The Stokes phenomenon and the Lerch zeta function

TL;DR

This paper investigates the asymptotic behavior of the Lerch zeta function for large complex arguments, revealing how exponentially small terms activate across Stokes lines and differ along the imaginary axes, supported by numerical validation.

Contribution

It provides a detailed analysis of the Stokes phenomenon for the Lerch zeta function, including the activation of exponential terms and their asymmetric transitions, which was not previously understood.

Findings

Exponential terms switch on across Stokes lines at arg a = ±π/2.

Transitions across the imaginary axes are generally unequal in scale.

Numerical calculations confirm the theoretical asymptotic predictions.

Abstract

We examine the exponentially improved asymptotic expansion of the Lerch zeta function $L(\lambda,a,s)=\sum_{n=1}^\infty \exp (2\pi ni\lambda)/(n+a)^s$ for large complex values of $a$, with $\lambda$ and $s$ regarded as parameters. It is shown that an infinite number of subdominant exponential terms switch on across the Stokes lines $\arg\,a=\pm\pi/2$. In addition, it is found that the transition across the upper and lower imaginary $a$-axes is associated, in general, with unequal scales. Numerical calculations are presented to confirm the theoretical predictions.