Mathematica Eterna
Open Access

Abstract

The Stokes phenomenon and the Lerch zeta function

R. B. Paris

We examine the exponentially improved asymptotic expansion of the Lerch zeta function L(λ, a, s) = P∞ n=0 exp(2πniλ)/(n + a) s for large complex values of a, with λ and s regarded as parameters. It is shown that an infinite number of subdominant exponential terms switch on across the Stokes lines arg a = ± 1 2 π. 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.