THE SEVENTH CHINA-JAPAN SEMINARON NUMBER THEORY
Plowing and starring through the high wave forms

Kyushu University
October 28-November 1, 2013

Organized byMasanobu Kaneko, Kyushu University
Shigeru Kanemitsu, Kinki University
Jianya Liu, Shandong University

Supported by
JSPS and NSF of China under the auspices of Kyushu University
Monday, October 28, 2013
Registration, opening and photo session
Address: Ito Guest House, Kyushu University （Moto-oka, Fukuoka,Japan
Monday, October 28, 2013
13:00-13:30 Opening ceremony (Masanobu Kaneko, Shigeru Kanemitsu, andJianya Liu).
• Monday Afternoon Session (Chair: Masanobu Kaneko)
October 28, 13:50-14:30

Lin Weng (Kyushu Univ.)

Motivic Euler product and its applications

October 28,14:50-15:30

Shigeki Akiyama (Tsukuba Univ.)

On the mean divisibility of multinomial sequences
October 28, 15:45-16:00

Tomokazu Onozuka (Nagoya Univ.)

Mean values of Mordell-Tornheim double zeta-functions (a joint workwith Takuya Okamoto)• Tuesday Morning Session (Chair: Katsuya Miyake)
October 29, 10:00-10:40 Yoshiyuki Kitaoka (Meijo Univ.),
The distribution of roots of a polynomial modulo prime powers
October 29, 11:00-11:40

Guangshi Lv (Shandong Univ.),

On Fourier coeﬃcients of automorphic forms• Tuesday Afternoon Session (Chair: Yoshiyuki Kitaoka )October 29, 13:30-14:10 Ryotaro Okazaki (Doshisha Univ.)
Geometry for totally imaginary quartic Thue equation

October 29, 14:30-15:10
Tech.),

Wenguang Zhai (China Univ. of Mining andOn the Dirichlet divisor problem in short intervals

October 29, 15:30-15:45

Tomoya Kiyuna (Kyushu Univ.)

A certain diﬀerential equation for Jacobi forms

October 29, 15:45-16:00

Takahiro Wakasa (Nogoya Univ.)

The explicit upper bound of the multiple integral of S(t) on the Riemann Hypothesis• Wednesday Morning Session (Chair: Jianya Liu)
October 30, 10:00-10:40

Andrzej Schinzel (PAN)

On integer-valued polynomials
October 30, 11:00-11:40 Katsuya Miyake(emeritus professor of Tokyo Metropolitan Univ.)
Complex multiplication in the sense of Abel

• Thursday Morning Session (Chair: Koichi Kawada)
October 31, 10:00-10:40 Kohji Matsumoto (Nagoya Univ.) A joint
composite hybrid strong universality theorem

October 31, 11:00-11:30 Hiroki Aoki (Tokyo Univ. of Sci.) On Jacobi
forms of fractional weight• Thursday Afternoon Session (Chair: Isao Walabayashi) October 31,
13:30-14:10 Shigeki Egami (Shibaura Inst. of Tech.) On analytic
continuation of some Dirichlet series

October 31, 14:30-15:10 Xiumin Ren (Shandong Univ.) Asymptotic
expansions of Voronoi’s summation formulas for GL(m) and applications

October 31, 15:30-16:10 Tianxin Cai (Zhejiang Univ.) Congruent
numbers on the right trapezoid

October 31, 16:20-16:35 Ade Irma Suriajaya (Nagoya Univ.) On thezeros of the second derivative of the Riemann zeta function under theRiemann hypothesis• Friday Morning Session (Chair: Shigeki Egami)
November 1, 10:30-11:10 Zhi-Wei Sun (Nanjing Univ.), Some newproblems involving primes and permutations
November 1, 11:30-11:40

Closing

ABSTRACTS
1. Shigeki Akiyama
University: Tsukuba University
E-mail: akiyama@math.tsukuba.ac.jp
Talk Title: On the mean divisibility of multinomial sequences.
Abstract: A sequence a1 , a2 , . . . of non-zero integers is divisible if an divides akn for a positive
this notion and deﬁne almost∏ninteger k. We generalize∏nk-mean divisibility: i=1 an divides C i=1 akn with a positive constant C.
Then we show that under some condition, sequences deﬁned by multinomial
coeﬃcients enjoy this property although they are not divisible. The proof
depends on an integral inequality which seems to be interesting of its own.
2. Hiroki Aoki
University: Tokyo University of ScienceE-mail: aoki hiroki@ma.noda.tus.ac.jp
Talk title: On Jacobi forms of fractional weight
Abstract: On elliptic modular forms, modular forms of fractional (not halfintegral) weights had been studied by Ibukiyama, Bannnai et al. In this talk,
we introduce Jacobi forms of fractional weights and fractional indices, fromthe view point of structure theorem. If time is allowed, we will also explain
Siegel modular forms of fractional weights.
3. Shigeki Egami

University: Shibaura Institute of Technology
E-mail: egami-sg@shibaura-it.ac.jp
Talk title: On analytic continuation of some Dirichlet series
Abstract: In this talk we discuss the analytic continuation of the following
two types of Dirichlet series: (1) Dirichlet series associated to linear recurrence series. (2) Variants of Hardy-Littlewood-Hecke series.
4. Yoshiyuki Kitaoka
University: Meijo University
E-mail: kitaoka@ccmfs.meijo-u.ac.jp
Talk title: The distribution of roots of a polynomial modulo prime
powersAbstract: Let f be a monic polynomial with integer coeﬃcients. The aim isto state several observations on the distribution of roots of f modulo prime
powers. They give new diﬃcult problems.
5. Guangshi Lü
University: Shandong University
E-mail: gslv@sdu.edu.cn
Talk title: On Fourier coeﬃcients of automorphic forms
Abstract: Fourier coeﬃcients of classical automorphic forms are interestingand important objects in number theory. In this talk, I shall introduce somerecent results on Fourier coeﬃcients of automorphic forms.
6. Katsuya Miyake
University: Emeritus professor of Tokyo Metropolitan University
E-mail: miyakek@bz-csp.tepm.jp
Talk title: Complex multiplication in the sense of Abel
Abstract: Starting from the deﬁnition of the most classical complex multiplication, we launch on expounding modernized one for elliptic curves andAbelian varieties. Also included is some comments from the coming book “Asurvey of elliptic functions”.
7. Ryotaro Okazaki
University: Doshisha University
E-mail: rokazaki@mail.doshisha.ac.jp
Talk title: Geometry for totally imaginary quartic Thue equation
Abstract: Let F be a binary quartic form of integer coeﬃcients. Assume
F has no real linear factors. We consider the Thue equation by equating
F with unity. This is one of the so-called trivial case of Thue equations.

However, the precise universal upper bound on the number of its solution isstill open. In this talk, we try to identify those F with 3 solutions module(x, y) ↔ (−x, −y). This will eventually leads to completion of a project
initiated by Nagell.
8. Kohji Matsumoto
University: Nagoya University
E-mail: kohjimat@math.nagoya-u.ac.jp
Talk title: A joint composite hybrid strong universality theorem
Abstract: After the discovery of the universality property of the Riemannzeta-function by Voronin (1975), the universality theory of zeta and Lfunctions has been studied quite extensively. Various reﬁnements and/orgeneralizations, such as joint universality, composite universality, hybrid universality, and strong universality were introduced and discussed. In this talk
we ﬁrst survey the history of universality theory, especially explain the above
four notions. Then we report a new result, obtained jointly with A. Laurincikas and J. Steuding, which combines all of these four notions.
9. Xiumin Ren
University: Shandong University
E-mail: yangbo-ye@uiowa.edu
Talk title: Asymptotic expansions of Voronoi’s summation formulasfor GL(m) and applications
Abstract: Let m ≥ 3 and f a Maass cusp form for GL(m) with Fouriercoeﬃcients Af (n1 , · · · , nm−1 ). I will talk about Voronoi’s summation formulafor Af (n1 , · · · , nm−1 ), give asymptotic expansions of the formula and outline
some applications.
10. Andrzej Schinzel
University: Polish Academy of Sciences
E-mail: schinzel@impan.pl
Talk title: On integer-valued polynomials
Abstract: This is a survey lecture but it contains new results of mine on ﬁxed
divisors of forms
11. Zhi-Wei Sun
University: Nanjing University
E-mail: zwsun@nju.edu.cn
Talk title: Some new problems involving primes and permutations

Abstract: In this talk we introduce various new problems on primes andpermutations. For example, we conjecture that for any positive integer nthere is a circular permutation a(0), a(1), ..., a(n) of 0, 1, · · · , n such that all
the n+1 adjacent sums a(0)+a(1), a(1)+a(2), · · · , a(n−1)+a(n), a(n)+a(0)are among those integers k with 6k − 1 and 6k + 1 twin primes.
12. Lin Weng
University: Kyushu University
E-mail: weng@math.kyushu-u.ac.jp
Talk title: Motivic Euler product and its applications
Abstract: We begin with a construction of motivic abelian and non-abelian
zeta functions for curves over any base ﬁeld, using moduli stacks of semistable bundles. Based on them, we deﬁne motivic Euler products. As applications, we oﬀer a pair of intrinsic relations between total motivic mass ofprincipal bundles and its semi stable parts, following Harder-Narashimhan,Zagier and Lamoun-Rapoport, with the help of Arthur and Laﬀorgue truncations. If time is allowed, we will also explain how these works are naturally
related with Atiyah-Bott’s question on Tamagawa measures associated tobundles over Riemann surfaces and our work with Zagier on the RiemannHypothesis for non-abelian zeta functions for elliptic curves over ﬁnite ﬁelds.
Short communications
13. Tomoya Kiyuna
University: Kyushu University
E-mail: t-kiyuna@math.kyushu-u.ac.jp
Talk title: A certain diﬀerential equation for Jacobi forms
Abstract: Kaneko and Zagier introduced a certain diﬀerential equation for
elliptic modular forms. The diﬀerential equation was studied by Kaneko andKoike. In this talk, we carry out a similar study for Jacobi forms.
14. Tomokazu Onozuka
University: Nagoya University
E-mail: m11022v@math.nagoya-u.ac.jp
Talk title: Mean values of Mordell-Tornheim double zeta-functions(a joint work with Takuya Okamoto (Ritsumeikan Univ.))
Abstract: Matsumoto and Tsumura proved the mean value theorems for
the Euler-Zagier double zeta-function. In this talk, we consider mean value
theorems for the Mordell-Tornheim double zeta-function.

15. Ade Irma Suriajaya
University: Nagoya University
E-mail: adeirmasuriajaya@yahoo.com
Talk title: On the zeros of the second derivative of the Riemann zetafunction under the Riemann hypothesis
Abstract: The number of zeros and the distribution of the real part of nonreal zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson,and Montgomery investigated the general case, meanwhile Akatsuka gave
sharper estimates for the ﬁrst derivative of the Riemann zeta function under
the truth of the Riemann hypothesis. In this talk, we shall introduce theextension of the results of Akatsuka to the second derivative of the Riemannzeta function.
16. Takahiro Wakasa
University: Nagoya University
E-mail: d11003j@math.nagoya-u.ac.jp
Talk title: The explicit upper bound of the multiple integral of S(t)on the Riemann Hypothesis
Abstract: We prove explicit upper bounds of the function Sm (t), deﬁned by
the repeated integration of the argument of the Riemann zeta-function. The
explicit upper bound of S(t) and S1 (t) have already been obtained by A.
Fujii. Our result is a generalization of Fujii’s results.