The discrete second moment of the Riemann zeta-function along powers of Gram points
Abstract
Let tk be the Gram points, and let d≥1 be a fixed integer. Put ηd=1/(2d+1−2), σd=1−ηd/d, and σd*=7/8 for d=2 and σd*=σd otherwise. We prove that, uniformly on compact subsets of σd*<ℜ s<1, 1/N∑k=2N|ζ(s+itkd)|2 ⟶ ζ(2ℜ s). The underlying result is a mean-square approximation by finite Euler products. For the squares of Gram points, grouping equal frequency ratios and applying the double large sieve improves the general derivative-test boundary 11/12 to 7/8. Assuming the Lindelöf hypothesis, both the Euler-product approximation and the second-moment formula hold on every compact subset of 1/2<ℜ s<1, for every fixed d≥1. As an application, we obtain a discrete universality theorem along the same Gram-power shifts, in the same unconditional and conditional regions.
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BibTeX
@misc{dundulis2026discrete,
title = {{The discrete second moment of the Riemann zeta-function along powers of Gram points}},
author = {Marius Dundulis and Ramūnas Garunkštis and Antanas Laurinčikas and Raivydas Šimėnas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22107},
eprint = {2610.22107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The discrete second moment of the Riemann zeta-function along powers of Gram points
AU - Marius Dundulis
AU - Ramūnas Garunkštis
AU - Antanas Laurinčikas
AU - Raivydas Šimėnas
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22107
UR - https://arxiv.org/abs/2610.22107
AB - Let t_k be the Gram points, and let d≥1 be a fixed integer. Put η_d=1/(2^d+1−2), σ_d=1−η_d/d, and σ_d^*=7/8 for d=2 and σ_d^*=σ_d otherwise. We prove that, uniformly on compact subsets of σ_d^*<ℜ s<1, 1/N∑_k=2^N|ζ(s+it_k^d)|^2 ⟶ ζ(2ℜ s). The underlying result is a mean-square approximation by finite Euler products. For the squares of Gram points, grouping equal frequency ratios and applying the double large sieve improves the general derivative-test boundary 11/12 to 7/8. Assuming the Lindelöf hypothesis, both the Euler-product approximation and the second-moment formula hold on every compact subset of 1/2<ℜ s<1, for every fixed d≥1. As an application, we obtain a discrete universality theorem along the same Gram-power shifts, in the same unconditional and conditional regions.
ER -