The asymptotic expansion of a sum appearing in an approximate functional equation for the riemann zeta function (2106.01371v1)

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.