The asymptotic expansion of a sum appearing in an approximate functional equation for the riemann zeta function

TL;DR

This paper derives the asymptotic expansion of a sum related to the Riemann zeta function for large n, using steepest descent, and verifies the results numerically.

Contribution

It provides a new asymptotic expansion for a sum in the functional equation of the zeta function, applying steepest descent to an integral representation.

Findings

Asymptotic expansion for A(n,s) as n→∞ with t=an

Numerical validation of the asymptotic approximation

Application of steepest descent method to zeta function sum

Abstract

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right) \frac{(-)^k}{(k+1)^s}.\] In this note we examine the asymptotics of $A(n,s)$ as $n\to\infty$ when $t=an$, where $a>0$ is a fixed parameter, by application of the method of steepest descents to an integral representation. Numerical results are presented to illustrate the accuracy of the expansion obtained.