Approximate functional equations for the Hurwitz and Lerch zeta-functions

As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunk\v{s}tis, A. Laurin\v{c}ikas, and J. Steuding (in [1]) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function $\zeta_L (s, \alpha, \lambda ) = \sum_{n=0}^\infty e^{2\pi i n \lambda}(n + \alpha)^{-s}$. In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions. R. Garunk\v{s}tis, A. Laurin\v{c}ikas, and J. Steuding (in \cite{GLS2}) obtained the results on the mean square values of $\zeta_L (\sigma + it, \alpha , \lambda)$ with respect to $t$. We obtain the main term of the mean square values of $\zeta_L (1/2 + it, \alpha , \lambda)$ using a simpler method than their method in [2].Comment: 13 page

Zeta functions on tori using contour integration

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla-Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well