Approximate functional equations for the Hurwitz and Lerch zeta-functions

Abstract

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].