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Small values of the Riemann zeta-function near the critical line

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Hung M. Bui, Alexandra Florea, André Heycock

Number Theory · Submitted 20 Oct 2026 · 43 references

Pith's verdict: A strong conditional paper that resolves a key case of Gonek's conjecture and advances the study of small values of zeta.

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arXiv ID2610.27591 (version 1)
Pith ID4WVZRD2C, Pith's permanent record of this paper and its review
CitationAlexandra Florea, Andr\'e Heycock, Hung M. Bui. "Small values of the Riemann zeta-function near the critical line." arXiv:2610.27591 (2026). https://pith.science/paper/4WVZRD2C
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The paper

The paper claims

Under the Riemann Hypothesis, the paper proves sharp negative-moment bounds for the Riemann zeta-function down to shifts of order 1/log T, confirming a conjecture of Gonek in that range.

The reading

The paper studies how small the Riemann zeta-function can get just to the right of the critical line. Assuming the Riemann Hypothesis, it proves that for every fixed 0

What carries the argument

A Gaussian large-deviation upper bound for log|ζ(1/2+α+it)|^{-1}, obtained by building on the recursive scheme of Arguin, Bailey, and Roberts. For small α, the contribution of zeros in the explicit formula is significant and is controlled by a delicate multi-scale analysis of prime sums. The paper separates the prime contributions into blocks and uses localization estimates (Propositions 5.1 and 5.2) to bound the probability that the sum exceeds a given level, keeping the dependence on α precise enough to recover the full Gaussian tail.

Extended reading notes

Core claim

For every fixed 0<k<1/2, under the Riemann Hypothesis, the negative moment I_{-k}(α,T) = (1/T) ∫$_T^{{2T}}$ |ζ(1/2+α+it)|^{-2k} dt satisfies I_{-k}(α,T) ≪ $α^{{-k^2}}$ for all α=o(1) with α ≫ 1/log T. This confirms Gonek's conjectured upper bound for the first time in this range. For k=1/2, a weaker bound is obtained that is sufficient to improve the known bound on the partial sums of the Möbius function to ∑_{n≤x} μ(n) ≪ √x exp(C √(log x / log log x)). For the minimum on intervals of length 1, the paper proves that for α small but not too close to 0, the minimum min_{|h|≤1} |ζ(1/2+α+i(t+h))| equals $α^{{1+o(1)}}$ with probability tending to 1.

Load-bearing premise

The Riemann Hypothesis (RH) is assumed throughout; the entire argument depends on RH to control the contribution of zeros in the explicit formula and to apply the large-deviation estimates. If RH is false, none of the main results are known to hold.

Editorial extensions

If this is right

  • For 0<k<1/2, the negative moment I_{-k}(α,T) is now known to have order α^{-k^2} when 1/log T ≪ α = o(1), matching Gonek's prediction.
  • The improved bound for the Möbius function ∑_{n≤x} μ(n) ≪ √x exp(C √(log x / log log x)) removes a log log x factor from the exponent compared to earlier conditional results.
  • The minimum of |ζ| on typical short intervals of bounded length is shown to be of order α^{1+o(1)} when α is slightly larger than (log T)^{-c}, supporting the Fyodorov–Hiary–Keating-type conjecture for the minimum.
  • The large-deviation estimate P(W_α > V) ≪ (1/√(log(1/α))) exp(-V^2 / log(1/α)) holds for V up to (1/2) log(1/α) when α log T is large, and for all bounded V/ log(1/α) when α is larger.
  • The methods extend to give sharp negative-moment bounds for k > 1/2 when α is large enough, and to negative moments in almost all short intervals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The Gaussian tail bound for log|ζ|^{-1} near the line suggests that the distribution of extreme negative values of ζ follows the same log-correlated field structure as the positive side, now with variance (1/2) log(1/α). This could be used to formulate precise fluctuation conjectures for the minimum, analogous to those for the maximum.
  • The multi-scale recursive technique used to handle zeros at very small α (down to 1/log T) may be adaptable to other L-functions where the explicit formula has a similar zero contribution, such as Dirichlet L-functions or automorphic L-functions, potentially yielding sharp negative moments there as well.
  • The improvement in the Möbius bound, from √x exp((log x)^{k/(k+1)} (log log x)^{7/(k+1)+ε}) to √x exp(C (log x / log log x)^{k/(k+1)}), hints that the true exponent for the Möbius sums under RH might be √(log x / log log x), which is the best possible up to the constant factor under current methods.
  • The paper leaves open the exact order of I_{-1/2}(α,T) when α ≪ 1/log T, as well as the second-order term in the minimum. Future work could aim for bounds with optimal constants or unconditional results for the negative moments.
Original 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 $$ \frac1T\int_T^{2T} \Big|\zeta(\tfrac12+\alpha+it)\Big|^{-2k}\,dt \ll \alpha^{-k^2}, $$ for $\alpha=o(1)$ and $\alpha\gg 1/\log T$. 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 M\"obius function. Namely, we show that $$ \sum_{n\leq x}\mu(n) \ll \sqrt{x}\exp \bigg(C\sqrt{\frac{\log x}{\log\log x}}\bigg) $$ 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 \[ \min_{|h|\leq1} \Big|\zeta\Big(\tfrac12+\alpha+i(t+h)\Big)\Big| =\alpha^{1+o(1)} \] in probability, provided that $\alpha \ll (\log T)^{-c}$ for some fixed $c >0$, and $\alpha\log T/\log\log T\to\infty$. 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 $\log|\zeta(1/2+\alpha+it)|^{-1}$, obtained by building on work of Arguin, Bailey, and Roberts.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-21: 1 request, 8,189 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.62 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.02 · states quantitative result 0.98 · compares with prior results 0.98 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 $$ \frac1T\int_T^{2T} \Big|\zeta(\tfrac12+\alpha+it)\Big|^{-2k}\,dt \ll \alpha^{-k^2}, $$ for $\alpha=o(1)$ and $\alpha\gg 1/\log T$. 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 M\"obius function. Namely, we show that $$ \sum_{n\leq x}\mu(n) \ll \sqrt{x}\exp \bigg(C\sqrt{\frac{\log x}{\log\log x}}\bigg) $$ 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 \[ \min_{|h|\leq1} \Big|\zeta\Big(\tfrac12+\alpha+i(t+h)\Big)\Big| =\alpha^{1+o(1)} \] in probability, provided that $\alpha \ll (\log T)^{-c}$ for some fixed $c >0$, and $\alpha\log T/\log\log T\to\infty$. 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 $\log|\zeta(1/2+\alpha+it)|^{-1}$, obtained by building on work of Arguin, Bailey, and Roberts.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line. contribution 0.85 0.90 0.07 asserted 0.62
s1 For every fixed $0<k<1/2$, we establish the conjecturally sharp bound $$ \frac1T\int_T^{2T} \Big|\zeta(\tfrac12+\alpha+it)\Big|^{-2k}\,dt \ll \alpha^{-k^2}, $$ for $\alpha=o(1)$ and $\alpha\gg 1/\log T$. result 0.99 0.96 0.53 asserted 1.00
s2 We also obtain almost sharp upper bounds slightly to the left of $1/\log T$. result 0.96 0.93 0.08 asserted 1.00
s3 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. interpretation 0.58 0.83 0.15 asserted 1.00
s4 A separate bound for the moment at $k=1/2$ yields an improved bound for cancellation in the M\"obius function. result 0.80 0.92 0.08 asserted 1.00
s5 Namely, we show that $$ \sum_{n\leq x}\mu(n) \ll \sqrt{x}\exp \bigg(C\sqrt{\frac{\log x}{\log\log x}}\bigg) $$ for some $C>0$, improving on previous results of Soundararajan, Balazard and de Roton, and of the first two authors. result 0.99 0.96 0.14 asserted 1.00
s6 For $t$ chosen uniformly from $[T,2T]$, we also prove that \[ \min_{|h|\leq1} \Big|\zeta\Big(\tfrac12+\alpha+i(t+h)\Big)\Big| =\alpha^{1+o(1)} \] in probability, provided that $\alpha \ll (\log T)^{-c}$ for some fixed $c >0$, and $\alpha\log T/\log\log T\to\infty$. result 1.00 0.96 0.13 asserted 1.00
s7 This agrees with the leading order predicted by a minimum analogue of the Fyodorov--Hiary--Keating Conjecture. interpretation 0.98 0.45 0.08 asserted 0.57
s8 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 $k>1/2$. result 0.65 0.93 0.07 asserted 0.98
s9 A main ingredient in the proof of some of these results is a Gaussian large-deviation upper bound for $\log|\zeta(1/2+\alpha+it)|^{-1}$, obtained by building on work of Arguin, Bailey, and Roberts. method 0.98 0.83 0.05 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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