The Fourth China-Japan Conference on Number Theory
Advisory Committee:
Tao Zhan, Shandong University, China
Organizing Committee:
Shigeru Kanemitsu, Kinki University, Japan
Jianya Liu, Shandong University, China
Guest Speakers:
Krishnaswami Alladi, University of Florida, USA
Vladimir N. Chubarikov, Moscow State University, Russia
Winfried Kohnen, Universitaet Heidelberg, Germany
Andrzej Schinzel, Polish Academy of Sciences, Poland
Igor Shparlinski, Macquarie University, Australia
Daqing Wan, University of California, USA
Trevor D. Wooley, University of Michigan, USA
Yangbo Ye, University of Iowa, USA
Shou-Wu Zhang, Columbia University, USA
Speakers:
Shigeki Akiyama, Niigata University, Japan
Takashi Aoki, Kinki University, Japan
Tianxin Cai, Zhejiang University, China
Yonggao Chen, Nanjing Normal University, China
YoungJu Choie, Pohang University of Science and Technology, Korea
Chaohua Jia, The Chinese Academy of Sciences, China
Koichi Kawada, Iwate University, Japan
Yoshiyuki Kitaoka, Meijo University, Japan
Yuk-Kam Lau, The University of Hong Kong, Hong Kong
Zhiguo Liu, East China Normal University, China
Kohji Matsumoto, Nagoya University, Japan
Katsuya Miyake, Waseda University, Japan
Leo Murata, Meiji-Gakuin University, Japan
Kenji Nagasaka, Hosei University, Japan
Yoshinobu Nakai, Yamanashi University, Japan1

Xiumin Ren, Shandong University, China
Masaki Sudo, Seikei University, Japan
Zhi-Wei Sun, Nanjing University, China
Yoshio Tanigawa, Nagoya University, Japan
Haruo Tsukada, Kinki University, Japan
Isao Wakabayashi, Seikei University, Japan
Tianze Wang, Henan University, China
Honggang Xia, Shandong University, China
Wenguang Zhai, Shandong Normal University, China
Wenpeng Zhang, Northwest University, China
Short Communications:
Kentaro Ihara, Kinki University, Japan
Takako Kuzumaki Kobayashi, Gifu University, Japan
Hailong Li, Weinan Teacher’s College, China
Huaning Liu, Northwest University, China
Guangshi Lü, Shandong University, China
Masayuki Toda, Kinki University, Japan
Deyu Zhang, Shandong University, China
Time: August 30 - September 3, 2006
Venue: Shandong University Academic Center (Weihai)
Supported by JSPS, NSFC, and Shandong University

2

Scientific Program
Wednesday, August 30, 2006

Opening Address
The Organizers
Time: 9:00-9:40

Morning Session
Chair: Jianya Liu
Time: 10:00-10:40
Speaker: Andrzej Schinzel
The Number of Solutions in a Box of a Linear Homogeneous Congruence
Time: 11:00-11:40
Speaker: Shou-Wu Zhang
Periods Integrals and Special Values of L-series

Lunch: 12:00 Shandong University Academic Center

Afternoon Session
Chair: Krishnaswami Alladi
Time: 14:30-15:00
Speaker: Zhi-Wei Sun
Curious Identities and Congruences Involving Bernoulli Polynomials
Time: 15:20-15:40
Speaker: Shigeki Akiyama
Rational Based Number System and Mahler’s Problem
Time: 15:50-16:10
Speaker: YoungJu Choie
Broué-Enguehard Maps and Atkin-Lehner Involutions
Time: 16:20-16:40
Speaker: Xiumin Ren
3

Estimates of Exponential Sums over Primes and Applications in Number
TheoryTime: 16:50-17:05
Speaker: Haruo Tsukada
A General Modular Relation Associated with the Riemann Zeta-function
Time: 17:10-17:25
Speaker: Kenji Nagasaka
Density, Probability and Applications
Time: 17:30-17:40
Speaker: Takako Kuzumaki Kobayashi
A Transformation Formula for Certain Lambert Series

Dinner: 18:30 Shandong University Academic Center

4

Thursday, August 31, 2006

Morning Session
Chair: Vladimir N. Chubarikov
Time: 9:00-9:40
Speaker: Krishnaswami Alladi
New Approaches to Jacobi’s Triple Product Identity and a Quadruple Product Extension
Time: 10:00-10:40
Speaker: Trevor D. Wooley
Waring’s Problem in Function Fields
Time: 11:00-11:40
Speaker: Yangbo Ye
A New Bound for Rankin-Selberg L-functions

Lunch: 12:00 Shandong University Academic Center

Afternoon Session
Chair: Yoshio Tanigawa
Time: 14:30-15:00
Speaker: Wenpeng Zhang
The Mean Value of Dedekind Sum and Cochrane Sum in Short Intervals
Time: 15:20-15:50
Speaker: Isao Wakabayashi
Some Thue Equations and Continued Fractions
Time: 16:10-16:30
Speaker: Yuk-Kam Lau
The Error Terms in Dirichlet’s Divisor Problem, the Circle Problem andthe Mean Square Formula of the Riemann Zeta-function
Time: 16:40-17:00
Speaker: Wenguang Zhai
On the Fourth Power Moments of ∆(x) and E(t)Time: 17:10-17:20
Speaker: Tianxin Cai / Xia Zhou
On the Height of Happy Numbers
5Time: 17:25-17:35
Speaker: Masayuki Toda
On Gauss’ formula for ψ and finite expressions for the L-series at 1
Time: 17:40-17:50
Speaker: Kentaro Ihara
On the Structure of Algebra of Multiple Zeta Values

Dinner: 18:30 Shandong University Academic Center

6

Friday, September 1, 2006
Excursion and Shopping (TBA)

Saturday, September 2, 2006

Morning Session
Chair: Kohji Matsumoto
Time: 9:00-9:40
Speaker: Vladimir N. Chubarikov
Trigonometric Sums in Number Theory, Analysis and Probability Theory
Time: 10:00-10:40
Speaker: Winfried Kohnen
Sign Changes of Fourier Coefficients and Hecke Eigenvalues of Cusp Forms
Time: 11:00-11:40
Speaker: Igor Shparlinski
Exponential and Character Sums with Combinatorial Sequences

Lunch: 12:00 Shandong University Academic Center

Afternoon Session
Chair: Wenpeng Zhang
Time: 14:30-15:00
Speaker: Chaohua Jia
On a Conjecture of Yiming Long
Time: 15:20-15:40
Speaker: Koichi Kawada
On the Sum of Five Cubes of Primes
Time: 15:50-16:10
Speaker: Honggang Xia
On Zeros of Cubic L-functions
Time: 16:20-16:40
Speaker: Zhiguo Liu
A Theta Function Identity to the Quintic Base
Time: 16:50-17:05
7Speaker: Masaki Sudo
On the Exponential Equations ax − by = c (1 ≤ c ≤ 300)
Time: 17:25-17:40
Speaker: Yoshinobu Nakai
TBA
Time: 17:45-17:55
Speaker: Deyu Zhang
Zero Density Estimates for Automorphic L-functions

Dinner: 18:30 Shandong University Academic Center

8

Sunday, September 3, 2006

Morning Session
Chair: Winfried Kohnen
Time: 9:00-9:40
Speaker: Daqing Wan
L-functions of Infinite Symmetric Powers
Time: 10:00-10:30
Speaker: Yoshiyuki Kitaoka
Distribution of Units of an Algebraic Number Field
Time: 10:50-11:20
Speaker: Kohji Matsumoto
The Riesz Mean of the Convolution Product of Von Mangoldt Functions
and the Related Zeta-function
Time: 11:30-12:00
Speaker: Katsuya Miyake
Twists of Hessian Elliptic Curves and Cubic Fields

Lunch: 12:00 Shandong University Academic Center

Afternoon Session
Chair: Shigeru Kanemitsu
Time: 14:30-15:00
Speaker: Leo Murata
On a Property of the Multiplicative Order of a (mod p)
Time: 15:20-15:50
Speaker: Yonggao Chen
On the Prime Power Factorization of n!
Time: 16:00-16:20
Speaker: Yoshio Tanigawa
Kronecker’s Limit Formula and the Hypergeometric Function
Time: 16:25-16:35
Speaker: Guangshi Lü
Some Results in Classical Analytic Number Theory
Time: 16:40-16:50
9Speaker: Huaning Liu
Mean Value of Dirichlet L-functions and Applications to Pseudorandom
Binary Sequences in Cryptography
Time: 16:55-17:05
Speaker: Hailong Li
The Structural Elucidation of Eisenstein’s FormulaConference Dinner: 18:30 Shandong University Academic Center

10

ABSTRACT
Shigeki Akiyama; Niigata University, Japane-mail: akiyama@math.sc.niigata-u.ac.jp
Title: Rational Based Number System and Mahler’s Problem
Abstract: We introduce a number system with a fixed rational number
base. For instance, it is based by powers of 3/2 with digits in {0, 1, 2}
with some restrictions. This is also produced by a boosted adding machine.
The system gives a way to expand positive integers. Though the associatedlanguage is not even context free, the odometer is given by an automaton.
The system can be “compactified” by extending to right. Each real number is aperiodic by this expansion. Up to countable exceptions, the expansion is unique. We characterized this countable exceptions and found aninteresting connection with Mahler’s problem on the distribution of fractional parts of (3/2)n in [2].
This talk is based on a joint work [1] with Ch. Frougny (Paris 7) and
J. Sakarovitch (CNRS, ENST).
References[1] S. Akiyama, Ch. Frougny, and J. Sakarovitch, Powers of rationals modulo 1 andrational base number systems, submitted.
[2] K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (1968),
313–321.

Krishnaswami Alladi; University of Florida, USAe-mail: alladi@mail.math.ufl.edu
Title: New Approaches to Jacobi’s Triple Product Identity and A Quadruple Product Extension
Abstract: Jacobi’s triple product identity for theta functions is one of themost fundamental in the theory of special functions and q-hypergeometricseries. We will describe new combinatorial approaches to Jacobi’s identity
via reformulations of an important partition theorem of Göllnitz. Recently
Andrews, Berkovich and I obtained a deep four parameter extension of thetheorem of Göllnitz. We will show how specializations of this four parameter
theorem can be used to construct a quadruple product extension of the triple
product identity.
Tianxin Cai; Zhejiang University, Chinae-mail: txcai@mail.hz.zj.cn
Title: On the Height of Happy Numbers
11Abstract: (Joint work with Xia Zhou) In this paper the authors extend acurious harmonic congruence to arbitrary length for any prime p > 3 andpositive integer n ≤ p − 2:(X−(n − 1)! Bp−n (mod p)
if 2 - n,
1≡,n
2− 2(n+1) n! Bp−n−1 p (mod p ) if 2|n.
l l · · · ln
l +l +···+ln =p 1 2
12
l1 ,··· ,ln >0

We also give a general method to find the least happy number of any given
height theoretically, which related to a problem asked by R.Guy.
Yonggao Chen; Nanjing Normal University, Chinae-mail: ygchen@njnu.edu.cn
Title: (Joint work with Wei Liu) On the Prime Power Factorization of n!
Abstract: Let p1 , p2 , · · · be the sequence of all primes in ascending order.
ep (n)For a positive integer n, let epi (n) be the nonnegative integer with pi i | n
ep (n)+1- n. In 1997, D. Berend proved a conjecture of Erdős andand pi i
Graham by showing that for every positive integer k there exist infinitely
many positive integers n with
ep1 (n!) ≡ 0(mod 2), ep2 (n!) ≡ 0(mod 2), · · · , epk (n!) ≡ 0(mod 2).
In 2003, Y. G. Chen proved Sander’s conjecture: for any given positive
integer k and any εi ∈ {0, 1}(i = 1, 2, · · · , k), there exist infinitely many
positive integers n such that
ep1 (n!) ≡ ε1 (mod 2), ep2 (n!) ≡ ε2 (mod 2), · · · , epk (n!) ≡ εk (mod 2).
In 2003, F. Luca and P. Stănică posed a conjecture:
Conjecture (F. Luca and P. Stănică). Let p1 , · · · , pk be distinct
primes, m1 , · · · , mk be arbitrary positive integers (> 2), and 0 6 ai 6 mi −1
for i = 1, · · · , k be arbitrary residue class modulo mi . ThenN
| {0 6 n < N : epi (n) ≡ ai (mod mi ), 1 6 i 6 k} |∼as N → ∞.
m1 · · · mk
We will talk about the progress on this topic.
YoungJu Choie; Pohang University of Science and Technology,
Korea
e-mail: yjc@postech.ac.kr
Title: Broué-Enguehard Maps and Atkin-Lehner Involutions
Abstract: (Joint work with P. Solé) Let ` be one of the ten integers suchthat the sum of their divisors divide 24. For each such `, (except 15) we give12

a map from an algebra of polynomial invariants of some finite group to thealgebra of modular forms invariant under the Atkin-Lehner group of level `.
These maps are motivated and inspired by constructions of modular lattices
from self-dual codes over rings . This work generalizes Broué-Enguehard
work in level one and three obtained from binary and ternary codes.
Vladimir N. Chubarikov; Moscow State University, Russia
e-mail: chubarik@mech.math.msu.su
Title: Trigonometric Sums in Number Theory, Analysis and Probability
TheoryAbstract: TBA
Kentaro Ihara; Kinki University, Japane-mail: k− ihara72@yahoo.co.jp
Title: On the Structure of Algebra of Multiple Zeta Values
Abstract: (Joint work with H. Ochiai) The multiple zeta value (MZV) isa number defined by convergent seriesX1
ζ(k1 , k2 , . . . , kn ) =
∈ R,
k1 k2
kn
m
m
·
·
·
m
1
2
n
m1 >m2 >···>mn >0
where k1 , k2 , . . . , kn are positive integers with k1 >L
1. Here n is the depth andk = k1 +· · ·+kn the weight of the MZV. Let Z = k≥0 Zk be the graded Qvector space generated by MZVs, where Zk is the component of weight k. It(n) (n0 )(n+n0 )
is known Z has a filtered graded Q-algebra structure: Zk Zk0 ⊂ Zk+k0 ,(n)where Zk is the subspace spanned by MZVs of weight k and depth 6 n.
The algebra Z is deeply connected with the theory of mixed motives andGalois representation on the fundamental group of a curve. The main problem is what the structure of Z is. Let Dk,n be the minimum number ofalgebra generators of Z in weight k and depth n.
h 2 i
Theorem For even d, we have Dd+3,3 6 d 48−1 .
In [1], Kaneko, Zagier and Ihara define a Q-vector space DShn (d) (called
double shuffle space) whose dimension gives an upper bound of the numbers
Dd+n,n for each n, d. The DShn (d) consists of homogeneous polynomials of degree d in n variables with Q-coefficients and which satisfy certain
identities. These identities are connected with two kinds of product structures (shuffle products) in Z. For example, f ∈ DSh3 (d) ⊂ Q[x1 , x2 , x3 ]satisfies the relations f (x1 , x2 , x3 ) + f (x2 , x1 , x3 ) + f (x2 , x3 , x1 ) = 0 andf ] (x1 , x2 , x3 ) + f ] (x2 , x1 , x3 ) + f ] (x2 , x3 , x1 ) = 0, where f ] (a, b, c) = f (a +13

b + c, b + c, c). It is shown in [1] that DShn (d) = {0} for odd d, consequently
Dk,n = 0 for k, n with distinct parity. For case n = 2, it is known that
Dd+2,2 6 [ d6 ] for even d. For case n = 3, Goncharov shows the inequality2
Dd+3,3 6 [ d 48−1 ] for even d in [2]. Our theorem above gives the same estimate,
but by quite different method, using the theory of invariants for reflectiongroups. As a corollary we have an expression of the space DSh3 (d) in termsof the invariants under the reflection groups.
References[1] K. Ihara, M. Kaneko, D. Zagier, Derivations and double shuffle relations for multiple
zeta values, Compositio Math. 142 (2006), 307–338.
[2] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math.
Res. Lett. 5 (1998), 497–516.

Chaohua Jia; The Chinese Academy of Sciences, Chinae-mail: jiach@math.ac.cn
Title: On a Conjecture of Yiming Long
Abstract: TBA
Koichi Kawada; Iwate University, Japane-mail: kawada@iwate-u.ac.jp
Title: On the Sum of Five Cubes of Primes
Abstract: Last year, Mikawa and Peneva proved that if c > 2/3 and N
is sufficiently large, then all but o(N c ) odd natural numbers in the interval[N, N + N c ] can be written as the sum of five cubes of primes. Working
jointly with them, I recently showed the corresponding conclusion with theweaker constraint c > 7/12, and I intend to talk on this slight improvement.
Yoshiyuki Kitaoka; Meijo University, Japane-mail: kitaoka@ccmfs.meijo-u.ac.jp
Title: Distribution of Units of an Algebraic Number Field
Abstract: We study distribution of units modulo a rational prime numbersof an algebraic number field.
Takako Kuzumaki Kobayashi; Gifu University, Japane-mail: kuzumaki@cc.gifu-u.ac.jp
Title: A Transformation Formula for Certain Lambert Series
14Abstract: The purpose of the present talk is to deduce the transformation
formula for certain Lambert series under the action of the full modulargroup from the functional equation for the appropriate zeta-functions withsign.character in the spirit of Goldstein-de la Torre.
Winfried Kohnen; Universitaet Heidelberg, Germany
e-mail: winfried@mathi.uni-heidelberg.de
Title: Sign Changes of Fourier Coefficients and Hecke Eigenvalues of Cusp
Forms
Abstract: This is a survey talk in which we will report on recent results
regarding changes of signs of Fourier coefficients and Hecke eigenvalues ofholomorphic cusp forms in one variable.
Yuk-Kam Lau; The University of Hong Kong, Hong Konge-mail: yklau@maths.hku.hk
Title: The Error Terms in Dirichlet’s Divisor Problem, the Circle Problem
and the Mean Square Formula of the Riemann Zeta-function.
Abstract: We are concerned with some recent work, contributed by various
researchers, on the error terms in the title, which are respectively definedasX∆(x) :=
d(n) − x(log x + 2γ − 1),n≤x

P (x) :=

X

r(n) − πx,

n≤x

µ
¶
t
|ζ(1/2 + iu)| du − t log+ 2γ − 1 .
2π
0

Z t
E(t) :=

2

In addition, we shall discuss on the generalizations of ∆(x) and E(t).
Hailong Li; Weinan Teacher’s College, Chinae-mail: lihailong@wntc.edu.cn
Title: The Structural Elucidation of Eisenstein’s Formula
Abstract: We shall give a complete structural description of the classicalEisenstein formula (and generalizations thereof) that expresses the first periodic Bernoulli polynomial as a finite combination of cotangent values, asa relation between two bases of the vector space of periodic Dirichlet series.
We shall also determine the limiting behavior of them, giving rise to Gauss’
famous closed formula for the values of the digamma function at rational
15

points on the one hand and elucidation of Eisenstein-Wang’s formulas on
the other.
Huaning Liu; Northwest University, Chinae-mail: hnliu@nwu.edu.cn
Title: Mean Value of Dirichlet L- functions and Applications to Pseudorandom Binary Sequences in Cryptography
Abstract: Let q ≥ 2, m ≥ 1 and n ≥ 1 be positive integers withm ≡ n mod 2. Set ²m,n = 1 if m ≡ n ≡ 1 mod 2 and ²m,n = 0 ifm ≡ n ≡ 0 mod 2. Using generalized Dedekind sums, Bernoulli polynomials and Bernoulli numbers, we prove a formula for the mean value ofL(m, χ)L(n, χ) as following:X
2
L(m, χ)L(n, χ)φ(q) χ mod q
χ(−1)=(−1)mm−n(−1) 2 (2π)m+n
=
2m!n!

!
²
m,n
Bm Bn φm+n−1 (q) ,rm,n,l φl (q)q l−m−n −
q
l=0

Ãm+n
X

where
rm,n,l = Bm+n−l¡m¢¡n¢¡ a+b+1 ¢

m X
n
X

Bm−a Bn−b

a=0 b=0a+b≥m+n−l

a

b

m+n−l

a+b+1

,

µ ¶
m
m!
Bm is the Bernoulli number, and=.
aa!(m − a)!
Furthermore, for an odd prime p, define½(−1)n+n ,if n is a quadratic residue mod p;
en =
n+n+1
(−1)
, if n is a quadratic nonresidue mod p,
where n is the multiplicative inverse of n modulo p such that 1 ≤ n ≤ p − 1.
We show that the sequence {en } is a “good” pseudorandom sequence, byusing the properties of exponential sums, character sums, Kloosterman sums
and mean value theorems of Dirichlet L- functions.
Zhiguo Liu; East China Normal University, Chinae-mail: zgliu@math.ecnu.edu.cn
Title: A Theta Function Identity to the Quintic Base
Abstract: We prove an identity connecting a theta function and a sum ofEisenstein series by using the complex variable theory of elliptic functions
and one theta function identity of S. McCullough and L.-C. Shen, whichcontains as a special case a famous identity of Ramanujan connected with16

partitions modulus 5. This identity allows us to develop a theory for Eisenstein series to the quintic base, which is much simpler than that of Chanand Liu [2006, Pacific J. Math 226, 53-64]. By the way, we find a recurrence
relation for the Dirichlet series L(s, χ5 ) at even points, which can be used
to compute the special values of L(s, χ5 ) . An identity for the Bernoulli
numbers is also derived. Finally, combining this identity with two identities
in Ramanujan’s lost notebook, we set up a curious Eisenstein series identity
related to the Rogers-Ramanujan continued fraction.
Guangshi Lü; Shandong University, Chinae-mail: gslv@sdu.edu.cn
Title: Some Results in Classical Analytic Number Theory
Abstract: In this talk I want to report one new estimate for exponential
sums over primes in short intervals and its several applications. These
results are as good as what one can obtain from the Generalized RiemannHypothesis.
Kohji Matsumoto; Nagoya University, Japane-mail: kohjimat@math.nagoya-u.ac.jp
Title: The Riesz Mean of the Convolution Product of von Mangoldt Functions and the Related Zeta-function
Abstract: Let Λ(n) be the von Mangoldt function, that is, Λ(n) = log p if
n is a power of a prime p and Λ(n) = 0 otherwise. The functionXΛ(k)Λ(m)G2 (n) =k+m=n

is interesting in connection with the famous conjecture of Goldbach, and has
been studied by Hardy-Littlewood andP Fujii. Fujii proved an asymptotic
formula for the summatory function n≤x G2 (n), under the assumption of
the Riemann hypothesis (RH).
In this talk, I will report some results, proved in a recent joint work with
Shigeki Egami, on the Dirichlet series∞X
G2 (n)
Φ2 (s) =.
snn=1This series can be regarded as a kind of double zeta-functions, and theMellin-Barnes integral formula can be applied. We show (under RH) that
Φ2 (s), originally convergent absolutely in <s > 2, can be continued meromorphically to <s > 1. Moreover, under certain plausible conjectures, weshow that <s = 1 is the natural boundary of Φ2 (s).
17As an application, we prove an asymptotic formula for Riesz means of
G2 (n), whose error estimate is probably best-possible.
Katsuya Miyake; Waseda University, Japane-mail: miyakek@aoni.waseda.jp
Title: Twists of Hessian Elliptic Curves and Cubic Fields
Abstract: In 1840’s Hesse investigated plane curves in a series of papers,
and found an interesting family of elliptic curves, Hµ : U 3 + V 3 + W 3 =3µU V W , with points of order 3 on the projective plane P2 (U : V : W ). Ourconcern in this talk is to introduce twists of the curves with µ in the rational
number field Q over quadratic fields, cubic fields and the Galois closures ofthe latters. We utilize a cubic polynomial, R(t; X) := X 3 + tX + t, t ∈Q − {0, −27/4}, to parametrize all of quadratic fields and cubic ones; we
exclude 0 and −27/4 as values of t to save R(t; X) from multiple roots;
indeed, its discriminant is −t2 (4t + 27). Let ξ be a root of R(t; X) = 0 in
the complex number field C and put Kt = Q(ξ); if R(t; X) = 0 has a root r
in Q, then we pick it up as ξ = r and have Kt = Q. Let K̃t be the splitting
field of the cubic polynomial R(t; X) over Q. If t runs over all such rational
values as Kt = Q, then K̃t covers all quadratic fields beside Q.
First we define a family H̃(µ, t), µ, t ∈ Q, of curves of genus 1 which is a
twist of Hµ as an algebraic curve over the splitting field K̃t . Put


0 1 0
Ξ =  0 0 1−t −t 0and M (x, y, z) := x13 + yΞ + zΞ2 where 13 is the unit matrix of size 3. Then
we define our H̃(µ, t) by
H̃(µ, t) : Tr(M (x, y, z)3 ) = 3µDet(M (x, y, z))where Tr and Det are the trace and the determinant of matrices, respectively. In general, the curve H̃(µ, t), though defined over Q, may not have
any rational points over Q. If Kt = Q, however, it has at least one rational
point over Q, and hence is an elliptic curve defined over Q; it is a quadratic
twist of Hµ if K̃t is a quadratic field.
We also show that the curve H(µ, t) : 2u3 + 6d0 uv 2 + w√3 = 3µ(u2 −d0 v 2 )w, d0 = −(4t + 27), is a quadratic twist of Hµ over Q( d0 ) and that
it is also isomorphic to H̃(µ, t) over Kt .
In the special case of µ = 0 we can show
Proposition 1. Suppose that the cubic polynomial R(t; X) is irreducible
over Q for t ∈ Q − {0, −27/4}. Then the curve H̃(0, t) has a rational point
over Q if and only if t = h3 /(2h − 1)2 for some h ∈ Q − {0, 1/2}.
18

We have some one parameter subfamilies of H̃(µ, t) which have rational
points over Q.
Proposition 2. Suppose that the cubic polynomial R(t; X) is irreducible
over Q for t ∈ Q − {0, −27/4}. Then H̃(µ, t) has a rational point over Q
in the following three cases, (1) t = 3(µ − 1), (2) t = (1 − µ)/3, and (3)t = 3(1 − µ)/2.
In deed, it is clear that, for each t ∈ Q − {0, −27/4}, we have infinitely
many such µ as H̃(µ, t) has a rational point over Q, and vice versa.
Leo Murata; Meiji-Gakuin University, Japane-mail: leo@gen.meijigakuin.ac.jp
Title: On a Property of the Multiplicative Order of a (mod p)
Abstract: (Joint work with Koji Chinen and and C. Pomerance) For a
fixed natural number a ≥ 2, let Da (p) denote the residual order of a in
(Z/pZ)∗ , P and N be the set of all prime numbers and the set of all natural
numbers, respectively. It is known that the map Da is (almost) surjectivefrom P to N, with the property ”|{Da−1 (n)}| is always finite”. In order tostudy the property of the function Da more closely, here we consider two
types of sets :Qa (x; k, l) = {p ≤ x; p ∈ P, Da (p) ≡ l(mod k)},0 ≤ l < k ∈ N,

M2 (x) = {p ≤ x; p ∈ P, D2 (p) ∈ P}.
Under the assumption of Generalized Riemann Hypothesis, for any residue
class in N, Qa (x; k, l) has the natural density ∆a (k, l), and this value is effectively computable (joint work with K. Chinen). We discuss about somenumber theoretical properties of ∆a (k, l) as a number theoretical functionof k and l.
And, as for the second set, we obtain an estimate |M2 (x)| ¿ x(log x)−2
which seems to be a best possible estimate.
Kenji Nagasaka; Hosei University, Japane-mail: JZH04324@nifty.com
Title: Density, Probability and Applications
Abstract: (Joint work with Greco Grecos, Rita Juliano) Densities are, generally speaking, useful tools to solve or to represent mathematical problems,
especially to number theoretical problems. Without Schnirelemann densitiy
notion, which, at a glance, seems curiously defined, additive number theory,in particular, the theory of base, would not develop as the present situation.
We would like to review several densities, and how they are applied andefficient.
19

We also report the definition of probabilities, that, indeed, is able to consider a kind of density, and several relating problems, such as, Solobay’s
measurable cardinals, Banach-Tarski’s paradox, occurring events of probability zero, etc. We conclude our presentation by returning Benford’s law
and Steinhauss’ Problem.
Yoshinobu Nakai; Yamanashi University, Japane-mail: nakai@edu.yamanashi.ac.jp
Title: TBA
Abstract: TBA
Xiumin Ren; Shandong University, Chinae-mail: xmren@sdu.edu.cn
Title: Estimates of Exponential Sums over Primes and Applications in
Number Theory
Abstract: Exponential sums appear naturally in number theory. Nontrivial estimates of such sums lead to various equi-distribution theorems, aswell as solutions to problems in additive number theory. For example, thesolution of the famous Waring-Goldbach problem depends on estimates of
exponential sums over primes. A celebrated example is Vinogradov’s proofof the ternary Goldbach conjecture.
In this talk, by using different methods, we will give such estimates for
exponential sums over κ-th powers of primes, where κ is a positive integer orκ ∈ (0, 1). Our theorem covers that of Vinogradov in the case 0 < κ ≤ 1, andare new in all the other (nonlinear) cases. Applications of these estimatesin number theory will be illustrated.
Andrzej Schinzel; Polish Academy of Sciences, Polande-mail: A.Schinzel@impan.gov.pl
Title: The Number of Solutions in a Box of a Linear Homogeneous Congruence
Abstract: Theorem. Let ai , bi (i = 1, . . . , k) be integers, bi > 0, n =lQαqj j , qj distinct primes, αj > 0. If
j=1

l
X
1
j=1

qj

≤1+

min{l, 2l − 5},n20then the number of solutions of the congruence a1 x1 +. . .+ak xk ≡ 0 (mod n)kQsuch that 0 ≤ xi ≤ bi is at least 21−n (bi + 1).
i=1

Igor Shparlinski; Macquarie University, Australia
e-mail: igor@ics.mq.edu.au
Title: Exponential and Character Sums with Combinatorial Sequences
Abstract: We give a survey of recent results, obtained in collaboration
with Moubariz Garaev and Florian Luca, on bounds of exponential andcharacter sums with various sequences of combinatorial interests such asµµ ¶¶∞
2n• middle binomial coefficients B =,µµ ¶¶n∞ n=1
12n• Catalan numbers C =
,
n+1 n
n=1
• factorials F = (n!)∞
n=1 .
More precisely, given a sequence U = (un ) and a prime p, we define
exponential sumsNX

χ(un ),

n=1

M X
N
X

exp(2πiaum un /p),NX

χ(un )

n=1

m=1 n=1

with an integer a relatively prime to p and a nontrivial multiplicative character χ modulo p.
We present several recent estimates obtained for these sums with theabove sequences. For example, we prove thatNX

χ(un ) = O(N 3/4 p1/8 (log p)1/4 ),1 ≤ N < p,n=1where U = (un ) is one of the sequences B, C or F. which is nontrivial forN ≥ p1/2+ε . We also discuss some results about the number of solutionsof various congruences with these sequences B, C and F which could be of
independent interest.
Finally we indicate some applications, for example, to the Waring problem
with factorials and outline some open problems.
Masaki Sudo; Seikei University, Japane-mail: sudo@ge.seikei.ac.jp
Title: On the Exponential Equations ax − by = c (1 ≤ c ≤ 300)
Abstract: M. A. Bennett obtained the following theorem in 2001. If a, b, c
are positive integers with with a, b ≥ 2 and 1 ≤ c ≤ 100, then the equation21

ax − by = c has at most one solution in positive integers x and y except theten exceptional cases. Now we will study the same problem in the case of
1 ≤ c ≤ 300. We follow Bennett’s method.
Zhi-Wei Sun; Nanjing University, Chinae-mail: zwsun@nju.edu.cn
Title: Curious Identities and Congruences Involving Bernoulli Polynomials
Abstract: In this talk we first tell the story how the developments ofsome curious identities concerning Bernoulli polynomials finally led to thefollowing unified symmetric relation (due to Z. W. Sun and H. Pan): If n
is a positive integer, r + s + t = n and x + y + z = 1, then we have
µ
¶
µ
¶
µ
¶
s t
t r
r s
r
+s
+t
x=0
x y n
y z n
z x n
where

µ

s t
x y

¶
:=
n

n
X
k=0

µ ¶µ
¶
s
t
(−1)
Bn−k (x)Bk (y).
kn−kkLet p be a prime, and let n > 0 and r are integers. We will also talkabout some congruences involving the Felck quotient
X µn ¶−b(n−1)/(p−1)c
Fp (n, r) := (−p)(−1)k ∈ Zkk≡r ( mod p)

and Bernoulli polynomials, obtained by p-adic methods. Finally we mention some related applications to Stirling numbers of the second kind andhomotopy exponents of special unitary groups.
Yoshio Tanigawa; Nagoya University, Japane-mail: tanigawa@math.nagoya-u.ac.jp
Title: Kronecker’s Limit Formula and the Hypergeometric Function
Abstract: We show a new proof of Kronecker’s (first) limit formula using
the hypergeometric function. First, we divide the Epstein zeta function into
two parts. Each part can be written by the integral of the hypergeometricfunction,but the argument of one part lies in a branch different from thestandard one. To treat this part, we make use of the connection formula ofthe hypergeometric function.
Masayuki Toda; Kinki University, Japane-mail: kty@stingray.fsci.fuk.kindai.ac.jp22Title: On Gauss’ Formula for ψ and Finite Expressions for the L-series at
1
Abstract: We show that Gauss’ closed formula for the values of the digammafunction at rational points is equaivalent to the finite expression for theL(1, χ), which appears in the class number formula for quadratic fields. We
also give several equivalent expressions for N (q) (Theorem 2) introduced by
Lehmer.
Haruo Tsukada; Kinki University, Japane-mail: tsukada@fuk.kindai.ac.jp
Title: A General Modular Relation Associated with the Riemann Zetafunction
Abstract: In this talk, we would like to present a modular relation of an
extremely general form, associated with the Riemann zeta-function. This
formula includes the Bochner’s formula, the Riesz sum formula, a K-Bessel
expansion, the incomplete Gamma expansion, and a J-Bessel expansion as
special cases, and enables us to treat wide variaties of similar formulas in asystematic way.
Isao Wakabayashi; Seikei University, Japane-mail: wakabayashi@st.seikei.ac.jp
Title: Some Thue Equations and Continued Fractions
Abstract: We consider the parameterized families of cubic and quartic
Thue equations
bx3 − ax2 y − (a + 3b)xy 2 − by 3 = k,bx4 − ax3 y − 6bx2 y 2 + axy 3 + by 4 = k.
We solve these equations when k is small and a is sufficiently large comparedwith b. We use Pad’e approximation method to bound from above the sizeof the solutions (x, y).
Daqing Wan; University of California, USAe-mail: dwan@math.uci.edu
Title: L-Functions of Infinite Symmetric Powers
Abstract: This is an expository lecture on L-functions of function fields.
For a finite dimensional l-adic representation, the L-function is theoreticallywell understood by celebrated works of Grothendieck and Deligne. Our
goal here is to understand the p-adic variation and its limit (if exists) of
the sequence of L–function of the k-th symmetric power of a geometric23

representation as k goes to infinity but converges p-adically to a p-adic
integer. In the universal elliptic curve example, the p-adic limit is essentiallythe theory of overconvergent p-adic modular forms. In general, it plays anessential role in the proof of Dwork’s conjecture. It should relate closely
to the conjectual theory of p-adic automorphic forms which is yet to be
developed.
Trevor D. Wooley; University of Michigan, USAe-mail: wooley@umich.edu
Title: Waring’s Problem in Function Fields
Abstract: (Joint work with Yu-Ru Liu (Waterloo)) Let F be a finite fieldof characteristic p, let K denote the polynomial ring F [t], and write J[t] forthe additive closure of the set of kth powers of polynomials in K. Define
Gq (k) to be the least integer s satisfying the property that every polynomial
in J[t] of sufficiently large degree admits a strict representation as a sumof s kth powers. We employ a version of the Hardy-Littlewood method
involving the use of smooth polynomials inorder to establish a bound of the
shape Gq (k) ≤ Ck log k + O(k log log k)). Here, the coefficient C is equal to1 when k < p, and C is given explicitly in terms of k and p when k > p,but in any case satisfies C ≤ 4/3. There are associated conclusions for thesolubility of diagonal equations over K, and for exceptional set estimates in
Waring’s problem.
Honggang Xia; Shandong University, Chinae-mail: xia@math.ohio-state.edu
Title: On Zeros of Cubic L-functions.
Abstract: We study the distribution of zeros of cubic L-functions andobtain a zero density theorem by large sieve method. By Patterson’s workon cubic Gauss sums, we get an estimation on the moments of corresponding
L-functions and result of class numbers of a certain type of number fields
follows as an application.
Yangbo Ye; University of Iowa, USAe-mail: yey@math.uiowa.edu
Title: A New Bound for Rankin-Selberg L–functions
Abstract: This talk will report a subconvexity bound for L(1/2, f × g)in the weight eigenvalue aspect of f , recently proved by Yuk-Kam Lau,
Jianya Liu, and the speaker. The new bound is O(k 2/3+² ), while the trivial, convexity bound in this case is O(k). This result is believed to have24

reached the second barrior, after the convexity bound, toward the Lindelof Hypothesis. Its applications include an estimate for the fourth power
moment of L(1/2 + it, g) over a short interval of t. The techniques usedin the proof include the Kuznetsov trace formula, spectral large sieving,Good’s estimation, spectral analysis and meromorphic continuation of ashifted convolution sum, and weighted stationary phase.

Wenguang Zhai; Shandong Normal University, Chinae-mail: wgzhai@163.com
Title: On the Fourth Power Moments of ∆(x) and E(t)Abstract: Let ∆(x) and E(t) denote the error term in the Dirichlet divisor
problem and the error term in the mean square of ζ(1/2 + it), respctively.
It is an important problem in the analytic number theory to study thepower moments of these two error terms. In this talk, we will give two
new results about the fourth power moments of these two functions, whichfurther improve previous results.

Deyu Zhang; Shandong University, Chinae-mail: zdy− 78@mail.sdu.edu.cn
Title: Zero Density Estimates for Automorphic L-functions
Abstract: In this talk, I will give zero density
Nestimates of the large sieve
type for the automorphic L-functions L(s, fχ), where f is a holomorphic cusp form for the group SL2 (Z) and χ(modq) is a primitive Dirichlet
character.

Shou-Wu Zhang; Columbia University, USAe-mail: szhang@math.columbia.edu
Title: Periods Integrals and Special Values of L-series
Abstract: I will survey some basic results about periods integrals of automorphic forms and their relations with special values of L-series.

Wenpeng Zhang; Northwest University, Chinae-mail: wpzhang− nwu@126.com
Title: The Mean Value of Dedekind Sum and Cochrane Sum in Short
Intervals
25

Abstract: For a positive integer q and an arbitrary integer h, the classical
Dedekind sum S(h, q) and Cochrane sum C(h, q) are defined by¶¶q µµ ¶¶ µµXaah
S(h, q) =
,
q
q
a=1
where
1
((x)) = x − [x] − , if x is not an integer; ((x)) = 0 if x is an integer,2¶¶q µµ ¶¶ µµX0aahC(h, q) =,qq
a=1
where a is defined by the equation aa ≡ 1 mod q andqX0

denotes thea=1summation over all 1 ≤ a ≤ q such that (a, q) = 1. What is the meanvalue distribution properties of the Dedekind sum or Cochrane sum in ashort interval [1, λp], where λ ∈ (0, 1). In fact, it is very difficult to get a
asymptotic formula for the mean value in the case of one variety as
X
X
S(a, p)andC(a, p).
a< p4

a< p4

The main purpose of this paper is to study a special kind of mean value
properties of the Dedekind sum and Cochrane sum by using the mean value
theorems of the Dirichlet L-functions, and prove the following results:XX¡¢
3
S(ab̄, p) = p2 + O p1+²64
ppa< 4 b< 4

andXX
a< p4 b< p4

C(ab, p) =

¡
¢
45 2
p + O p1+² .
1024

26