Analytic continuation of the Hurwitz Zeta Function with physical application

TL;DR

This paper introduces a new formula for the analytic continuation of the Hurwitz zeta function, linking it to the Euler gamma and polylogarithmic functions, with applications in spectral zeta function regularization in physics.

Contribution

It provides explicit formulas for derivatives of the Hurwitz zeta function at negative integers and applies these results to compute physical quantities like pair production rates.

Findings

Explicit formulas for derivatives at negative integers

Application to pair production rate calculation

Enhanced understanding of Hurwitz zeta function in physics

Abstract

A new formula relating the analytic continuation of the Hurwitz zeta function to the Euler gamma function and a polylogarithmic function is presented. In particular, the values of the first derivative of the real part of the analytic continuation of the Hurwitz zeta function for even negative integers and the imaginary one for odd negative integers are explicitly given. The result can be of interest both on mathematical and physical side, because we are able to apply our new formulas in the context of the Spectral Zeta Function regularization, computing the exact pair production rate per space-time unit of massive Dirac particles interacting with a purely electric background field.