Small values of the Riemann zeta-function near the critical line
Abstract
Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line. For every fixed 0<k<1/2, we establish the conjecturally sharp bound 1/T∫T2T |ζ(1/2+α+it)|−2k dt ≪ α−k2, for α=o(1) and α≫ 1/log T. We also obtain almost sharp upper bounds slightly to the left of 1/log T. This improves the upper bounds of the first two authors for shifts as small as 1/log T, confirming a conjecture of Gonek in this range. A separate bound for the moment at k=1/2 yields an improved bound for cancellation in the Möbius function. Namely, we show that ∑n≤ xμ(n) ≪ √xexp (C√((log x)/(loglog x))) for some C>0, improving on previous results of Soundararajan, Balazard and de Roton, and of the first two authors. For t chosen uniformly from [T,2T], we also prove that min|h|≤1 |ζ(1/2+α+i(t+h))| =α1+o(1) in probability, provided that α≪ (log T)−c for some fixed c >0, and αlog T/loglog T→∞. This agrees with the leading order predicted by a minimum analogue of the Fyodorov--Hiary--Keating Conjecture. Further results include bounds for minima on intervals of logarithmically growing length, for negative moments in almost all short intervals, and an extension of the range of sharp negative-moment bounds for k>1/2. A main ingredient in the proof of some of these results is a Gaussian large-deviation upper bound for log|ζ(1/2+α+it)|−1, obtained by building on work of Arguin, Bailey, and Roberts.
Related
- One-level density of zeros of the Riemann zeta function weighted by its special values
- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
- Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line
- Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
BibTeX
@misc{bui2026small,
title = {{Small values of the Riemann zeta-function near the critical line}},
author = {Hung M. Bui and Alexandra Florea and André Heycock},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27591},
eprint = {2610.27591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27591},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Small values of the Riemann zeta-function near the critical line
AU - Hung M. Bui
AU - Alexandra Florea
AU - André Heycock
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27591
UR - https://arxiv.org/abs/2610.27591
AB - Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line. For every fixed 0<k<1/2, we establish the conjecturally sharp bound 1/T∫_T^2T |ζ(1/2+α+it)|^−2k dt ≪ α^−k^2, for α=o(1) and α≫ 1/log T. We also obtain almost sharp upper bounds slightly to the left of 1/log T. This improves the upper bounds of the first two authors for shifts as small as 1/log T, confirming a conjecture of Gonek in this range. A separate bound for the moment at k=1/2 yields an improved bound for cancellation in the Möbius function. Namely, we show that ∑_n≤ xμ(n) ≪ √xexp (C√((log x)/(loglog x))) for some C>0, improving on previous results of Soundararajan, Balazard and de Roton, and of the first two authors. For t chosen uniformly from [T,2T], we also prove that min_|h|≤1 |ζ(1/2+α+i(t+h))| =α^1+o(1) in probability, provided that α≪ (log T)^−c for some fixed c >0, and αlog T/loglog T→∞. This agrees with the leading order predicted by a minimum analogue of the Fyodorov--Hiary--Keating Conjecture. Further results include bounds for minima on intervals of logarithmically growing length, for negative moments in almost all short intervals, and an extension of the range of sharp negative-moment bounds for k>1/2. A main ingredient in the proof of some of these results is a Gaussian large-deviation upper bound for log|ζ(1/2+α+it)|^−1, obtained by building on work of Arguin, Bailey, and Roberts.
ER -