symmetry S S Article The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function Janyarak Tongsomporn 1, *, Saeree Wananiyakul 1 and Jörn Steuding 2   Citation: Tongsomporn, J.; Wananiyakul, S.; Steuding, J. The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function. Symmetry 2021, 13, 2410. https://doi.org/10.3390/ sym13122410 Academic Editor: Ioan Ras , a Received: 29 October 2021 Accepted: 9 December 2021 Published: 13 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand; s.wananiyakul@gmail.com 2 Department of Mathematics, Würzburg University, Am Hubland, 97218 Würzburg, Germany; steuding@mathematik.uni-wuerzburg.de * Correspondence: janyarak.to@wu.ac.th Abstract: In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally. Keywords: zeta-functions; Riemann hypothesis 1. Introduction The periodic zeta-function, introduced by Berndt and Schorenfeld [1] in 1975, is defined by F(s; α)= ∑ n≥1 e(nα)n −s , for ℜ(s) > 1 and a real parameter α where the abbreviation e(α)= exp(2πi α) is used; the naming reflects the periodicity F(s; α + 1)= F(s; α) for which we may assume α ∈ (0, 1] in the sequel. The periodic zeta-function admits a meromorphic continuation to the whole complex plane (details about this in the following section). Of particular interest are special values of the parameter, F(s;1)= ζ (s) and F  s; 1 2  =  2 1−s − 1  ζ (s), (1) where ζ (s) is the Riemann zeta-function. We shall prove that for no other values of α than 0 or 1/2 mod 1 the quotient F(s; α)/ζ (s) is an entire function. As usual, we denote the nontrivial (non-real) zeros of ζ (s) as ρ = β + i γ. The number of nontrivial zeros ρ = β + i γ of ζ (s) satisfying 0 < γ < T is by the Riemann–von Mangoldt formula asymptotically given by N(T) : = #{ρ = β + i γ :0 < γ ≤ T} = T 2π log T 2πe + O(log T) (here, every multiple zero would be counted according to its multiplicity; however, there is no multiple ζ -zero known so far). The Riemann hypothesis states that all nontrivial zeros ρ lie on the critical line ℜ(s)= 1/2. Recently, Sowa [2] obtained under assumption of the truth of this unproven conjecture the limit lim T→∞ 1 N(T) ∑ 0<γ<T F(ξ + i γ; α)= exp(2πi α) (2) where the convergence type is (i) distributional whenever ξ > 0, (ii) in L 2 -norm whenever ξ > 1, and (iii) uniform whenever ξ > 3/2. His approach relies on Fourier analysis, and Symmetry 2021, 13, 2410. https://doi.org/10.3390/sym13122410 https://www.mdpi.com/journal/symmetry