Masanori Katsurada | Searcharxiv

arXiv2023

Asymptotic expansions for a class of generalized holomorphic Eisenstein series, Ramanujan's formula for $ζ(2k+1)$, Weierstrass' elliptic and allied functions

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new variants of the celebrated formulae of Euler and of Ramanujan for specific values of the Riemann zeta-function (Theorem~4 and Corollaries~4.1--4.5), and to various modular type relations for the classical Eisenstein series of any even integer weight (Corollary~4.6) as well as for Weierstraß' elliptic and allied functions (Corollaries~4.7--4.9). Crucial r{ô}les in the proofs are played by certain Mellin-Barnes type integrals, which are manipulated with several properties of confluent hypergeometric functions.

math.NT↗

arXiv2022

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

For a complex variable $s$ and real parameters $a$ and $λ$ with $a>0$, let $ϕ(s,a,λ)$ denote the Lerch zeta-function with a complex variable, $ϕ^{\ast}(s,a,λ)$ a slight modification of $ϕ(s,a,λ)$ defined by extracting the (possible) singularity of $ϕ(s,a,λ)$ at $s=1$, and $(ϕ^{\ast})^{(m)}(s,a,λ)$ 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 $(ϕ^{\ast})^{(m)}(s,a,λ)$, transformed through the Laplace-Mellin and Riemann-Liouville operators (say, $\mathcal{LM}_{z;τ}^α$ and $\mathcal{RL}_{z;τ}^{α,β}$, respectively) in terms of the variable $s$. We shall show that complete asymptotic expansions exist if $a>1$ for $\mathcal{LM}_{z;τ}^α(ϕ^{\ast})^{(m)}(s+τ,a,λ)$ and $\mathcal{RL}_{z;τ}^{α,β}(ϕ^{\ast})^{(m)}(s+τ,a,λ)$ (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 $(s,z)=(σ,it)$ as $t\to\pm\infty$ (Corollaries~2.1,~4.1,~6.1 and~8.1).

math.NT↗