On Louchard's Asymptotic Series
Michael E. Hoffman

TL;DR
This paper explores the asymptotic series related to a specific integral, revealing cancellations of complex zeta values and proposing conjectures about the structure of the coefficients in terms of zeta functions.
Contribution
It extends Louchard's asymptotic series analysis, providing new formulas for higher-order terms and conjecturing a general pattern involving zeta values.
Findings
Cancellation of alternating multiple zeta values in coefficients
Explicit formulas for $I_n$ for 6 ≤ n ≤ 9
Conjecture of a rational polynomial structure in zeta values
Abstract
Recently G. Louchard obtained an asymptotic series for the integral as , and computed for in terms of values of the Riemann zeta function. An interesting feature of the computation is that the are first obtained in terms of alternating multiple zeta values, but then everything except products of ordinary zeta values cancels out. We obtain similar formulas for , , and conjecture a general formula for in terms of alternating multiple zeta values. We also conjecture that is a rational polynomial in the ordinary zeta values.
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Taxonomy
TopicsAdvanced Mathematical Identities · Mathematics and Applications · Advanced Combinatorial Mathematics
