Year: 2026
Type: preprint
Abstract: We study the Laurent coefficients of the Barnes multiple zeta function with complex parameters in a common open half-plane and a fixed holomorphic determination of the logarithm. At the highest pole, we derive explicit limit formulae for every regular Laurent coefficient in terms of finite multiple sums and logarithmic correction terms. At the lower possible poles, we establish all-order relations with the Taylor coefficients at the origin and their parameter derivatives. We also determine the large-order behavior by subtracting all principal parts. The remaining function is entire, so Cauchy's estimate separates explicit residue contributions from a remainder that decays faster than any fixed geometric rate. The neighboring residues yield the limiting even and odd subsequences, with vanishing residues and cancellations accounted for. The constant-term cases recover known finite-part representations; the main focus is their extension to higher Laurent coefficients.
Source: arXiv (Cornell University)
Author: Miyagawa, Takashi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213404447