Rokas Tamošiūnas | Searcharxiv

arXiv2019

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

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.

math.NT↗

arXiv2019

Symmetry of zeros of Lerch zeta-function for equal parameters

For most values of parameters $λ$ and $α$, the zeros of the Lerch zeta-function $L(λ, α, s)$ are distributed very chaotically. In this paper we consider the special case of equal parameters $L(λ, λ, s)$ and show by calculations that the nontrivial zeros either lie extremely close to the critical line $σ= 1/2$ or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.

math.NT↗