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.
Categories: math.CA