1. Hiroki Aoki
University:
Tokyo University of Science
E-mail:
aoki_hiroki@ma.noda.tus.ac.jp
Talk title:
On the convergence of formal Fourier Jacobi expansion
Abstract: It is well known that a Siegel modular form has a Fourier Jacobi expansion. Each
coefficient of this expansion is a Jacobi form. Namely, each coefficient satisfies the translation
formula of Jacobi forms. It is also well known that the translation formula of Siegel modular
forms comes from the translation formula of Jacobi forms and the involution. The involution is
very simple formula we can check easily. We will discuss about the convergence of formal Fourier
Jacobi expansion which satisfies the involution.
2. Valentin Blomer
University:
University of Goettingen
E-mail:
blomer@uni-math.gwdg.de
Talk title:
Smooth values of binary cubic forms
Abstract: Let F be an integral binary cubic form. We are interested in the least value of y such
that a positive proportion of values F(a, b) with a, b not exceeding x have no prime factor
exceeding y. We show that y = x^a is admissible for a > 0.61 in general and for any a > 0 if F is
reducible. The proof uses various sieve methods. This is joint work with A. Balog, C. Dartyge and
G. Tenenbaum.
3. Jorg Brudern
University:
University of Gottingen
E-mail:
bruedern@uni-math.gwdg.de
Talk title:
Incomplete convolutions
Abstract:
We consider the class of multiplicative functions with values in the unit disc, and
study the maximal function of the sum over the small divisors of these functions. A moment
estimate for these maximal functions will be proved that gives new insight, for example, in the
value distribution of the Moebius function
4. Tianxin Cai
University:
Zhejiang University
E-mail:
txcai@zju.edu.cn
Talk title:
A new kind of Waring’Problem
Abstract:
In this paper， we raise a new kind of Waring’s Problem, i.e., n= x_1+…+x_s，with
the product x_1…x_s=x^k, which is the generalization of Waring’s Problem, similarly we define
g’(k),G’（k）…….
5. Yingchun Cai
University:
Tongji University
E-mail:
yingchuncai@tongji.edu.cn
Talk title:
Some results in additive number theory
Abstract: With the developments of analytic number theory, especially the sieve methods and the
circle methods, many achievements occurred and much more work remain to be done. In this talk
I will report some results in additive number theory obtained by the speaker.
6. Shichao Chen
University:
Henan University
E-mail:
schen@henu.edu.cn
Talk title:
The partition function and modular forms
Abstract: We introduce some resluts on the distribution of the ordinate partiiton function p(n)
mod prime powers via the theory of modular forms.
7. Yong-Gao Chen
University:
Nanjing Normal University
E-mail:
ygchen@njnu.edu.cn
Talk title:
On representation functions
Abstract: Let $\mathbb{N}$ be the set of all nonnegative integers. For a set $A\subseteq
\mathbb{N}$, let $R_1(A,n)$, $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions of
$a+a'=n, a,a'\in A$; $a+a'=n, a,a'\in A,a<a'$ and $a+a'=n, a,a'\in A,a\leqslant a'$ respectively. We
will present recent progress on representation functions $R_1(A,n)$, $R_2(A,n)$ and
$R_3(A,n)$ and related research.
8. Shuichi Hayashida
University:
Osaka University
E-mail:
hayashida@math.sci.osaka-u.ac.jp
Talk title:
Cohen type Eisenstein series of degree two and a relation among Fourier
coefficients
Abstract:
Eisenstein series is important for study of modular forms. Siegel-Eisenstein series
of half-integral weight of degree two was investigated by Y.Tanigawa in 1986. He obtained
certain relations among Fourier coefficients of this series.
In this talk I would like to show the same relation for generalized Cohen type Eisenstein series
of degree two, which is a certain Siegel modular form of half-integral weight. Our result contains
also a relation for p=2. Usually p=2 is a bad prime for modular forms of half-integral weight.
However, generalized Cohen type Eisenstein series is essentially "level 1". So, this form has a
good property also for p=2.
9. Shigeru Kanemitsu
University:
Kinki University
E-mail:
omnikanemitsu@yahoo.com
Talk title:
Some applications of circulant matrices
Abstract: In this talk, we are going to present some results obtained in a joint paper with Jing,
Kaida and Waldshcmidt about applications of circulant matrices, esp. to the Blahut theorem, Lin.
Feedback shift reg. and top. Matrices for carbon-hydrites.
10. Koichi Kawada
University:
Iwate University
E-mail:
kawada@iwate-u.ac.jp
Talk title:
Some additive problems with smooth numbers
Abstract: This talk shall concern some additive problems involving smooth numbers (or,
numbers without larger prime factor). In particular, we discuss sums of cubes of smooth numbers.
11. Angel Kumchev
University:
Towson University
E-mail:
akumchev@towson.edu
Talk title:
Sums of almost equal squares of primes
Abstract:
I will report on some recent joint work with Taiyu Li on representations of
integers as sums of three, four and five almost equal squares of primes. For example, we show that
every sufficiently large integer $n \equiv 5 \!\! \mod{24}$ can be represented as a sum $p_1^2 +
\dots + p_5^2$, where the $p_i$'s are primes with $|p_i - (n/5)^{1/2}| < n^{\theta}$ and $\theta <
9/20$, thus improving on earlier work by Liu, L\"u and Zhan. We also obtain several estimates for
the exceptional sets for similar representations as sums of three and four almost equal squares of
primes. In those cases, some of our results improve on earlier work and some appear to be new.
12. Yuk-Kam Lau
University:
The University of Hong Kong
E-mail:
yklau@maths.hku.hk
Talk title:
Average values of Fourier coefficients of holomorphic primitive forms in
arithmetic progressions
Abstract: Let a(n) be the n-th Fourier coefficient of a normalized holomorphic Hecke eigenform
(or a primitive form) for the full modular group. Denote by S(X,q,b) the summation of a(n) where n
runs over b,b+q,b+2q,… and does not exceed X. We shall discuss the variance of the value of
S(X,q,b) , i.e. the sum of the square of S(X,q,b) over b=1,2,…,q.
13. Wenzhi Luo
University:
The Ohio State University
E-mail:
wluo@math.ohio-state.edu
Talk title:
On the distribution of integer points on spheres
Abstract:
In this talk, we apply the works of Boecherer and Schulze-Pillot on the Yoshida lift
to evaluate asymptotically the variance of the Linnik distribution, which involves the spherical
harmonics and makes use of the Jacquet-Langlands correspondence. We show the variance is
proportional to the scalar product twisted by certain critical L-values.
14. Guangshi Lv
University:
Shandong University
E-mail:
gslv@sdu.edu.cn
Talk title:
General divisor problems related to certain L-functions
Abstract: The estimation of the coefficients of L-functions is studied extensively in number
theory. In this talk we shall introduce some recent progress on the general divisor problems related
to the Hecke L-function associated with a cusp form, the Dedekind zeta function of an algebraic
number field, and the Epstein zeta function for a positive definite quadratic form.
15. Katsuya Miyake
University:
Tokyo Metropolitan University
E-mail:
miyakek@bz-csp.tepm.jp
Talk title:
The Twists of Hessian Elliptic Curves over Splitting Fields of Cubic
Polynomials and the related Elliptic 3-Folds
Abstract: We constructed a twist $\tilde{H}(\mu, t)$ of Hesse's elliptic curve $H_{\mu} : U^3 +
V^3 + W^3 = 3\mu UVW, \mu \in \mathbb{Q} - \{1\}$ over the splitting field of the cubic
polynomial $R(t; X) := X^3 + tX + t, t \in \mathbb{Q} - \{0, -27/4 \}$ as an algebraic curve. It
may not, however, have any rational point over $\mathbb{Q}$.
Our main aim of the talk is to give a necessary and sufficient condition for the twist to have a
rational point over $\mathbb{Q}$, that is, to be an elliptic curve over $\mathbb{Q}$.
Our method does not depend on the specific values of the parameters $\mu$ and $t$ in
$\mathbb{Q}$. Hence we are able to give an elliptic 3-fold over an affine plane from
$\tilde{H}(\mu, t)$ under the essential part of our sufficient condition. In case where the splitting
field is cyclic, we also give a necessary and sufficient condition for the twist to have a rational
point over $\mathbb{Q}$, and give an elliptic 3-fold over an elliptic surface in this case.
16. Ryotaro Okazaki
University:
Doshisha University
E-mail:
rokazaki@mail.doshisha.ac.jp
Talk title:
Weber's class number problem and Pohst's lower bound on units
Abstract: Let $V_n = athbf{Q}(2 os(2 i/2^{n+2}))$,
which is the $n$-th layer of the
cyclotomic $ athbf{Z}_2$- extension of $ athbf{Q}$. Weber asked if the class number
$h_n$ of $V_n$ is always equal to $1$. This problem is an important variant of Gauss conjecture
on class numbers of real quadratic fields. For a real quadratic field, its units are important. In
short, a real quadratic field with class number $1$ should not have small units.
Excluding small units is also important for Weber's class number problem. There are several
ways of measuring the size of units. Horie used the house function of Mahler, which is estimated
from below by Kronecker's Theorem. The Mahler measure is often applied in Diophantine
problems. In recent years, Schinzel's lower bound for the Mahler measure turned out useful.
Pohst's lower bound is a similar lower bound for ``$L_2$''-norm of logarithms of the unit. It
has been applied in algebrac number theory, and particularly in computation of invariants of
algebraic number fields.
In this talk, the speaker explains how to apply Pohst's lower bound in Weber's class number
problem.
17. Hao Pan
University:
Nanjing University
E-mail:
haopan79@yahoo.com.cn
Talk title:
The additive complements of primes and Goldbach's problem
Abstract: We extend two results of Ruzsa and Vu on the additive complements of primes.
18. Hourong Qin
University:
Nanjing University
E-mail:
hrqin@nju.edu.cn
Talk title:
CM Elliptic Curves and Quadratic Polynomials Represent Primes
Abstract: We will talk about the relationship between the Local $L$-series of CM Elliptic
Curves and the Hardy-Littlewood Conjecture.
19. Zhi-Wei Sun
University:
Nanjing University
E-mail:
zwsun@nju.edu.cn
Talk title:
Some of my number-theoretic conjectures and related progress
Abstract: In recent years the author raised many number-theoretic conjectures some of which
involve representations of integers, binary quadratic forms, congruences for sums of binomial
coefficients, and series for 1/π. In this talk we will introduce my main conjectures and related
progress.
20. Yoshio Tanigawa
University:
Nagoya University
E-mail:
tanigawa@math.nagoya-u.ac.jp
Talk title:
On the general divisor problems and the sum of Chowla and Walum
Abstract: We study the relation between the discrete and the continuous mean value of
¥Delta_a^2(x), where ¥Delta_a(x) is the error term in the general divisor problems. We try to find
this formula in a sufficiently explicit form. As an application, we give the formula of the mean
value in the range -1<a<1. We also study the integral containing the error term in the weighted
two-dimensional divisor problems.
21. Isao Wakabayashi
University:
Seikei University
E-mail:
wakaba.isao@gmail.com
Talk title:
Number of solutions of simple Thue equations of degree 4
Abstract: Let a and b be non-zero integers, and put F(x,y)=bx^4-ax^3y-6bx^2y^2+axy^3+by^4.
We prove that the number of integer solutions of the Thue equation F(x,y)=1,-1 is 0 or 4 except for
a few cases. For the proof, we suppose that the equation has a solution, and we use Pade
approximation method in order to obtain an upper bound for the second solution. In order to
obtain a lower bound for the second solution, we use continued fractions with rational partial
quotients.
22. Sardar Mohib Ali Khan
University:
University of Education, Lahore
E-mail:
ali.uno@gmail.com
Talk title:
On a selberg type divisor problem associated with cusp forms
Abstract: In this 10 minute talk, we will present results obtained in a joint paper with S.
Kanemitsu and A. Laurincikas. We have considered the Selberg type divisor problem for a pair of
cusp forms. We form the Dirichlet convolution of the coefficients of cusp forms and present an
asymptotic formula for the summatory function.
23. Nianliang Wang
University:
ShangLuo University
E-mail:
wangnianliangshangluo@yahoo.com.cn
Talk title:
On the short-interval character sums with polynomial Weight
Abstract: In this short talk, we shall establish the counterpart of Szmidt, Urbanowicz and
Zagier’s formula in the sense of the Hecker correspondence. The motivation is the derivation of
the values of the Riemann zeta-function at positive even integral arguments from the partial
fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last
is equivalent to the functional equation, we may view their elegant formula as one for the Lambert
series, and comparing the Laurent coefficients, we may give a functional equational approach to
the short-interval character sums with polynomial weight.
24. Trevor.D.Wooley
University:
University of Bristol
E-mail:
matdw@bristol.ac.uk
Talk title:
Vinogradov’s mean value theorem via efficient congruencing
Abstract: Exponential sums of large degree play a prominent role in the analysis of problems
spanning the analytic theory of numbers. In 1935, I. M. Vinogradov devised a method for
estimating their mean values very much more efficient than the methods available hitherto due to
Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free
region of the Riemann zeta function, in Diophantine approximation, and in Waring's problem.
Recent applications from the 21st century include sum-product estimates in additive
combinatorics, and the investigation of the geometry of moduli spaces.
Over the past 75 years, estimates for the moments underlying Vinogradov's mean value
theorem have failed to achieve those conjectured by a factor of roughly log k in the number of
implicit variables required to successfully analyse exponential sums of degree k. In this talk we
will sketch the ideas underlying our recent work which comes within a stone's throw of the best
possible conclusions. Some applications will be addressed, as well as some of the latest
developments.
25. Yangbo Ye
University:
The University of Iowa
E-mail:
yangbo-ye@uiowa.edu
Talk title:
Resonance of a Maass form for SL(3,Z)
Abstract: Let $f$ be a Maass cusp form for $SL(3,\Bbb Z)$ with Fourier coefficients
$A_f(m,n)$. Consider the sum $\sum_{X<n\leq 2X}A_f(m,n)e(\alpha n^\beta)$ and its smoothly
weighted version $\sum_{n>0}A_f(m,n)\phi(n/X)e(\alpha n^beta)$, where $\phi\in
C_c^\infty(0,\infty)$. Bounds for them will be proved for various $\alpha$ and $\beta$. These
bounds manifest a strange vibration nature of Maass forms.
26. Wenguang Zhai
University:
China University of Mining and Technology(Beijing)
E-mail:
zhaiwg@hotmail.com
Talk title:
Sign changes of the error term in Weyl‘s law for the Heisenberg manifolds
Abstract:
Let R(t) denote the error term in the Weyl's law of the eigenvalues of the
Laplacian on the Heisenberg manifold with rational meric. Two results on sign changes of R(t) are
established， which are analogous to corresponding results of the error term of the Gauss circle
problem.
27. Haigang zhou
University:
Tongji University
E-mail:
haigangz@tongji.edu.cn
Talk title:
Jacobi forms and Diophantine equations system
Abstract: Determining the exact formulae of the number of ways to represent a number as a sum
of squares is very classical in number theory. In this talk, we will give an exact formula for the
number of Hurwitz quaternion integers with norm n and trace m, which is equivalent to a problem
of Diophantine equations system. As corollaries, we obtain a formula of the number of solutions
of Diophantine equation system with a quadratic equation and a linear equation and obtain Gauss’s
famous formula of sums of three squares again. The tool we use here is the theory of Jacobi forms.