A note on a conjecture of Ng

Andrew Pearce-Crump

A note on a conjecture of Ng

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Under the Riemann Hypothesis and simple zeros, a second moment of a zeta-function ratio summed over non-trivial zeros is bounded below by half the conjectured value.

desk verdict This note derives a conditional lower bound reaching exactly half of Ng's conjectured value for the second moment of a zeta ratio summed over zeros. read the letter →

arxiv 2606.12376 v1 pith:OJHGDXVQ submitted 2026-06-10 math.NT

Andrew Pearce-Crump This is my paper · ORCID

classification math.NT

keywords Riemannzetafunctionnon-trivialzerossecondmomentNgconjectureHypothesissimple

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 proves a conditional lower bound on the second moment formed by summing a ratio of zeta functions over the non-trivial zeros of the Riemann zeta function. The bound equals half the size predicted by Ng's conjecture. The result requires the Riemann Hypothesis plus the assumption that every non-trivial zero is simple. A reader would care because the moment controls the average size of the ratio at the zeros themselves, which in turn governs how zeta behaves near its own zeros. The work therefore supplies a partial step toward confirming the full conjectured size of that moment.

What carries the argument

The summed second moment over non-trivial zeros ρ of a ratio of zeta functions evaluated at shifts of ρ.

What would settle it

An explicit multiple zero of the zeta function, or a numerical computation of the moment that falls below the stated lower bound while satisfying the assumptions, would contradict the result.

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Extended reading notes

Core claim

Assuming the Riemann Hypothesis and that all non-trivial zeros are simple, the sum over those zeros of the squared modulus of a fixed ratio of zeta functions is at least half the value Ng conjectured for the same sum.

Load-bearing premise

The proof requires that the Riemann Hypothesis holds and that every non-trivial zero of the zeta function is simple.

Editorial extensions

If this is right

Ng's conjecture receives support at least up to a factor of one-half under the stated hypotheses.
The average squared size of the ratio at the zeros cannot be smaller than the proved threshold.
Any further unconditional improvement would have to exceed this conditional half-size bound.
The same method may apply to related moments involving derivatives or higher powers of the ratio.

Reading between the lines

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

If the full conjecture holds, the lower bound proved here is asymptotically tight up to the factor of two.
The result may combine with upper bounds already in the literature to pin the moment between half and the full conjectured value.
Removing the simplicity assumption would require handling contributions from multiple zeros separately.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The punchline is that Pearce-Crump shows, under RH plus simplicity of all zeros, that the second moment in question is at least half the size Ng conjectured. That specific half-bound is the new piece; prior work apparently did not record it.

The paper does the straightforward job of stating the assumptions up front and delivering the inequality without extra claims. It is short and precise about what it reaches and what it does not.

The main limitation is that the bound stops at half the target. That is not a flaw in execution but it does keep the result incremental. The derivation itself is not visible from the abstract, so one cannot yet check the error terms or how the ratio is handled, but nothing in the statement suggests circularity or hidden fitting. The assumptions are the usual ones for this area and are not hidden.

This is for analytic number theorists who track moment conjectures for zeta and L-functions. Someone already following Ng's work or related ratios over zeros will get a small but usable data point. It is not broad enough for a general audience.

I would send it to a referee. The result is cleanly stated, the assumptions are explicit, and the contribution is modest but honest; a specialist can judge the details in review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to establish a conditional lower bound (under the Riemann Hypothesis and the assumption that all non-trivial zeros of the zeta function are simple) for the second moment of a ratio of zeta functions summed over the non-trivial zeros of the Riemann zeta function; the bound obtained is half the size of the value conjectured by Ng.

Significance. If correct, the result supplies a partial, conditional confirmation of Ng's conjecture by recovering 50% of the expected main term. This constitutes incremental progress in the study of moments of zeta ratios at zeros, but the factor of 1/2 and the restrictive hypotheses limit its immediate impact; the work is presented explicitly as a note rather than a full resolution.

minor comments (2)

[Abstract] The abstract refers to 'a ratio of zeta functions' without specifying the precise form (e.g., which shifts or which functions appear); a single sentence clarifying the ratio would improve readability.
The manuscript should include a brief comparison, even if only heuristic, explaining why the method yields exactly half the conjectured constant rather than the full value.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for recommending minor revision. The referee's summary correctly captures the content and scope of our note. Since no specific major comments were listed under the MAJOR COMMENTS section, we have no points requiring detailed rebuttal or manuscript changes. The limitations noted in the significance section (factor of 1/2, conditional hypotheses) are already explicitly stated in the manuscript and reflect the incremental nature of the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper derives a conditional lower bound (half the conjectured size) for the indicated second moment of a zeta ratio summed over zeros, under the external assumptions of RH and simplicity of all non-trivial zeros. No equations or steps are described that reduce the bound to a fitted input, self-definition, or self-citation chain; the result is presented explicitly as a partial inequality rather than a full resolution or unification. The derivation chain is self-contained against the stated assumptions, with no load-bearing renamings, ansatzes, or uniqueness claims imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on two strong domain assumptions stated in the abstract; no free parameters or invented entities are visible.

assumptions (2)

domain assumption Riemann Hypothesis
Explicitly required for the lower bound to hold.
domain assumption All non-trivial zeros are simple
Explicitly required for the lower bound to hold.

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Cite this review

Pith. "Pith review of A note on a conjecture of Ng." pith.science (2026). https://pith.science/paper/OJHGDXVQ

@misc{pith2026260612376,
  author       = {Pith},
  title        = {Pith review of: A note on a conjecture of Ng},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJHGDXVQ}},
  note         = {Machine review of arXiv:2606.12376}
}

read the original abstract

In this note we give a lower bound for the second moment of a ratio of zeta functions summed over the non-trivial zeros of the Riemann zeta function that is half the size of the conjectured value. Our result is conditional on the assumption of the Riemann Hypothesis and that all the non-trivial zeros of the zeta function are simple.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Negative discrete second moments of Dirichlet $L$-functions
math.NT 2026-06 unverdicted novelty 6.0 of 10

Assuming GRH and simple zeros, lower bounds are established for ∑ |L'(ρ,χ)|^{-2} and ∑ |L(2ρ,χ²)/L'(ρ,χ)|² that recover half the conjectured asymptotic when q is fixed and degrade to 1/(2+A) when q = T^A.

Reference graph

Works this paper leans on

11 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Davenport,Multiplicative Number Theory, 3rd ed., revised by H

H. Davenport,Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics74, Springer-Verlag, New York, 2000

2000
[2]

S. M. Gonek,The second moment of the reciprocal of the Riemann zeta-function and its derivative, Talk at Mathematical Sciences Research Institute, Berkeley, June 1999

1999
[3]

Hamieh, H

A. Hamieh, H. Kadiri, G. Martin, and N. Ng,Comparative Prime Number Theory Problem List, arXiv:2407.03530 (2024)

work page arXiv 2024
[4]

M. B. Milinovich and N. Ng,A note on a conjecture of Gonek, Functiones et Approx- imatio46.2(2012), 177–187

2012
[5]

H. L. Montgomery and R. C. Vaughan,Hilbert’s inequality, J. London Math. Soc. (2) 8(1974), 73–82

1974
[6]

Ng,Limiting distributions and zeros of ArtinL-functions, Ph.D

N. Ng,Limiting distributions and zeros of ArtinL-functions, Ph.D. thesis, University of British Columbia, 2000

2000
[7]

Ng,Prime number error terms, preprint, arXiv:2505.11295 (2025)

N. Ng,Prime number error terms, preprint (2025), arXiv:2505.11295

work page arXiv 2025
[8]

Ng, private communication, 2026

N. Ng, private communication, 2026

2026

Show all 11 references

[9]

Rudnick and K

Z. Rudnick and K. Soundararajan,Lower bounds for moments ofL-functions, Proc. Natl. Acad. Sci. USA102(2005), 6837–6838

2005
[10]

E. C. Titchmarsh,The Theory of the Riemann Zeta-function, 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, 1986

1986
[11]

K. M. Tsang,SomeΩ-theorems for the Riemann zeta-function, Acta Arith.46 (1986), no. 4, 369–395. School of Mathematics, University of Bristol, Bristol, BS8 1UG, United Kingdom Email address:andrew.pearce-crump@bristol.ac.uk

1986

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Davenport,Multiplicative Number Theory, 3rd ed., revised by H

H. Davenport,Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics74, Springer-Verlag, New York, 2000

2000

[2] [2]

S. M. Gonek,The second moment of the reciprocal of the Riemann zeta-function and its derivative, Talk at Mathematical Sciences Research Institute, Berkeley, June 1999

1999

[3] [3]

Hamieh, H

A. Hamieh, H. Kadiri, G. Martin, and N. Ng,Comparative Prime Number Theory Problem List, arXiv:2407.03530 (2024)

work page arXiv 2024

[4] [4]

M. B. Milinovich and N. Ng,A note on a conjecture of Gonek, Functiones et Approx- imatio46.2(2012), 177–187

2012

[5] [5]

H. L. Montgomery and R. C. Vaughan,Hilbert’s inequality, J. London Math. Soc. (2) 8(1974), 73–82

1974

[6] [6]

Ng,Limiting distributions and zeros of ArtinL-functions, Ph.D

N. Ng,Limiting distributions and zeros of ArtinL-functions, Ph.D. thesis, University of British Columbia, 2000

2000

[7] [7]

Ng,Prime number error terms, preprint, arXiv:2505.11295 (2025)

N. Ng,Prime number error terms, preprint (2025), arXiv:2505.11295

work page arXiv 2025

[8] [8]

Ng, private communication, 2026

N. Ng, private communication, 2026

2026