Zeros of the Lerch zeta-function and of its derivative for equal parameters

TL;DR

This paper extends results relating zeros of the Riemann zeta-function and its derivative to the Lerch zeta-function with equal parameters, showing near symmetry of zeros despite the usual asymmetry.

Contribution

It establishes a Levinson-Montgomery type result for the Lerch zeta-function with equal parameters, highlighting near symmetry of zeros.

Findings

Zeros of the Lerch zeta-function with equal parameters are nearly symmetric about the critical line.

The distribution of zeros for the Lerch zeta-function with equal parameters resembles that of the Riemann zeta-function.

The analogue of the Riemann hypothesis is generally not true for the Lerch zeta-function, but symmetry properties are restored.

Abstract

A. Speiser proved that the Riemann hypothesis is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function left of the critical line. His result has been extended by N. Levinson and H.L. Montgomery to the statement that the Riemann zeta-function and its derivative have approximately the same number of non-real zeros left of the critical line. We obtain the Levinson-Montgomery type result for the Lerch zeta-function with equal parameters. For the Lerch zeta-function, the analogue of the Riemann hypothesis is usually not true and its zeros usually are distributed asymmetrically with respect to the critical line. However, for equal parameters, the symmetry of the zeros is almost restored.