Research on the special values of various zeta functions

我们的研究从正奇整数处黎曼ζ值的拉马努金公式开始。我们发现他的公式是基于Bochner的模关系和带移位参数的逆Mellin变换。作为这一新观点的应用，我们导出了各种ζ值的解析表达式，例如整数点上的Dirichlet l -函数值，有理点上的多个Hurwitz ζ值，所有正整数上的自同构l -函数值。我们还发现Ramanujan公式关于参数x的推导得到了全纯Eisenstein级数的Guinand公式和自同构。我们也尝试用g和h函数推广模关系。马德隆常数在晶体化学中起着重要的作用。我们将它们视为Epstein zeta函数的特殊值，并对Glasser、Zucker、Chaba-Pathria等人的前人著作进行了阐述。应用各种二次型公式，如Chowla-Selberg公式，推导了马德隆常数之间的各种关系。进一步研究了一般狄利克雷级数的均方公式，并对误差项进行了详细估计。我们还研究了值|L(1,x)|的均方，其中我们采用了经典的双伽马函数公式，并推导了一个非常接近的公式。最后利用指数和理论研究了一般型多重zeta函数的解析延拓，以及Euler-Zagier型二重zeta函数的阶估计。

Our research started from the Ramanujan formula for the Riemann zeta-values at positive odd integers. We found that his formula is based on Bochner's modular relations and the Inverse Mellin transform with shifted arguments. As an application of this new viewpoints, we derived the analytic expressions of various zeta-values, jfor examples, the Dirichlet L-function values at integer points, the multiple Hurwitz zeta-values at rational arguments, automorphic L-function values at all positive integers. We also found that the derivation of Ramanujan's formula with respect to the parameter x gives Guinand's formula and automorphy of holomorphic Eisenstein series. We also tried to generalize modular relations using G-and H-functions.The Madelung constants play an important role in crystal chemistry. We regard them as special values of Epstein zeta function and elucidated the previous works of Glasser, Zucker, Chaba-Pathria and others. Applying various formulas for quadratic forms, e.g. Chowla-Selberg formulas, we derived various relations among Madelung constants.Furthermore, we studied the mean square formula for general Dirichlet series with a functional equation and estimated the error term in great detail. We also studied the mean square of the values |L(1,x)|, where we employed the classical formula for digamma function, and derived a very close formula for it. Finally we studied the analytic continuation of multiple zeta function of general type, and order estimate of double zeta function of Euler-Zagier type using the theory of exponential sums.