Even and Odd Integer Values of the Secondary Zeta Function
Abstract
The secondary zeta function is defined by Z(s)=∑γ>01/γs, where 1/2+iγ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of Z(s) at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann ξ-function. Define L(x)=log((ξ(x+1/2))/ξ(1/2)), ℰ=xd/dx, 𝒟k=ℰ∏j=1k(ℰ−2j). Then, for k≥1, Z(2k)=((−1)k)/(2k(k−1)!)∫0∞(𝒟k L(x))/(x2k+1) dx and Z(2k+1)=((−1)k)/(π (2k−1)!!)∫0∞(𝒟k L(x))/(x2k+2) dx. In the last part, we write these formulas out explicitly for Z(k), 2≤ k≤ 7, both in terms of L(x) and in terms of ξ(x+1/2) for reference. We also derive a limit formula for γ1, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.
Related
- One-level density of zeros of the Riemann zeta function weighted by its special values
- A generating function identity for finite and symmetric multiple zeta values via Dougall's bilateral 2H2 summation
- Large values of the Hurwitz zeta function with rational parameter
- Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function
- Lower-order terms in mean-square formulas for the Euler--Zagier double zeta-function and regularized multiple zeta values
BibTeX
@misc{kawalec2026even,
title = {{Even and Odd Integer Values of the Secondary Zeta Function}},
author = {Artur Kawalec},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19175},
eprint = {2610.19175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19175},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Even and Odd Integer Values of the Secondary Zeta Function
AU - Artur Kawalec
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19175
UR - https://arxiv.org/abs/2610.19175
AB - The secondary zeta function is defined by Z(s)=∑_γ>01/γ^s, where 1/2+iγ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of Z(s) at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann ξ-function. Define L(x)=log((ξ(x+1/2))/ξ(1/2)), ℰ=xd/dx, 𝒟_k=ℰ∏_j=1^k(ℰ−2j). Then, for k≥1, Z(2k)=((−1)^k)/(2^k(k−1)!)∫_0^∞(𝒟_k L(x))/(x^2k+1) dx and Z(2k+1)=((−1)^k)/(π (2k−1)!!)∫_0^∞(𝒟_k L(x))/(x^2k+2) dx. In the last part, we write these formulas out explicitly for Z(k), 2≤ k≤ 7, both in terms of L(x) and in terms of ξ(x+1/2) for reference. We also derive a limit formula for γ_1, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.
ER -