Symmetry of zeros of Lerch zeta-function for equal parameters

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.

The size of the Lerch zeta-function at places symmetric with respect to the line $\Re(s)=1/2$

“…P r o o f. The lemma is proved by Garunkštis and Tamošiūnas in [11], Proposition 2. We reproduce the proof here for completeness.…”

Zeros of the extended Selberg class zeta-functions and of their derivatives

“…Such function has an almost symmetrical distribution of non-trivial zeros with respect to the line σ = 1/2. Insights which helped to overcome difficulties raised by "almost symmetricity" in [4] led to Theorem 1.1 of this paper, although ζ(s) has a strictly symmetrical zero-distribution.…”

“…See Yıldırım [23] for Dirichlet L-functions; Šleževičienė [20] for the Selberg class; Luo [12], Garunkštis [3], Minamide [13], [14], [15], Jorgenson and Smailović [6] for Selberg zeta-functions and related functions; Garunkštis and Šimėnas [5] for the extended Selberg class. In Garunkštis and Tamošiūnas [4] the Levinson and Montgomery result was generalized to the Lerch zeta-function with equal parameters. Such function has an almost symmetrical distribution of non-trivial zeros with respect to the line σ = 1/2.…”

“…Proof. We repeat the steps of the proof of Proposition 4 in [4]. Contrary to Proposition 4, here we do not need the upper bound for ε (see (2.6)).…”

“…Contrary to Proposition 4, here we do not need the upper bound for ε (see (2.6)). This is because the "symmetric" functional equation (2.1) leads to the convenient formula (2.2), while the "almost symmetric" functional equation of the Lerch zeta-function with equal parameters in [4] leads to a more restricted version of (2. In view of bounds (2.4) and (2.5), Lemma α from Titchmarsh [21, Section 3.9] gives that, for |s − s 0 | ≤ r j ,…”