ASYMPTOTIC EXPANSIONS FOR THE LAPLACE-MELLIN AND RIEMANN-LIOUVILLE TRANSFORMS OF LERCH ZETA-FUNCTIONS : PRE-ANNOUNCEMENT VERSION (Analytic Number Theory and Related Areas)

Abstract

This article summarizes the results appearing in the forthcoming paper [13]. For a complex variable s, and real parameters a and $lambda$ with a>0, the Lerch zetafunction $phi$(s, a, $lambda$) is defined by the Dirichlet series displaystyle sum_{l=0}^{infty}e($lambda$ l)(a+l)^{-s}({rm Re} s>1), and its meromorphic continuation over the whole s-plane, where e($lambda$)=e^{2 $pi$ i $lambda$}, and the domain of the parameter a can be extended to the whole sector |mathrm{a}xmathrm{g}z|< $pi$. It is treated in the present article several asymptotic aspects of the Laplace-Mellin and Riemann-Liouville (or Erdély-Köber) transforms of $phi$(s, a, $lambda$), together with its slight modification $phi$^{*}(s, a, $lambda$), both applied with respect to the (first) variable s and the (second) parameter a. We shall show that complete asymptotic expansions exist for these objects when the pivotal parameter z of the transforms tends to both 0 and infty through the sector |mathrm{a}xmathrm{g}z|< $pi$ (Theorems 1−8). It is ffirther shown that our main formulae can be applied to deduce certain asymptotic expansions for the weighted mean values of $phi$^{*}(s, a, $lambda$) through arbitrary vertical half lines in the s-plane (Corollaries 2.1 and 4.1), as well as to derive several variants of the power series and asymptotic series for Euler s gamma and psi functions (Corollaries 8.1−8.8).