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\begin{document}
\baselineskip=17pt
\title{Abstract of talks to be presented at the 5th Japan-China seminar, Aug. 27-31, 2008, Kinki University,
Higashi-osaka, Japan}
\author{   }
\date{}

\maketitle

\author{Professor Chaohua Jia (Academia Sinica)}

\title{\bf{Recent Progresses on Dynamics of a Special
Arithmetic Function}}

\begin{abstract}
Let $P(n)$ denote the largest prime factor of integer
$n>1$ and $p_i$ denote the prime number. For $n=
p_1p_2p_3$, we define the $w$ function by
$$
w(n)=P(p_1+p_2)P(p_1+p_3)P(p_2+p_3).
$$

In 2006, Wushi Goldring proposed some problems on 
dynamics of the $w$ function and made some conjectures.
In this talk, the speaker will introduce some recent
progresses on dynamics of  the $w$ function and 
relative study.


\end{abstract}

\author{Akinari Hoshi and Katsuya Miyake (Waseda University)}

\title{\bf Some Diophantine problems arising from the isomorphism
problem of generic polynomials}

\begin{abstract}
Let k be a field and G a finite group. A k-generic polynomial for G covers all G-Galois extensions
over K (�� k) by specializing parameters.
We study the field isomorphism problem of k-generic polynomials for various G's to determine
the 'moduli' of G-Galois extensions given by them over K and announce some results of the cubic case.
In cubic and quartic cases over the rational number field with the least possible number of parameters,
we see by Hilbert's Irreducible Theorem that the generic polynomials give the same number fields
for infinitely many specialized rational values of the parameters and also by Siegel's Theorem
that there are only finitely many integral values of the parameters which may give the same number fields.
We will also show a correspondence between integral solutions of some parametrized Thue equations
and isomorphism classes of Shanks' simplest cubic fields. 
\end{abstract}








\author{Professor Wenguang Zhai (Shandong Normal University)}

\title{\bf{On the error term in Weyl's law for the Heisenberg manifolds}}

\begin{abstract} 
For a fixed integer $l\geq 1$  , let $R(t)$ denote
the error term in the Weyl's law of  a $(2l+1)$-dimensional
Heisenberg manifold with the  metric $g_l.$ In this paper we shall
prove the asymptotic formula of the $k$-th power moment for any
integers $3\leq k\leq 9.$ We shall also prove that the function
$t^{-(l-1/4)}R(t)$ has a distribution function.
\end{abstract}

\author{Professor Trevor Wooley (University of Bristol)}

\title{\bf{Progress on Diophantine problems in function fields}}

\begin{abstract}
Let k be a finite field of characteristic p. We discuss
what is known concerning Diophantine problems involving dth
powers in k[t], and in particular analogues of Waring's problem.
Much, but not all, of this work rests on a version of the
Hardy-Littlewood (circle) method applicable to k[t]. Recent progress
 makes use of work joint with Yu-Ru Liu that delivers mean value
estimates for exponential sums over smooth polynomials, and an
analogue of Vinogradov's mean value theorem, in this function
fields setting. We will describe recent progress, and dream a
little of future developments.
\end{abstract}

\author{Professor Zhi-Wei Sun (Nanjing University)} 

\title{\bf{On representations of integers involving triangular numbers}}

\begin{abstract}
Triangular numbers are integers of the form $T_x=x(x+1)/2\ (x\in\Bbb Z)$.
We will talk about some recent problems and results on representations of integers involving triangular numbers.
In particular, we will introduce the recent work of Ben Kane and the speaker on mixed sums of squares and triangular numbers, and the speaker's conjectures
on sums of primes and triangular numbers.
\end{abstract}


Professor Yumiko Hironaka (Waseda University)

\title{\bf{Spherical functions and hermitian Siegel series on certain
homogeneous spaces of $U(n,n)$}}

\begin{abstract}
For each nondegenerate hermitian form $T$, we consider the space $X_T$
which is isomorphic
to $U(n,n)/U(r) \times U(2n-r)$ over the algebraic closed field, where
$r$ is the size of $T$.
Then we study functional equations for spherical functions (with respect
to Borel subgroup of $U(n,n)$) on $X_T$. On the other hand, spherical
functions on $X_T$ with repect to Siegel parabolic subgroup have close
relation to hermian Siegel series via those integral representation.
Hence we have functional equations of hermitian Siegel series from those
of spherical functions.
\end{abstract}

Professor Yoshiyuki Kitaoka (Meijo University)

\title{\bf{A statistical relation of roots of a polynomial in different local fields}}

\begin{abstract}
Let $f(x)$ be a monic polynomial in $Z[x]$.
We observe a statistical relation of roots of $f(x)$ in different
local fields $Q_p$, where $f(x)$ decomposes completely.
Based on it, we propose several conjectures.
\end{abstract}

Professor W.-P. Zhang (North West University)
\title{On the fourth power mean of the character sums over a short interval}
\begin{abstract}
We will prove the asymptotic formula for the discrete fourth power mean
of weighted character sums in an short interval:
$$
S_k=\sum_{\chi \mod q}\left|\sum_{n\le N} n^k \chi(n)\right|^4.
$$

{\bf Theorem}  Let $q > 2$ be an integer and let $k\geq 0$ be any
fixed real number. Then for any real number $N$ with $1 < N <
\sqrt{q}$, we have the asymptotic formula
$$
S_k=\frac{12}{\pi^2}\cdot \frac{\phi^4(q)}{q^3}\cdot
\frac{N^{4k+2}}{(2k+1)^3}\cdot \prod_{p|q}\frac{p}{p+1}\cdot\ln
N+O\left( \phi(q)\cdot N^{4k+2}\cdot 2^{\omega(q)}\right),
$$
where $\phi(q)$ and $\omega(q)$ have their usual meaning and $\displaystyle \prod_{p|q}$ signifies 
the product over all prime divisors of $q$.
\end{abstract}


Professor Yong-Gao Chen (Nanjing Normal University) 

\title{\bf{Fermat numbers and integers of the form $a^k+ a^l+ p^{\alpha}$}} 


\begin{abstract}
In 1849, A. de Polignac conjectured that every odd number larger than 3 can be written as the sum of an odd prime 
and a power of 2. In 1934, N. P. Romamoff proved that the set of positive odd integers 
which can be expressed in the form $2^n + p$ has positive lower asymptotic density, 
where $n$ is a nonnegative integer and $p$ a prime. In 1950, J. G. van der Corput proved 
that the counterexamples of de Polignac's conjecture form a set of positive lower density. 
Using a covering system, P. Erd\"{o}s proved that there is an infinite arithmetic progression of positive odd integers
 each of which has no representation of the form $2(tm) + p$. 

Refining the argument of Erd\"{o}s, R. Crocker proved that there are in-finitely many positive odd integers 
not representable as the sum of a prime and two positive powers of two. P. Z. Yuan proved 
that there are infinitely many positive odd integers not representable as the sum of a prime power 
and two positive powers of two. 
Before this Z, W. Sun and M. H. Le handled the integers of the form $c(2^a + 2^b) + p^{\alpha}$ 
for many values of the constant $c$. 

All these articles are intimate with Fermat numbers $F_n = 2^{2^{n}} + 1$ . 
We do not know whether there are infinitely many Fermat primes and 
whether there are infinitely many Fermat composite numbers. 
On the other hand, we cannot exhibit an odd number of reasonable size which cannot be represented 
as the sum of a prime power and two powers of two. For example we have checked there are no such numbers 
between 3 and $l0^7$. One of our results implies that if the number of odd numbers, 
which are less than $x$ and cannot be represented as the sum of a prime power and two powers of two, is 
$o(\sqrt{x})$, then there are infinitely many Fermat primes. If the number of odd numbers, 
which are less than $x$ and cannot be represented as the sum of a prime power and two powers of two, is 
$O(x^{\frac{1}{4}})$, then all sufiicient large Fermat numbers are primes. 
\end{abstract}

Professor Dr. W. Kohnen (Universitaet Heidelberg)

\title{\bf{Generalized modular functions and their Fourier coefficients}}
\begin{abstract}
Abstract: Generalized modular functions are holomorphic functions on the 
complex upper half-plane that satisfy the usual definition of a modular 
function, except that the character need not be unitary. They occur for 
example in physics in connection with vertex operator algebras. We will show 
(under certain conditions) that they Fourier coefficients have a "bad" 
arithmetic behavior, unless they really are classical modular functions. 
This is joint work with G. Mason. 
\end{abstract}

Professor T.-X. Cai and Dr. Zhou Xia (Jiezhang University)

\title{\bf{Decomposition and its applications to multiple zeta functions}}

\begin{abstract}
Abstract. We introduce the definition of decomposable function and establish its decomposition. 
As applications, we prove several new or known results on reduction and evaluation for various kinds of 
multiple zeta functions, such as Tornheim sums and Witten zeta functions.
\end{abstract}

Professor Andrzei Schinzel (Polska Akadmia Nauk (PAN))
\title{\bf{The number of non-zero coefficients of a polynomial---solved and unsolved}}

Full paper is available on request.

\end{document}
