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. Garunkštis, A. Laurinčikas, and J. Steuding (in [1]) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function ${\zeta }_L(s,\alpha ,\lambda )={\sum }_{n=0}^{\infty }{e}^{2\pi in\lambda }(n+\alpha {)}^{-s}$. In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions. R. Garunkštis, A. Laurinčikas, 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].

13 pages

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