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REVIEW 2 major objections 4 minor 17 references

Complex moments of the derivative of the Riemann zeta function

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper conjectures that complex moments of the derivative of the Riemann zeta function at its non-trivial zeros follow a single asymptotic formula for every exponent with real part greater than −3.

desk verdict New general conjecture for complex moments of zeta'(rho) with two clean random matrix derivations; the load-bearing splitting heuristic is the only real caveat. read the letter →

arxiv 2509.07788 v1 pith:RVVHI25J submitted 2025-09-09 math.NT

Christopher Hughes , Andrew Pearce-Crump This is my paper · ORCID
classification math.NT MSC 11M0611M2611M50
keywords RiemannzetafunctionderivativeatzeroscomplexmomentsdiscreterandommatrixtheoryhybridproductofprimeandzerofactorsToeplitzdeterminantsgamma
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

This paper proposes a leading-order formula for the complex moments of the derivative of the Riemann zeta function evaluated at its zeros. For every complex exponent k with real part greater than −3, the normalised sum over zeros is conjectured to approach (log(T/2π))^k / Γ(k+2), using the branch of the complex power fixed by the hybrid product of prime and zero factors. This is the first general conjecture for these complex moments, and it reproduces the two known cases k=1 and k=−1. Its notable prediction is that no arithmetic factor survives at leading order, in contrast to the corresponding absolute-value moments. The evidence comes from two independent random-matrix computations and a proof that the underlying splitting assumption holds for k=1.

What carries the argument

The central mechanism is the hybrid factorisation ζ(s) ≈ P_X(s) Z_X(s), where P_X is a truncated product over primes and Z_X is a truncated product over zeros. Differentiating at a zero gives ζ′(ρ) ≈ P_X(ρ) Z′_X(ρ), and the pseudo-independence of P_X and Z_X, the Splitting Conjecture, turns the moment of ζ′(ρ)^k into the product of the moment of P_X(ρ)^k and the moment of Z′_X(ρ)^k. The paper evaluates those two moments by exact methods: a multi-dimensional beta integral gives the characteristic-polynomial analogue, a Toeplitz determinant with one singularity at the origin gives the zero-factor analogue, and a zero-sum formula gives the prime-factor mean. Both factor analogues produce N^k /

What would settle it

Numerically test the normalised sum for a non-integer k with Re(k)>−3, say k=1/2, using a large set of zeros up to height T and the branch defined by the hybrid product; if the ratio to (log(T/2π))^{1/2} / Γ(5/2) does not approach 1 as T grows, the conjecture or the splitting heuristic fails. A cheaper check is to examine k=−1/2 or k=3/2, since the k=1 case is already proven.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Conjecture 1: assuming the Riemann Hypothesis, for Re(k)>−3, the average of ζ′(1/2+iγ)^k over zeros γ with 0<γ≤T is asymptotically (log(T/2π))^k / Γ(k+2). The derivation shows that a Haar-averaged unitary characteristic polynomial has exact complex derivative moments e^{iπk/2} Γ(N+k+1) / (N! Γ(k+2)), asymptotic to e^{iπk/2} N^k / Γ(k+2); the same leading constant, without the factor i^k once θ-differentiation is translated to t-differentiation, is obtained for the zero factor in the hybrid model. The prime factor separately has average 1 over the zeros. The branch is not obtained by continuous variation of log ζ′(s), but by the product represent

Load-bearing premise

The conjecture stands on the pseudo-independence of the truncated prime product P_X and the truncated zero product Z_X at the zeros of zeta; this splitting has been verified only for k=1, and if it fails for another k with Re(k)>−3 the formula would be wrong even though the random-matrix computations are individually correct.

Editorial extensions

If this is right

  • Every exponent Re(k)>−3 is covered by one closed formula; no new arithmetic constant is introduced.
  • The k=−2 case is predicted to have a vanishing leading term, with a heuristic bound O(T^{−1/2+ε}); this zero is a direct consequence of the pole of the gamma factor at k=−2.
  • The known k=1, proven unconditionally, and k=−1, known conditionally on simple zeros, arise as special cases, so the conjecture interpolates the existing evidence.
  • Because ζ′(ρ) is complex, numerical or analytic tests must adopt the hybrid-product branch rather than continuous variation of log ζ′(s); the paper identifies this as the correct convention.
  • The same hybrid-plus-random-matrix route is announced as extending to mixed products of higher derivatives of ζ at zeros.

Reading between the lines

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

  • The proof of the zero-factor theorem depends on the Fourier coefficients of the smoothing factor vanishing for all non-positive modes, so the leading constant should be independent of the particular smoothing function in the hybrid product; testing two different smoothings numerically would probe universality.
  • If the Splitting Conjecture fails, the first symptom would be a missing lower-order constant in the averaged ratio, not a visible failure at k=1; computing the normalised sum for k=1/2 at increasing heights is a sharper test than the k=1 verification.
  • The same gamma-only leading constant would plausibly carry over to other L-functions with unitary symmetry, where the discrete zeros play the role of eigenangles; the paper does not state this, but the mechanism is generic.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper formulates Conjecture 1, asserting that for Re(k)>-3 the discrete complex moments of zeta'(rho) at zeros satisfy (1/N(T)) sum_{0<gamma<=T} zeta'(1/2+i gamma)^k ~ (1/Gamma(k+2)) (log(T/2pi))^k as T->infty, with a branch convention based on the hybrid Euler-Hadamard product. The support consists of two independent random matrix calculations: Theorem 4 uses Selberg's integral for the derivative of unitary characteristic polynomials, and Theorem 6 uses a Toeplitz-determinant/Fisher-Hartwig computation for the hybrid model Z'_X. Theorem 7 shows that the average of P_X(rho)^k is 1 for complex k under RH, and Theorem 12 uses a twisted first moment to verify the resulting factorization for k=1 only.

Significance. If Conjecture 1 is correct, it provides the first conjectured leading-order asymptotics for complex moments of zeta' at its zeros, covering all Re(k)>-3 and unifying the known k=1 and k=-1 cases. The two random matrix computations are rigorous and mutually consistent, and they give the same constant and the same analytic region Re(k)>-3. The proof that the P_X factor contributes a mean of 1 for every complex k is a valuable independent result. The paper is also transparent about the role of the unproved Splitting Conjecture, although the main conjecture depends on it essentially.

major comments (2)
  1. [Section 3, after Theorem 7] The passage 'It is believed that P_X(s) and Z_X(s) operate pseudo-independently ... so the moments of zeta are products of moments of P_X and Z_X' is the load-bearing step. To obtain Conjecture 1 for every k with Re(k)>-3, one must multiply the averages from Theorem 6 and Theorem 7, i.e. assume E[P_X(rho)^k Z'_X(rho)^k] = E[P_X(rho)^k] E[Z'_X(rho)^k] for all such k. The only internal verification is k=1 (Theorem 12); the k=-1 case is cited from the literature but is not shown to follow from the same splitting, and no other k is tested. If the factorization fails for some k, say k=2, the leading constant in Conjecture 1 would acquire an arithmetic correction. This is not a flaw in the RMT derivations themselves, but it means Conjecture 1 as stated is stronger than the paper's evidence. The conjecture should either be stated as conditional on the Splitting Conjecture, or additional checks
  2. [Section 1 and Conjecture 1] The complex power zeta'(rho)^k is not well-defined without a branch choice, and the paper explicitly notes that the appropriate branch is 'not the same as that which comes from continuously varying log zeta'(s)' and is instead tied to the hybrid product. However, Conjecture 1 itself never states this branch convention. For non-integer k, a different branch changes the left side by a phase e^{2 pi i n k}, while the right side has a fixed phase from (log(T/2pi))^k. Since the conjecture is central, the branch definition (e.g. via the partial Hadamard product Z_X and the limiting procedure described in Section 3) must be part of the conjecture statement, not only an informal remark preceding it.
minor comments (4)
  1. [Proof of Theorem 7, Section 6] The proof begins 'In the case when Re(s)=1/2 and X=(log T)^{2-epsilon}', which is broader than the theorem's assumption X=O(log T). This is not a substantive problem, but the relation between the two choices of X should be clarified so the reader sees why the O(log T) condition is sufficient.
  2. [Theorem 6 statement] Typo: 'eignenangles' should be 'eigenangles'.
  3. [Section 5, proof of Theorem 6] The remark after the proof states the result holds for k not in {-3,-4,-5,...}. This set appears to omit k=-2, where Gamma(k+2) has a pole but the leading asymptotic is interpreted as zero. A sentence explaining the limiting interpretation at k=-2 would remove ambiguity.
  4. [Conjecture 1] The displayed formula for N(T) is easy to misread. Consider adding parentheses: N(T) = (T/(2pi)) log(T/(2pi e)) + O(log T).

Circularity Check

0 steps flagged · score 2.0 of 10

No circular step: the conjecture is derived from independent Selberg and Toeplitz computations with a transparent, explicitly labeled Splitting Conjecture; score 2 reflects minor self-citation, not circularity.

full rationale

The derivation chain is not circular. Conjecture 1 rests on three separate computations: Theorem 4 evaluates the characteristic-polynomial derivative moment exactly by Selberg's integral; Theorem 6 evaluates the hybrid Z'_X model by a Toeplitz determinant (Ehrhardt–Silbermann) with Fourier coefficients computed in Lemma 8; Theorem 7 evaluates the P_X factor via Landau/Gonek zero sums. These computations are independent and do not assume the target asymptotic. The only place where a product of averages is used is Section 3: 'It is believed that P_X(s) and Z_X(s) operate pseudo-independently ... and so the moments of zeta are products of moments of P_X and Z_X.' This is explicitly labeled a belief (the 'Splitting Conjecture' of GHK), not an established theorem, and the paper verifies it only for k=1 in Theorem 12 using an external twisted first-moment result of Benli–Elma–Ng. Reliance on an unproved heuristic is a correctness/assumption risk, not circularity. The self-citations to GHK/HKO are to published theorems with proofs (hybrid formula, RMT derivations) and do not assume Conjecture 1. The branch convention is defined via the hybrid product to match the RMT branch; this is a definitional choice, not an input masquerading as a prediction. The conjecture also matches the known k=1 and k=-1 benchmarks, providing external checks. No equation of the paper reduces by construction to a fitted parameter or to a prior result of the same authors, so no circular step is present. Score 2 is assigned only because several load-bearing ingredients come from the first author's prior work and because the Splitting Conjecture is inherited from that same circle; this is proximity and transparency, not circularity.

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

No numerical parameters are fitted: N=log(T/2pi) is the standard matrix-size matching, the hybrid truncation X=O(logT) is arbitrary and cancels at leading order, and the smoothing function u cancels in Lemma 8. The moment order k is the variable, not a fitted constant. The load-bearing content is the domain heuristics (RMT mapping, splitting conjecture, RH, branch choice) rather than any fitted value.

assumptions (7)
  • domain assumption Riemann Hypothesis
    Assumed at the outset (Section 1); needed to write zeros as rho=1/2+i gamma and used in Theorems 7 and 12 via Lemma 11.
  • domain assumption Keating-Snaith random matrix modeling heuristic
    The conjecture is obtained by replacing zeta with characteristic polynomials of Haar-distributed unitary matrices (Sections 2 and 5).
  • domain assumption Gonek-Hughes-Keating Splitting Conjecture
    Assumed pseudo-independence of P_X and Z_X to multiply their moments for arbitrary k; proven only for k=1 in Theorem 12.
  • domain assumption Hybrid Euler-Hadamard product formula (Theorem 5)
    Stated as a theorem following [7]; used to define Z_X' and to justify the branch in (3.4).
  • standard math Selberg integral and convergence
    Used in Theorem 4; the paper corrects the convergence regime in Mehta's book without proof.
  • standard math Ehrhardt-Silbermann Toeplitz determinant asymptotics
    Theorem 2.5 in [4] with one Fisher-Hartwig singularity, used in Theorem 6.
  • domain assumption Branch convention for complex powers
    The choice of arg(1-e^{i theta}) is explicit in the random matrix setting, but the corresponding zeta-side branch is described only via the product and is explicitly said not to be the naive continuous variation of log zeta'.

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Cite this review

Pith. "Pith review of Complex moments of the derivative of the Riemann zeta function." pith.science (2026). https://pith.science/paper/RVVHI25J

@misc{pith2026250907788,
  author       = {Pith},
  title        = {Pith review of: Complex moments of the derivative of the Riemann zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVVHI25J}},
  note         = {Machine review of arXiv:2509.07788}
}
read the original abstract

We conjecture results about the complex moments of the derivative of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this via two different random matrix computations. In the first, we find an exact formula for the complex moments of the derivative of the characteristic polynomials of unitary matrices averaged over Haar measure using Selberg's integral. In the second, we consider the hybrid approach for zeta, first proposed by Gonek, Hughes and Keating.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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