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. Garunktis, A. Laurinikas, and J. Steuding (in 1) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function L (s, , )= n=0 e 2in (n+ ) -s .In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions.R. Garunktis, A. Laurinikas, and J. Steuding (in 2) obtained the results on the mean square values of L ( + it, , ) with respect to t.We obtain the main term of the mean square values of L (1/2 + it, , ) using a simpler method than their method in 2.