Approximate Functional Equations for the Hurwitz and Lerch Zeta-functions

摘要

As one of the asymptotic formulas for the zeta-function, Hardy and Lit-tlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunkstis, A. Laurincikas, and J. Steuding (in [1]) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function ζ_L(s,α,λ) =∑~∞_(n=0)e~(2πinλ)(n+α)~(-s). In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions. R. Garunkstis, A. Laurin?ikas, and J. Steuding (in [2]) obtained the results on the mean square values of ζ_L(σ + it, α, λ) with respect to t. We obtain the main term of the mean square values of ζ_L（1 /2 + it，α, λ) using a simpler method than their method in [2].