Consecutive moderate gaps between zeros of the Riemann zeta function

Anurag Sahay; Steven M. Gonek

REVIEW 3 major objections 4 minor 14 references

Consecutive moderate gaps between zeros of the Riemann zeta function

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Under RH and pair correlation, a positive proportion of zeta zeros start r consecutive moderate gaps.

desk verdict The consecutive-moderate-gaps counting argument is new and mostly right, but the proof of Theorem 2.1 has a fixable gap in deducing simplicity from the stated Pair Correlation Conjecture; the unconditional theorem is the cleanest part. read the letter →

arxiv 2412.15481 v1 pith:JN5UEVI4 submitted 2024-12-20 math.NT

Steven M. Gonek , Anurag Sahay This is my paper · ORCID

classification math.NT MSC 11M2611M0611M50

keywords RiemannzetafunctionzerosgapspaircorrelationconjectureHypothesismoderateXi-functionshortintervals

topics The Riemann Hypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the ordinates of Riemann zeta zeros can be followed by arbitrarily many consecutive gaps all of size at least a fixed multiple of the average gap. Its main conditional theorem says yes: assuming the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, for every fixed r ≥ 1 there is a c > 0 such that a positive proportion of zeros γ_n ≤ T have r consecutive gaps ≥ 2πc/log T. The proof is a short counting argument: the zeros whose first 'small' gap occurs j steps later are each bounded by f(c)N(T), where f(c) is the pair-correlation function. The paper also proves an unconditional result that, in most short intervals of length 2πm/log T, each of r consecutive subintervals contains a moderate gap, and applies this to bound ξ''/ξ' on the critical strip.

What carries the argument

The counting device is the partition of all zeros into sets S_j(T,c): zeros whose current and next j-1 gaps are moderate but whose (j+1)-st gap is small. These sets are disjoint, yielding the exact identity N_r(T,c) = N(T) - Σ_{j=1}^r |S_j(T,c)|. Under the Pair Correlation Conjecture each |S_j(T,c)| is at most f(c)N(T) plus an error, because every such zero has some nearby follower within 2πc/log T; the function f(c) is the conjectured limiting count of close pairs. The unconditional theorem instead uses the classical formula N(t+h)-N(t) = (h/2π) log t + S(t+h)-S(t) together with Fujii's mean-square bound on S(t+h)-S(t) to show that in most short intervals the zero count is close to its expected value, forcing a moderate gap among each subinterval's zeros.

What would settle it

Compute the empirical pair-counting function on high zeros: for a fixed c ≤ c_r/2 and a large height T, form the ratio N_r(T,c)/N(T) and compare it with 1 - r f(c). If for some fixed r and c it is eventually below 1 - r f(c) by a positive amount, or if the raw pair count in (5) deviates by a fixed proportion from f(c)N(T), then Theorem 2.1's conclusion is not derivable from the stated hypotheses and the conjecture behind it must be revisited.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.1: under RH and the Pair Correlation Conjecture, N_r(T,c), the number of ordinates γ_n ≤ T whose next r gaps are all at least 2πc/log T, satisfies N_r(T,c) ≥ (1 - r f(c) + o(1)) N(T), where f(c) = ∫_0^c (1 - (sin πu/(πu))^2) du. Because f(c) → 0 as c → 0+, this exceeds (1+o(1))N(T) times a positive constant for small c, so a positive proportion of zeros begin r consecutive moderate gaps. The paper gives explicit c_r values for r up to 1000, and shows unconditionally that every sufficiently long interval of the form (T,2T] contains a large set of heights t for which r consecutive subintervals of length 2πm/log T each contain a moderate gap.

Load-bearing premise

The load-bearing premise is Montgomery's Pair Correlation Conjecture, which asserts that the number of pairs of zeros within 2πc/log T of each other is asymptotically f(c)N(T); if this count differs, the proof's bound on each |S_j(T,c)| collapses.

Editorial extensions

If this is right

For every fixed r ≥ 1, Theorem 2.1 implies that for all sufficiently small c a positive proportion of zeros up to T are starting points of r consecutive moderate gaps.
The explicit bound f(c) ≤ (π/3)^2 c^3 for c ≤ 1/π yields a positive proportion whenever c < (3/π)^{4/3} r^{-1/3}; the table lists c_r for r = 1,...,1000.
Under the weaker Well-Spacing Hypothesis, the same argument gives N_r(T,c) ≥ (1 - r M c^δ + o(1))N(T), with a positive proportion whenever c is small enough relative to r, M, and δ.
Unconditionally, Theorem 3.1 gives heights t in a set of measure > (1-ε)T in (T,2T] such that each of r consecutive intervals (t+(j-1)h, t+jh] contains at least one pair of consecutive zeros with gap ≥ 4π/(3 log T).
Assuming RH, Theorem 4.1 supplies a sequence T_j → ∞ with T_{j+1}-T_j ≪ 1 on which ξ''/ξ'(σ+iT_j) ≪ (log T_j)^{C+1} uniformly for -1 ≤ σ ≤ 2, supporting estimates used in work on the derivative of ξ.

Reading between the lines

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

If the GUE Hypothesis predicts the joint distribution of r consecutive gaps, it should give the exact limiting density of zeros with r moderate gaps; the paper's f(c) bound is only a first-order lower bound obtained from pair data alone.
The disjoint decomposition N_r = N - Σ|S_j| is very soft; it would transfer verbatim to any sequence of points whose close-pair counting function reflects a similar level repulsion, so the positive-proportion conclusion should hold for other point processes with pair-correlation control.
A numerical check on high zeros would compare the empirical proportion of zeros with r runs of moderate gaps at c = c_r/2 against 1 - r f(c); systematic agreement would support the conjecture, while a serious deficit would indicate that pair correlations alone do not capture consecutive-gap structure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

We read the Gonek–Sahay note. The short version: the main conditional theorem is a nice idea but the proof as written has a gap that is real and repairable; the unconditional result in Section 3 is the most solid part; the application in Section 4 is the least polished.

What's new: the question of r consecutive moderate gaps hasn't been addressed before, and the decomposition N_r(T,c)=N(T)−sum|S_j| is exactly the right way to see it. The union bound over first-failure positions is simple and, once you have the right pair-correlation input, gives a positive proportion. The unconditional Theorem 3.1 is clean: using Fujii's mean-square bound, the authors show that on a set of t of density 1, each of r consecutive intervals of length 2πm/log T contains about m zeros, and therefore contains a gap of size ≫1/log T. That is a solid, self-contained contribution.

The problem: the proof of Theorem 2.1 claims that the Pair Correlation Conjecture (5) implies almost all zeros are simple. The stress-test note is correct. Equation (5) sums over pairs with 0<γ_n−γ_m<2πc/log T; a positive density of double zeros contributes nothing to that sum, yet each double zero has next gap zero and forces |S_1| to be a positive fraction of N(T). The standard Montgomery form with an even test function whose Fourier transform is positive does imply simplicity, so the theorem can be repaired by quoting the right form of the conjecture; but as written the proof's load-bearing step does not follow from the displayed conjecture. This is a genuine soft spot, not a nitpick, but it is fixable.

The application section has its own soft spots. The choice of T_j = γ3 + (log γ3)^{−C} and the claim that the zero of ξ' in the middle interval is at distance ≥2π/(3 log T) from an endpoint are asserted without proof. The estimates in Case 2 are sketched and some of the sums look like they need more care than the text gives. This is the least trustworthy part of the paper.

Overall: the paper deserves a serious referee. The main result is likely correct under the standard form of PCC; the unconditional result is genuinely useful. The referee should push for a rewritten proof of Theorem 2.1 using a pair-correlation statement that actually gives simplicity, and a fuller justification of Theorem 4.1. I'd send it out.

Referee Report

3 major / 4 minor

Summary. The paper studies the existence of r consecutive 'moderate' gaps between ordinates of nontrivial zeros of the Riemann zeta function, where a gap is moderate if it is at least 2πc/log T. Under the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, Theorem 2.1 claims that a positive proportion of zeros begin r consecutive moderate gaps; Corollary 2.2 gives explicit admissible ranges of c. A similar statement is proved under a weaker Well-Spacing Hypothesis. Section 3 gives an unconditional result: for every fixed r and ε, for large fixed m and h=2πm/log T, a set T⊂(T,2T] of measure >(1-ε)T exists such that each of r consecutive intervals of length h contains a moderate gap. Section 4 applies these ideas to prove, assuming RH, a bound for ξ''/ξ' on horizontal segments along a sequence T_j→∞ with bounded gaps, filling a gap noted in the literature.

Significance. If the proof gaps are repaired, this is a useful contribution. The main conditional result is a clean application of the pair-correlation heuristic, and the paper is transparent about hypotheses: it explicitly conditions on standard conjectures, fits no free parameters, and derives explicit constants recorded in Table 1. Theorem 3.1 is unconditional and its proof via Fujii's second-moment bound is elegant. The application in Section 4 addresses a stated gap in an earlier paper coauthored by the first author. However, as it stands, the proof of the central conditional theorem uses a consequence of the Pair Correlation Conjecture that is not contained in the stated version, and the proof of the application contains unresolved separation and sequence-selection issues. The significance is therefore real but conditional on a careful revision.

major comments (3)

[§2.3, Eq. (5) and proof of Theorem 2.1] The proof claims that the stated Pair Correlation Conjecture implies that almost all zeros are simple, but this does not follow from (5). The sum in (5) is restricted to pairs with 0<γ_n−γ_m<2πc/log T, so a positive density of multiple zeros contributes exactly zero to every such sum for every c>0. If a proportion δ>0 of zeros were double zeros, then each double zero would contribute at least one zero gap of size 0, so |S_1(T,c)|≥δ N(T)+o(N(T)), while the proof's bound would give |S_1(T,c)|≤(f(c)+o(1))N(T); since f(c)→0 as c→0+, these are incompatible for small c. The standard test-function form of the Pair Correlation Conjecture, with an even weight w having w(0)>0, does imply simplicity, so the theorem is likely repairable. Please replace (5) by such a form, or add an explicit hypothesis that all but o(N(T)) zeros are simple.
[§4, Theorem 4.1, sequence selection] The theorem asserts a sequence T_j→∞ with T_{j+1}−T_j≪1, but the proof does not justify this bounded-gap property. Theorem 3.1 only provides, for each large T, a set T⊂(T,2T] of measure >(1−ε)T. A set of large measure in a long interval need not contain points with bounded gaps: its complement could be a single interval of length εT, which is ≫1 for fixed ε. The proof says only 'there exists a set T ... Now let T_j=γ3+(logγ3)^{-C}', without explaining how the associated heights form a syndetic sequence. This can likely be repaired by showing that the chosen γ3's have positive density in [T,2T], but as written the stated conclusion is unsupported.
[§4, proof of Theorem 4.1, Case 2] The estimate in Case 2 that the sum over T_j−1≤γ≤γ2 is O((log T)^2) because 'γ−T_j ≫ 1/log T_j for each term' is not justified by the hypotheses of Theorem 3.1. The zero γ2 lies in the first interval (t,t+h], while γ3 lies in the second interval (t+h,t+2h]; nothing prevents γ2 from being extremely close to γ3. If γ3−γ2 is small, then T_j−γ2=(γ3−γ2)+(log γ3)^{-C} can be as small as (log γ3)^{-C}, which is far smaller than 1/log T_j, so the omitted sum is not O((log T)^2) and equation (18) does not follow. The preceding 'without loss of generality' about γ*3 does not fix this, since T_j is fixed to γ3+δ and the alternative γ4−γ*3≥2π/(3log T) is not used in the argument. A separation argument for γ2, γ3, and γ*3, or a different choice of T_j, is needed.

minor comments (4)

[§2.3, proof of Corollary 2.2] In the proof, the threshold is written as c < (3/π)^{4/3} r^{-1/3}; the calculation f(c)≤(π^2/9)c^3 gives c < (3/π)^{2/3} r^{-1/3}, matching the statement of the corollary. Please correct the exponent.
[§3, Theorem 3.1] The statement says each interval contains '< 3/2m ordinates', but the proof actually establishes the two-sided bound m/2 < M_j < 3m/2. The lower bound is needed for the average-spacing argument and should be included in the statement for clarity.
[§3, proof of Theorem 3.1] The intervals are written as '(t + (j − 1)h), t + jh]' with mismatched parentheses; they should be '(t+(j−1)h, t+jh]'.
[§4, proof of Theorem 4.1] The sentence 'We prove the first assertion only as the proof of the second is similar' does not correspond to the theorem as stated, which contains a single assertion; this appears to be a vestigial remark from an earlier version and should be removed or corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: main results are conditional on explicitly stated conjectures; the only self-citation is motivational and non-load-bearing.

full rationale

The paper's central conditional theorem (Theorem 2.1) is derived from the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, both stated explicitly as hypotheses rather than derived from the conclusion. The counting identity N_r(T,c) = N(T) - sum_j |S_j(T,c)| is definitional, and the bound |S_j(T,c)| <= (f(c)+o(1))N(T) follows formally once one accepts the pair-correlation bound on small nonzero gaps. The proof's assertion that 'the Pair Correlation Conjecture implies that almost all zeros are simple' is not obtained by assuming the target result; it is a separate inference, and as stated it may be a gap in the derivation (since equation (5) counts only nonzero separations), but that is a correctness concern, not circularity. The unconditional Theorem 3.1 is self-contained, using N(t) asymptotics and Fujii's mean-square bound, and the application in Theorem 4.1 explicitly re-proves a bound that had been stated without proof in Farmer–Gonek–Lee [8], acknowledging the missing justification rather than citing it as evidence. Self-citations to [8] are contextual or motivational, not load-bearing. No constants are fitted to data, no target result is assumed in its own proof, and no known result is merely renamed. The circularity burden is therefore minimal.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted; the conditional results are explicitly conditional on standard conjectures. The main new mathematical input is a counting argument using PCC and an unconditional fluctuation estimate due to Fujii.

assumptions (7)

domain assumption Montgomery's Pair Correlation Conjecture (equation (5), Section 2.3)
Used in Theorem 2.1 to count consecutive small gaps by f(c)N(T); unproved.
domain assumption Riemann Hypothesis
Assumed in Theorems 2.1, 2.3, 4.1 and in the Alternative Hypothesis discussion; unproved.
domain assumption Well-Spacing Hypothesis
Assumed in Theorem 2.3 to obtain N_r ≥ (1 - r M c^δ) N(T); unproved.
domain assumption Alternative Hypothesis (Baluyot's form)
Assumed in Section 2.1 to infer all but o(N) gaps are ≥ π/(2 log T); unproved and considered unlikely.
standard math Fujii's mean-square estimate ∫(S(t+h)-S(t))^2 dt ≪ T log(3+h log T)
Cited from [7] and used in Theorem 3.1 to control local fluctuations of N(t).
standard math Titchmarsh's explicit formula N(t) = (t/2π) log(t/2π) - t/2π + 7/8 + S(t) + O(1/t)
Cited from [14], used in Theorem 3.1.
standard math Davenport's formula (10) for ζ'/ζ(s) near the critical line
Cited from [5], used in the proof of Theorem 4.1.

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Pith. "Pith review of Consecutive moderate gaps between zeros of the Riemann zeta function." pith.science (2026). https://pith.science/paper/JN5UEVI4

@misc{pith2026241215481,
  author       = {Pith},
  title        = {Pith review of: Consecutive moderate gaps between zeros of the Riemann zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JN5UEVI4}},
  note         = {Machine review of arXiv:2412.15481}
}

abstract

Let $0<\gamma_1\leq \gamma_2 \leq \cdots $ denote the ordinates of nontrivial zeros of the Riemann zeta function with positive imaginary parts. For $c>0$ fixed (but possibly small), $T$ large, and $\gamma_n\leq T$, we call a gap $\gamma_{n+1}-\gamma_n$ between consecutive ordinates ``moderate'' if $\gamma_{n+1}-\gamma_n \geq 2\pi c/\log T$. We investigate whether infinitely often there exists $r$ consecutive moderate gaps between ordinates $\gamma_{n+1}-\gamma_n, \gamma_{n+2}-\gamma_{n+1}, \ldots , \gamma_{n+r}- \gamma_{n+r-1}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Arias de Reyna and B

J. Arias de Reyna and B. Rodgers, On convergence of points to limiting processes, with an appl ication to zeta zeros , Expo. Math. 42 (5) (2024), Paper No. 125588

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[2]

S. A. C. Baluyot, On the pair correlation conjecture and the alternative hypo thesis, J. Number Theory 169 (2016), 183-226

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Conrey and H

B. Conrey and H. Iwaniec, Spacing of zeros of Hecke L-functions and the class number pr oblem, Acta Arith., 103 (3) (2002), 259–312

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[7]

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[8]

D. W. Farmer, S. M. Gonek, and Y. Lee, Pair correlation of the zeros of the derivative of the Rieman n ξ-function, J. Lond. Math. Soc. (2), 90 (1) (2014), 241–269

work page 2014

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[9]

H. L. Montgomery. The pair correlation of zeros of the zeta function , In Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Lo uis, Mo., 1972), pages 181–193. 1973

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M. L. Mehta, Random matrices , Pure and Applied Mathematics, 142, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004

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A. M. Odlyzko, On the distribution of spacings between zeros of the zeta fun ction, Math. Comp. 48 (177) (1987), 273–308

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B. Rodgers and H. S. Vallabhaneni, Autocorrelations of characteristic polynomials for the Al ternative Circular Unitary Ensemble , Glasg. Math. J. 66 (1) (2024), 51–64

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Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Arias de Reyna and B

J. Arias de Reyna and B. Rodgers, On convergence of points to limiting processes, with an appl ication to zeta zeros , Expo. Math. 42 (5) (2024), Paper No. 125588

work page 2024

[2] [2]

S. A. C. Baluyot, On the pair correlation conjecture and the alternative hypo thesis, J. Number Theory 169 (2016), 183-226

work page 2016

[3] [3]

Conrey and H

B. Conrey and H. Iwaniec, Spacing of zeros of Hecke L-functions and the class number pr oblem, Acta Arith., 103 (3) (2002), 259–312

work page 2002

[4] [4]

J. B. Conrey and C. L. Turnage-Butterbaugh, On r-gaps between zeros of the Riemann zeta-function , Bull. Lond. Math. Soc., 50 (2) (2018), 349–356

work page 2018

[5] [5]

Davenport, Multiplicative Number Theory , Graduate Texts in Math

H. Davenport, Multiplicative Number Theory , Graduate Texts in Math. 74, Springer-Verlag, New York, 1980

work page 1980

[6] [6]

des Cloizeaux and M

J. des Cloizeaux and M. L. Mehta, Asymptotic behavior of spacing distributions for the eigen values of random matrices , J. Mathematical Phys. 14 (1973), 1648–1650

work page 1973

[7] [7]

Fujii, On the distribution of the zeros of the Riemann Zeta function i n short intervals , Bull

A. Fujii, On the distribution of the zeros of the Riemann Zeta function i n short intervals , Bull. Amer. Math. Soc. 81 (1975), 139–142

work page 1975

[8] [8]

D. W. Farmer, S. M. Gonek, and Y. Lee, Pair correlation of the zeros of the derivative of the Rieman n ξ-function, J. Lond. Math. Soc. (2), 90 (1) (2014), 241–269

work page 2014