On Louchard's Asymptotic Series

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 $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1n}dx$ as $n\to\infty$, and computed $I_j$ for $j\le 5$ in terms of values of the Riemann zeta function. An interesting feature of the computation is that the $I_j$ 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 $I_n$, $6\le n\le 9$, and conjecture a general formula for $I_n$ in terms of alternating multiple zeta values. We also conjecture that $I_n$ is a rational polynomial in the ordinary zeta values.