Research on the special values of various zeta functions

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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.