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

Riemann Zeta函数、Dirichlet L函数等Dirichlet级数的研究已有很长的历史。它们仍然是当今研究的主要对象，因为许多重要的数论性质都反映在狄里克莱级数的解析性质中，如解析延拓、函数方程、极点位置及其余项。为了研究Zeta函数的详细局部性质，Kiuchi和Tanigawa导出了E_σ(T)(=ζ^2(σ+it)和R(σ+it)均方公式的误差项)的短区间中值定理(=ζ^2(σ+it)和R(σ+it)(=ζ^2(S)近似函数方程的余项)。我们还研究了带特征的Dirichlet因子问题产生的误差项的平均值。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，我们推导了具有不完全伽马函数的近似函数方程。利用我们的公式，我们检验了黎曼假设对于几条椭圆曲线在Im(S)[小于或等于]400的范围内成立。此外，我们还研究了状态猜想与黎曼假设之间的关系。上述函数方程与模关系有关。Kanemitsu，Yoshimoto和Tanigawa从模关系的角度重建了许多与Ramanujan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Zeta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。我们还计算了非正整数的Zeta值。

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.