COMPLETE ASYMPTOTIC EXPANSIONS FOR THE TRANSFORMED LERCH ZETA-FUNCTIONS VIA THE LAPLACE-MELLIN AND RIEMANN-LIOUVILLE OPERATORS (PRE-ANNOUNCEMENT) (Problems and prospects in Analytic Number Theory)

Abstract

This is a pre-announcement version of the forthcoming paper [Complete asymptotic expansions for the transformed Lerch zeta-functions via the LaplaceMellin and Riemann-Liouville operators, preprint.]. For a complex variable s, and any real parameters a and λ with a > 0, the Lerch zeta-function φ(s, a, λ) is defined by the Dirichlet series Σ[∞][l=0]e(λl)(a + l)⁻[s] (Res> 1), and its meromorphic continuation over the whole s-plane, where e(λ) = e[2πiλ], and the domain of the parameter a can be extended to the whole sector |arg z| < π through the procedure in [M. Katsurada, Power series and asymptotic series associated with the Lerch zeta-function, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), 167-170.]. It is the principal aim of the present article to treat asymptotic aspects of the transformed functions obtained by applying the Laplace-Mellin and Riemann-Liouville operators (in terms of the variables), which are denoted by LM[α][z;T] and RL[α, β][z;T] respectively, to a slight modification, φ*(s, a, λ), of φ(s, a, λ). For any m ∈ ℤ, let (φ*)[(m)](s, a, λ) denote the mth derivative with respect to s if m ≥ 0, and the |m|th primitive defined with its initial point at s + ∞ if m < 0. We shall then show that complete asymptotic expansions exist, if a > 1, for .LM[α][z;T](φ*)[(m)](s+τ, a, λ) and for RL[α, β][z;T](φ*)[(m)](s+τ, a, λ) (Theorems 1-4), as well as for their severa.l iterated variants (Theorems 5-10), when the pivotal parameter z of the transforms tends to both O and oo through appropriate sectors. Most of our results include any vertical ha.If-lines in their respective regions of validity; this allows us to deduce complete asymptotic expansions for the relevant transforms through arbitrary vertical half-lines, upon taking (s, z) = (a, it) with any σ ∈ ℝ, when t → ±∞ (Corollaries 2.1, 4.1, 6.1 and 8.1).