PreprintCC BY 4.0arXiv
The sixth moment of the Riemann zeta function
Abstract
We prove new large value estimates for the Riemann zeta function on the critical line. For instance, we improve the upper bound on the measure of t∈ [T, 2T] with |ζ(1/2 + it)| ≥ T1/8 for the first time since Hardy-Littlewood (1923). Our results imply improved upper bounds on the k-th moment of zeta for every 4 < k ≤ 12. We show in particular that ∫0T |ζ(1/2 + it)|6 d t ≪ε T5/4 − 1/60 + ε. The main ingredient is a new large value estimate for exponential sums with square-root phases, and certain perturbations. Those arise in the expansion of the short second moment of zeta.
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BibTeX
@misc{faveri2026sixth,
title = {{The sixth moment of the Riemann zeta function}},
author = {Alexandre de Faveri and Mayank Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01035},
eprint = {2610.01035},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01035},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The sixth moment of the Riemann zeta function
AU - Alexandre de Faveri
AU - Mayank Pandey
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01035
UR - https://arxiv.org/abs/2610.01035
AB - We prove new large value estimates for the Riemann zeta function on the critical line. For instance, we improve the upper bound on the measure of t∈ [T, 2T] with |ζ(1/2 + it)| ≥ T^1/8 for the first time since Hardy-Littlewood (1923). Our results imply improved upper bounds on the k-th moment of zeta for every 4 < k ≤ 12. We show in particular that ∫_0^T |ζ(1/2 + it)|^6 d t ≪_ε T^5/4 − 1/60 + ε. The main ingredient is a new large value estimate for exponential sums with square-root phases, and certain perturbations. Those arise in the expansion of the short second moment of zeta.
ER -