Asymptotic expansions for the Laplace-Mellin and Riemann-Liouville transforms of Lerch zeta-functions

TL;DR

This paper derives complete asymptotic expansions for Laplace-Mellin and Riemann-Liouville transforms of modified Lerch zeta-functions, analyzing their behavior as parameters tend to zero or infinity, with applications along vertical lines in the complex plane.

Contribution

It provides new asymptotic expansion results for these transforms of Lerch zeta-functions, including their iterated variants, expanding understanding of their asymptotic behavior in complex analysis.

Findings

Existence of complete asymptotic expansions for the transforms when a>1.

Asymptotic expansions valid as the pivotal parameter z tends to 0 or infinity.

Results applicable along vertical lines in the complex plane for large imaginary parts.

Abstract

For a complex variable $s$ and real parameters $a$ and $\lambda$ with $a>0$, let $\phi(s,a,\lambda)$ denote the Lerch zeta-function with a complex variable, $\phi^{\ast}(s,a,\lambda)$ a slight modification of $\phi(s,a,\lambda)$ defined by extracting the (possible) singularity of $\phi(s,a,\lambda)$ at $s=1$, and $(\phi^{\ast})^{(m)}(s,a,\lambda)$ for any $m\in\mathbb{Z}$ the $m$th derivative with respect to $s$ if $m\geq0$, while if $m\leq0$ the $|m|$-th primitive defined with its initial point at $s+\infty$. The present paper aims to study asymptotic aspects of $(\phi^{\ast})^{(m)}(s,a,\lambda)$, transformed through the Laplace-Mellin and Riemann-Liouville operators (say, $\mathcal{LM}_{z;\tau}^{\alpha}$ and $\mathcal{RL}_{z;\tau}^{\alpha,\beta}$, respectively) in terms of the variable $s$. We shall show that complete asymptotic expansions exist if $a>1$ for $\mathcal{LM}_{z;\tau}^{\alpha}(\phi^{\ast})^{(m)}(s+\tau,a,\lambda)$ and $\mathcal{RL}_{z;\tau}^{\alpha,\beta}(\phi^{\ast})^{(m)}(s+\tau,a,\lambda)$ (Theorems~1--4), as well as for their iterated variants (Theorems~5--10), when the `pivotal' parameter $z$ (of the transforms) tends to both $0$ and $\infty$ through appropriate sectors. Most of our results include any vertical ray in their region of validity; this allows us to deduce complete asymptotic expansions along vertical lines $(s,z)=(\sigma,it)$ as $t\to\pm\infty$ (Corollaries~2.1,~4.1,~6.1 and~8.1).