Number theory arithmetic in Shangri-La : proceedings of the 6th China-Japan seminar

273 Pages·2013·1.74 MB·English

by  Kanemitsu

NUMBER THEORY
ARITHMETIC IN SHANGRI-LA
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Series on Number Theory and Its Applications ISSN 1793-3161
Series Editor: Shigeru Kanemitsu (Kinki University, Japan)
Editorial Board Members:
V. N. Chubarikov (Moscow State University, Russian Federation)
Christopher Deninger (Universität Münster, Germany)
Chaohua Jia (Chinese Academy of Sciences, PR China)
Jianya Liu (Shangdong University, PR China)
H. Niederreiter (National University of Singapore, Singapore)
Advisory Board:
A. Schinzel (Polish Academy of Sciences, Poland)
M. Waldschmidt (Université Pierre et Marie Curie, France)
Published
Vol. 1 Arithmetic Geometry and Number Theory
edited by Lin Weng & Iku Nakamura
Vol. 2 Number Theory: Sailing on the Sea of Number Theory
edited by S. Kanemitsu & J.-Y. Liu
Vol. 4 Problems and Solutions in Real Analysis
by Masayoshi Hata
Vol. 5 Algebraic Geometry and Its Applications
edited by J. Chaumine, J. Hirschfeld & R. Rolland
Vol. 6 Number Theory: Dreaming in Dreams
edited by T. Aoki, S. Kanemitsu & J.-Y. Liu
Vol. 7 Geometry and Analysis of Automorphic Forms of Several Variables
Proceedings of the International Symposium in Honor of Takayuki Oda
on the Occasion of His 60th Birthday
edited by Yoshinori Hamahata, Takashi Ichikawa, Atsushi Murase &
Takashi Sugano
Vol. 8 Number Theory: Arithmetic in Shangri-La
Proceedings of the 6th China–Japan Seminar
edited by S. Kanemitsu, H.-Z. Li & J.-Y. Liu
LaiFun - Number Theory-- Arithmetic.pmd 1 1/15/2013, 9:01 AM
Series on Number Theory and Its Applications Vol. 8
NUMBER THEORY
ARITHMETIC IN SHANGRI-LA
Proceedings of the 6th China–Japan Seminar
Shanghai, China 15–17 August 2011
Editors
Shigeru Kanemitsu
Kinki University, Japan
Hongze Li
Shanghai Jiao Tong University, China
Jianya Liu
Shandong University, China
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
8731_9789814452441_tp.indd 2 17/1/13 9:21 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Number Theory and Its Applications — Vol. 8
NUMBER THEORY: ARITHMETIC IN SHANGRI-LA
Proceedings of the 6th China–Japan Seminar
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-4452-44-1
Printed in Singapore.
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v
PREFACE
This present volume is the proceedings of the 6th China-Japan Semi-
nar on Number Theory “Arithmetic in Shangri-La” held during August
15-17,2011atShanghaiJiaoTongUniversityFacultyClub,XuhuiCampus
1954, Huashan Road, Shanghai (Shangri-La), PRC; the organizers being
Shigeru Kanemitsu and Jianya Liu with Professor Hongze Li as the local
organizer.
The volume is dedicated to the memory of the late Professor Jerzy
Urbanowicz who passed away on September 6, 2012. He was a good friend
to many of us.
Pami¸eci Doktora Jerzego Urbanowicza, naszego przyjaciela
Thetitlesoundssomewhatromanticorexoticbutquitenaturalbecause
thevenuewasinShanghai—Shang-hai,whichwasadoptedbyJ.Hiltonin
his“LostHorizon”asaUtopia–Shangri-La.Weenjoyeddiscussingthetitle
of the proceedings over dinner on the first day, and finally, it was Angel
whogavethis title, thus the title being heaven-sent.Althoughinthe novel,
the place is locatedsomewhere in Tibet, we do not mind andjust took the
name which is so well-known in the Western world as a Utopia (not in the
sense of Thomas Moor).
Nowaboutthecontentsoftheseminarandthepresentproceedings:The
talks ranged over such a wide spectrum of contemporary number theory.
As can be seen from the titles of papers as well as in the following brief
descriptionsinthisvolume,wesucceededinassemblingfar-reachingsurvey
papers in Analytic Number Theory (Classical and Modern with emphasis
on additive number theory: BKW, K; CX), Theory of Automorphic L-
functions (LLW, L, RY), and Miscellaneous.
The volume contains the following:
H. Aoki’s “On Jacobi forms with levels” contains a useful survey on
Jacobi forms, whose theory is not easily accessible, save for the book of
Eichler-Zagier.
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vi PREFACE
Bru¨dern-Kawada-Wooley’s“Additive representation in thin sequences,
VIII: Diophantine inequalities in review”, which is to substitute the pre-
vious paper with the same title published in the proceedings of the 5th
China-Japan Seminar, and surveys their recent results in an enlightening
way.
Their main concern is the diagonal form
λ xk+ +λ xk,
1 1 ··· s s
as x , ,x ranges over Z or a subset thereof, where s 2, k 1 and
1 s
··· ≥ ≥
λ , ,λ arenon-zerorealnumbers.IntheAnnextotheGallery,theyrefer
1 s
···
to some improvements via the third author’s new results, Wooley’s Cooley
results on Vinogradov’s mean value theorem via efficient congruencing.
In a paper by Z. Cui and B.-Q. Xue, “A note on the distribution of
primes in arithmetic progression”, the authors, inspired by the result of
Bauer and Wang on the ternary Goldbach problem, prove the following
result: For any A,B >0 and for (b,r)=1
N
logp= 1+O(log−AN) ,
2ϕ(r)
N/2<p≤NX,p≡b(modr) (cid:16) (cid:17)
holds for all primes r N1/5(logN)B with a specified exceptional set.
≤
S.KanemitsuandM.Waldschmidt’s “Matricesoffinite abeliangroups,
finite Fourier transform and codes” introduces a generalization of the pre-
vious investigations on FFT to the case of finite groups, abelian or not.
Especially, all that appears in the FFT in engineering has been elucidated
from the point of view of linear representations of finite groups, including
many worked-out examples for the readers in engineering disciplines.
InY.Kitaoka’s“AremarkonaresultofEichler”,the authorcompletes
the workof Eichler,by removingthe limitation p 13mod24.It is shown
≡
that Fourier coefficients of a weighted sum of theta series θ(z,F ) with
i
positive coefficients are a sum of class numbers of imaginary quadratic
fields.
A. Kumchev’s “Weyl sums over primes in short intervals” bounds ex-
ponential sums of the form
f (α;x,y)= Λ(n)e αnk ,
k
x<n x+y
X≤ (cid:0) (cid:1)
where k 2 is an integer, 2 y x, Λ(n) is von Mangoldt’s function.
≥ ≤ ≤
For y = xθ with θ < 1, they play a central role in applications of the
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PREFACE vii
Hardy–Littlewood circle method to additive problems with almost equal
prime unknowns. Wooley’s breakthrough works here to lead to improved
bounds.
T.KuzumakiandJ.Urbanowicz’s“Oncongruencesforcertainbinomial
coefficients of E. Lehmer’s type” is a posthumous joint work of the late
second author with the first, who completed the draft left with her.
Y.-K. Lau, J.-Y. Liu, and J. Wu’s “Sign changes of the coefficients of
automorphic L-functions” is a well-written survey on recent results on the
sign change problem (for the coefficients) of automorphic L-functions. It
is written in such a way that even beginners of the field can sometimes
read between lines and it will be ideal for surveillance and updating the
knowledge.
G.-S. Lu¨’s “On Fourier coefficients of automorphic forms”gives numer-
ous new results on integral power sums of Fourier coefficients λ (n) of
f
an automorphic form f of the author and others, sometimes more effective
thanthenumericalboundfortheRamanujanConjecture.Themethoduses
symmetric power L-functions and their Rankin-Selberg L-functions.
In the paper by K. Miyake, “The twists of Hessian elliptic curves over
splitting fields of cubic polynomials and the related elliptic 3-folds”, the
author has constructed a twist H˜(µ,t) of Hesse’s elliptic curve over the
splitting field of a certain cubic polynomial.
X.-M. Ren and Y.-B. Ye’s “Asymptotic Voronoi’s summation formula
and their duality for SL (Z)” is a laborious work which provides us with
3
complete proofs for the estimates of error terms in
3 Γ(1+s+2k+µf(j))
Ψk(x)=ZRes=σ(π3x)−sjY=1 Γ(−s+2µ2f(j)) ψ˜(−s−k)ds
which is essential for asymptotic expansions of Voronoi’s summation for-
mula for A (m,n). Here A (m,n) are the Fourier coefficients of a Maass
f f
cusp form for SL (Z).
3
We invited Professor Andrzej Schinzel [Schinzel, Jerzy Urbanowicz’s
publications on pure mathematics] to write on the late Professor J. Ur-
banowicz.
In Z.-W. Sun’s “Conjectures involving arithmetical sequences”, the au-
thor states many conjectures on combinatorialas well as number-theoretic
sequences.
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viii PREFACE
Finally,anoteofthanksisdue.OurseminarwassupportedbytheNSFC
(NationalScienceFoundationofChina)andShanghaiJiaoTongUniversity.
We wouldlike to expressour heartythanks to these organizationsfor their
generosity. We would like to thank Shanghai Jiao Tong University for its
generous permission of using its excellent facilities. The conference room
wasequippedwith modernconveniencesandwasveryusefulin conducting
the seminar.
As in the case of the last few proceedings, Professor Jing Ma from
Jilin University made a devoted help in its editing and we record here
our hearty thanks to her for her excellent and beautiful preparation of the
manuscript of the proceedings. Thanks are due to Ms. Kwong Lai Fun of
WorldScientificforherassistanceinthe productionandpublicationofthis
proceedings.
Asusualwecompletetheprefacebyapoem.ThistimeProfessorJianya
Liu has composed it.
Cover picture: It is a modification of the most famous lithograph of Hoku-
sai — modified because there is Shangri-La at the back of waves — which
is so compelling that it is well liked by many scientists and adopted as a
cover picture in many books. The lines and circles are for analyzing the
appealing image in terms of the golden section by R. Yanagi.
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