)
Dr. Jun Furuya (Nagoya University)
Professor Masayoshi Hata (Kyoto University)
Professor Mikihito Hirabayashi (Kanazawa Institute of Technology)
Mr. Park Kyung Ho (Saga University)
Dr. Taro Horie (Suzuka Inst. of Techn.)
Dr. Yumiko Ichihara (Nagoya University, now Waseda University)
Professor Chaohua Jia (Academia Sinica)
Professor Masanobu Kaneko (Kyushu University)
Professor Shigeru Kanemitsu (University of Kinki)
Professor Masanori Katsurada (Keio University)
Professor Kiyoshi Kikukawa (University of Kinki)
Professor !sao Kiuchi (Yamaguchi University)
Professor Dr. Winfried Kohnen (Universitat Heidelberg)
Dr. Hiroshi Kumagai (Kagoshima Inst. of Techn.)
Professor Minggao Lu (Shanghai University)
Ms. Keiko Mashimo (University of Kinki)
Professor Tapani Matala-Aho (University of Oulu)
Professor Kohji Matsumoto (Nagoya University)
Mr. Hidehiko Misho (Nagoya University)
Professor Katsuya Miyake (Tokyo Metropolitan University)
Professor Yasuo Motoda (Yatsushiro National College of Technology)
Professor Leo Murata (Meiji Gakuin University)
Professor Kenji Nagasaka (Hosei University)
Professor Makoto Nagata (RIMS-Kyoto University)
Mr. Minoru Nakashima (Saga University)
Dr. Hirofumi Nagoshi (Keio University, now RIMS-Kyoto University)
Professor Toru Nakahara (Saga University)
Professor Yoshinobu Nakai (Yamanashi University)
Professor Shoichi Nakajima (Gakushuin University)
Professor Yoshimasa Nakano (University of Kinki)
Professor Junjiro Noguchi (University of Tokyo)
Professor Ryotaro Okazaki (Doshisha University)
Professor Chengbiao Pan (The China Agricultural University)
Professor Ryuji Sasaki (Nihon University)
X
Professor Ken'ichi Sato (Nihon University)
Professor Iekata Shiokawa (Keio University)
Professor Shigeshi Shirasaka (Kagoshima lnst. of Techn.)
Professor Masaki Sudo (Seikei University)
Mr. Masatoshi Suzuki (Nagoya University)
Professor Yoshio Tanigawa (Nagoya University)
Professor Brunello Tirozzi (University of Rome)
Professor Haruo Tsukada (University of Kinki)
Mr. Hiroyuki Tsutsumi (Shimane University)
Ms. Kaoru Wada (University of Kinki)
Professor Isao Wakabayashi (Seikei University)
Professor Lin Weng (Nagoya University, now Kyushu University)
Mr. Mao Xiang Wu (Nagoya University)
Dr. Masami Yoshimoto (RIMS Kyoto University, now Nagoya University)
Professor Wenpeng Zhang (Northwest University)
Who's who in the picture 1. M. Nagata 2. H. Tsutumi 3. T. Horie 4. M. Nakashima 5.
M. Hata 6. Y. Motoda 7. Y. Nakai 8. P.-K. Ho 9. T. Matala-aho 10. T. Arakawa 11. C.-B.
Pan 12. L. Wen 13. M. Sudo 14. S. Egami 15. S. Akiyama 16. C.-H. Jia 17. T.-X. Cai 18. Y.
Tanigawa 19. C. Deninger 20. R. Okazaki 21. M. Kaneko 22. S. Kanemitsu (multi-national
Kanemitsu Corporation, cf. p. p. 118) 23. K. Miyake 24. K. Matsumoto 25. W.-P. Zhang
26. K. Nagasaka 27. M. Amou 28. W. Kohnen 29. M. Yoshimoto 30. B. Tirozzi 31. K.
Alladi 32. M. Hirabayashi 33. I. Kiuchi 34. I. Shiokawa 35. I. Wakabayashi 36. M. Suzuki
37. M. Katsurada 38. H. Nagoshi 39. M.-X. Wu 40. H. Misho 41 J. Furuya 42. Y. Ichikawa
43. K. Wada 44. K. Mashimo 45. H. Kumagai 46. K. Kikukawa
