Sum formulas for arithmetical functions and mean value theorem for the zeta functions

算术函数的求和公式以及 zeta 函数的中值定理

基本信息

批准号：
11640022
负责人：
TANIGAWA Yoshio
金额：
$ 1.02万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640022/
关键词：
Dirichlet series Mean value formula Arithmetical function Elliptic curve Modular relation Ramanujan formula Multiple zeta function Analytic continuation Bernoulli数 Rankin-Selberg 級数近似関数等式二乗平均

项目摘要

The Riemann zeta-function, the Dirichlet L-fucntion and other Dirichlet series have a long history of researches. They are still the main oobjects of researches at present day, since many important number theoretic properties are reflected in the analytic properties of Dirichlet series, such as the analtic continuation, functional equation, the place of poles and their residues.To study the detailed local behavior of zeta function, Kiuchi and Tanigawa derived mean value theorem for short intervals for E_σ (T)(=the error term of the mean square formula of ζ^2 (σ+it) and R (σ+it)(=the remainder term of the approximate functional equation of ζ^2 (s)). We also studied the mean value of eror term arising from the Dirichlet divisor problem with characters.Akiyama and Tanigawa considered the L-function associated to elliptic curves. In order to compute the numerical values, we derived the approximate functional equation with incomplete gamma functions. Using our formula, we checked that the Riemann hypothesis holds in the range Im (s) 【less than or equal】400 for several elliptic curves. Furthermore, we studied the relation between Sate-Tate conjecture and the Riemann hypothesis.The above mentioned functional equation is related to the modular relation. Kanemitsu, Yoshimoto and Tanigawa reconstruct many results related to the Ramanujan formulas from the point of modular relations. The new formula for ζ(2/3) of Ramanujan type is also obtained.The analytic continuation of Euler-Zagier's multiple zeta-function was established by Akiyama, Egami and Tanigawa. We noted for the first time that the points of indeterminacy appear for this function. We also computes the zeta-values at non-positive integers.

Riemann Zeta功能，Dirichlet L-Fucntion和其他Dirichlet系列具有悠久的研究历史。它们仍然是当今研究的主要原理，因为许多重要数量的理论特性反映在Dirichlet系列的分析特性中，例如Dirichlet系列的分析特性，例如隔离延续，功能等效性，杆子的位置及其残留的位置及其残留物。研究ZETA功能的详细局部行为，Zeta功能的均值，kiuchi和Tanigawa的均值（tanigawa for tem sere for）的均值（=）= for e_ for（= ζ^2（σ+IT）和R（σ+IT）（=ζ^2（s）的近似功能方程式），我们还研究了由字符的dirichlet分隔术引起的EROR项的平均值。使用我们的公式，我们检查了riemann假设在几个椭圆曲线的范围内（小于或相等）400。上述功能方程与模块化关系有关。 Kanemitsu，Yoshimoto和Tanigawa从模块化关系点重建了与Ramanujan公式有关的许多结果。还获得了Ramanujan类型的ζ（2/3）的新公式。我们首次注意到此功能的不确定性外观点。我们还计算非阳性整数处的Zeta值。