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

• 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

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.

黎曼ζ函数、狄利克雷l -函数等狄利克雷级数的研究历史悠久。由于Dirichlet级数的解析性质反映了许多重要的数论性质，如解析延拓、泛函方程、极点位置及其残数等，因此Dirichlet级数仍然是目前研究的主要对象。为了研究ζ函数的详细局部行为，Kiuchi和Tanigawa推导了E_σ (T)（= ζ^2 （σ+it）均方公式的误差项）和R (σ+it)（= ζ^2 (s)近似泛函方程的余数项）的短区间均值定理。我们还研究了带有字符的狄利克雷除数问题所引起的误差项的平均值。秋山和谷川考虑了与椭圆曲线相关的l函数。为了计算数值，我们导出了不完全函数的近似泛函方程。使用我们的公式，我们检查了黎曼假设在若干椭圆曲线的Im (s)【小于或等于】400范围内成立。进一步研究了黎曼假设与state - tate猜想之间的关系。上述泛函方程与模关系有关。Kanemitsu， Yoshimoto和Tanigawa从模关系的角度重构了许多与拉马努金公式相关的结果。得到了Ramanujan型ζ(2/3)的新公式。Akiyama、Egami和Tanigawa建立了Euler-Zagier多重ζ函数的解析延拓。我们第一次注意到这个函数出现了不确定点。我们也计算非正整数处的值。

项目成果

A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series"Math.Proc.Camb Phil Soc.. 127. 117-131 (1999)

A.Ivic、K.Matsumoto 和 Y.Tanikawa：“论 Rankin-Selberg 级数系数的 Riesz 均值”Math.Proc.Camb Phil Soc.. 127. 117-131 (1999)

S.Akiyama,S.Egami and Y.Tanigawa: "Analytic continuation of multiple zeta functions and their values at non-positive in-tegers"Acta Arith.. (to appear).

S.Akiyama、S.Egami 和 Y.Tanikawa：“多个 zeta 函数及其在非正整数处的值的解析延拓”Acta Arith..（待发表）。

S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves"Mathematics of Computation. 68. 1201-1231 (1999)

S.Akiyama 和 Y.Tanikawa：“与椭圆曲线相关的 L 函数值的计算”计算数学。

S.Akiyama, S.Egami and Y.Tanigawa: "Analytic continuation of multiple zeta functions and their values at non-positive integers"Acta Arith.. (to appear).

S.Akiyama、S.Egami 和 Y.Tanikawa：“多个 zeta 函数及其非正整数值的解析延拓”Acta Arith..（即将出现）。

I..Kiuchi and Y.Tanigawa: "The mean value theorem of the Riemann Zeta-function in the critical strip for short intervals"Number Theory and its Applications (ed.by Gyery and Kanemitsu). 231-240 (1999)

I..Kiuchi 和 Y.Tanikawa：“短区间临界带中黎曼 Zeta 函数的均值定理”数论及其应用（由 Gyery 和 Kanemitsu 编辑）。