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This paper conjectures that every product of shifted zeta functions at the nontrivial zeros has a full asymptotic expansion whose derivatives in the shifts yield all integer moments of mixed derivatives, including a new explicit formula for

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New conjectures give the full asymptotic expansion of discrete moments of mixed derivatives of the Riemann zeta function at its zeros, including a previously unknown formula for the second moment of ζ'(ρ).

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A genuinely new conjectural formula for the second moment of ζ′ at its zeros, built from the Ratios recipe; the main risk is unproved pole cancellation in the shift variables. the 2 major comments →

arxiv 2509.07792 v1 pith:IRL72LJZ submitted 2025-09-09 math.NT

Integer moments of the derivatives of the Riemann zeta function

Christopher Hughes , Andrew Pearce-Crump This is my paper
classification math.NT MSC 11M0611M50
keywords Riemann zeta functiondiscrete moments at zerosderivatives of zetaShanks conjectureRatios Conjecturerandom matrix theorycharacteristic polynomialsStieltjes constants
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 conjectures the complete asymptotic expansion of any product of Riemann zeta functions evaluated at the nontrivial zeros, with small shifts added to each argument. Because differentiating with respect to a shift turns ζ(1/2+iγ+α) into ζ^(n)(1/2+iγ), differentiating the conjecture yields the leading-order and full asymptotic expansions for integer moments of mixed derivatives. The authors reach the same conjecture from two independent routes: a random matrix model, where the same combinatorial identities appear for characteristic polynomials, and the ratios recipe, which converts the discrete sum into a Cauchy integral and then into an Euler-product main term. The flagship concrete output is a full asymptotic for the second moment of ζ′(1/2+iγ), with lower-order terms built from Stieltjes constants and products over primes, supported by numerical data up to the millionth zero. The entire structure depends on an unproved heuristic: that off-diagonal terms in the recipe's Dirichlet-series expansion oscillate away and contribute only to the error term.

Core claim

Under the Riemann Hypothesis and mild conditions on shifts α_j, Conjecture 1 states that the discrete shifted moment Σ_{0<γ≤T} ζ(1/2+iγ+α_1)···ζ(1/2+iγ+α_k) equals the δ-derivative at δ=0 of an integral over t of one zero-swap term Z_{α_1,...,α_k,δ} plus k one-swap terms (t/2π)^{−α_j−δ} Z_{...,−δ,...,−α_j}, together with (T/2π)log(T/2π), and an error O(T^{1/2+ε}). The Z functions are products of zeta values at 1+shift divided by ζ(1+α_j), times an Euler-product arithmetic factor A. Expanding the shifts and differentiating recovers Conjecture 3: the leading asymptotic for a product of n_j-th derivatives is (−1)^{Σ n_j + k} n_1!···n_k!/(Σ n_j +1)! times (T/2π)(log(T/2π))^{Σ n_j+1}. The k=2 cas

What carries the argument

The engine is a generating integral. Cauchy's theorem turns the discrete sum over zeros into a contour integral of (ζ′/ζ)(s) times the shifted product; the functional equation rewrites it so that the integrand becomes a d/dδ derivative of ζ(s+δ)/ζ(s) times the product. The ratios recipe then replaces ζ(s) by its Dirichlet series and each numerator ζ by its approximate functional equation, keeps only diagonal terms where h m = n_1···n_k, and resums the surviving Euler products into the rational functions Z and the arithmetic factor A. The zero- and one-swap terms are exactly the choices of χ factors that contain equal numbers of χ(s) and χ(1−s), so the oscillatory pieces drop out. Taking d/dδ

Load-bearing premise

The conjecture rests on the recipe's unproved diagonal-dominance step—after substituting the Dirichlet series and approximate functional equations, all terms with h times m not equal to n_1···n_k are assumed to oscillate in t and vanish into the O(T^{1/2+ε}) error—together with the asserted holomorphy of the surviving permutation sum in the shifts.

What would settle it

Compute Σ_{0<γ≤T} ζ′(1/2+iγ)^3 numerically for increasing T: Conjecture 3 predicts the leading term is (1/24)(T/2π)(log(T/2π))^4, so a persistent relative deviation larger than T^{−1/2+ε} at large T would rule out the conjecture. A more direct check is to compute the off-diagonal sums in the recipe's expansion (7.2) for small k and verify that they decay relative to the diagonal terms at the asserted rate.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If Conjecture 1 holds, Conjecture 3 gives the leading asymptotic for every mixed-derivative moment, and Conjecture 4 gives the kth moment of the nth derivative of ζ at the zeros.
  • The k=2 case yields Conjecture 5, an explicit full asymptotic for the previously open second moment of ζ′(1/2+iγ), including all lower-order terms.
  • The k=1 case recovers, in integral form, the full Shanks asymptotic for all higher derivatives of ζ at the zeros, matching an earlier result whose error term becomes power-saving under RH.
  • The random matrix analogue produces the same factorial coefficient for derivatives of characteristic polynomials evaluated at eigenvalues, strengthening the link between zero statistics and unitary eigenvalue statistics.
  • Any higher mixed moment can in principle be extracted from Conjecture 1 by Taylor expansion and differentiation, though the paper writes out only the leading term for k≥3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the authors do not spell out: applying the same δ-derivative device to ratios with denominator shifts should yield full asymptotics for correlations such as ζ^(m)(1/2+iγ+β)ζ^(n)(1/2+iγ−β), giving joint information about derivatives at nearby zeros.
  • The leading-order coefficient in Conjecture 3 comes entirely from Vandermonde and complete-homogeneous-symmetric-polynomial identities, independent of the Euler-product factor; this suggests the leading term is more robust than the full expansion, whose arithmetic terms depend on the diagonal-dominance assumption.
  • Conjecture 5 implies that the real part of Σ ζ′(1/2+iγ)^2 is positive on average with a specific polynomial shape; extending the numerical comparison far beyond the first million zeros would either sharpen confidence in the O(T^{1/2+ε}) error term or reveal its failure.
  • If the recipe's off-diagonal assumption fails, the random matrix derivation still predicts the leading factorial coefficients, so the two routes would diverge first in the lower-order arithmetic terms rather than in the leading term.
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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.

Referee Report

2 major / 4 minor

Summary. The paper formulates Conjecture 1, a Ratios-Conjecture-style asymptotic for the discrete shifted moments of products of Riemann zeta functions evaluated at the non-trivial zeros, assuming RH and small shifts. The main term is presented as the δ-derivative of an integral of a one-denominator ratio Z_{α_1,…,α_k,δ} plus one-swap terms, together with a T/(2π) log(T/(2π)) term and an O(T^{1/2+ε}) error. From this, the authors derive leading-order asymptotics for moments of mixed derivatives (Conjectures 3 and 4), which specialize to a new conjecture for the second moment of ζ′(1/2+iγ) (Conjecture 5) with numerical support. A parallel random-matrix model is developed, giving exact expressions for shifted characteristic-polynomial averages and leading-order derivative moments that match the zeta conjectures.

Significance. If Conjecture 1 holds, the paper provides a unified framework that generalizes the authors' earlier work on Shanks' conjecture to arbitrary mixed derivative moments and yields the first full asymptotic conjecture for the second moment of ζ′(ρ). The paper has several strong points: the k=1 case reproduces the proven asymptotic of [14]; the leading-order terms in Conjectures 3–4 agree with the independently obtained random-matrix results in Theorems 4–5; and the k=2 numerics in Section 9.2 show a small residual after subtracting the full conjectured polynomial, providing a falsifiable check. The derivations contain no fitted constants, and the arithmetic terms are explicitly defined Euler products. The main weaknesses are the unproved diagonal-dominance step in the Ratios recipe and an inadequately supported holomorphy assertion for the main term in the shifts; these are load-bearing for the derivative-moment consequences.

major comments (2)
  1. [§8, Remark 5 / Eq. (1.1)] The holomorphy of the right-hand side of (1.1) in the shifts is asserted in Remark 5 with a citation to Lemma 6.7 of [4], but this is not demonstrated for the specific expression here. After the δ-differentiation used in (8.1), the zero-swap term contains ζ′(1+α_j)/ζ(1+α_j) and the one-swap terms contain ζ(1+α_ℓ−α_j)/ζ(1+α_ℓ), so poles occur at α_j=0 and α_j=α_ℓ. Section 8 only verifies cancellation of the leading poles after replacing ζ(1+x) by 1/x and A by 1; no proof is given that the exact sum over j is analytic. Since Conjectures 3–5 are obtained by Taylor expansion about α_j=0 and term-by-term differentiation, a surviving pole would invalidate them. The expression is not the standard CFZ permutation sum, so a direct reduction to Lemma 6.7 of [4] is needed, or the conjecture must be stated with an explicit analytic-continuation caveat.
  2. [§7, Step 4] The diagonal-dominance step — discarding all terms with hm≠n_1⋯n_k after Eq. (7.2) — is the load-bearing heuristic that converts the oscillatory Cauchy integral into the algebraic main term Z_{α,δ} and A_{α}(δ). This is the Ratios Conjecture and is unproved; if off-diagonal contributions do not vanish at the asserted level, the main term of Conjecture 1 changes. The paper is transparent about this in the derivation, but the resulting conjectures (3–5) inherit this assumption and it should be stated explicitly when these results are advertised (e.g., in the abstract) rather than only in the derivation.
minor comments (4)
  1. [§7] The side of the rectangle at Re s = 1−c is called the 'left-hand side'; it should be the right-hand side.
  2. [§6] Typo 'requried' in the proof of Lemma 8; also 'truely' in Section 1 and 'occured' in the Acknowledgements.
  3. [§1] Reference [13] is listed as 'in preparation'; Conjecture 2 and the introduction rely on it. If possible, provide a stable reference or state its availability.
  4. [§9.2] The numerical evidence would be more compelling with a table of residual magnitudes at selected T, since the figures are qualitative; please also state the size of the imaginary part that is discarded.

Circularity Check

0 steps flagged

No significant circularity; main conjecture is a Ratios-recipe heuristic with independent benchmarks.

full rationale

Conjecture 1 is obtained by applying the CFZ Ratios recipe to the Cauchy contour integral (7.1). The main term's Z functions and arithmetic factor A are produced by the recipe's diagonal-sum steps (Dirichlet series, approximate functional equation, diagonal condition hm=n_1...n_k), not by fitting the target sums. No parameter is adjusted to data: the k=1 case reproduces the proven theorem of [14], and the k=2 prediction is compared with direct zero computations with arithmetic terms computed from primes, not fitted. Conjectures 3-5 follow by Taylor expansion and differentiation of Conjecture 1; the leading coefficients come from Laurent expansions of zeta and the independent combinatorial Lemma 2. The two frameworks (RMT Theorem 5 and Ratios Section 8) agree at leading order, providing an external cross-check. Self-citations [13] and [14] are contextual or benchmark, not load-bearing. Flagged weaknesses are (i) Section 7 Step 4's unproved diagonal-dominance heuristic, and (ii) Remark 5's citation of Lemma 6.7 of [4] for holomorphy without showing that the delta-derivative one-swap sum reduces to that lemma; both are epistemic risks, not circularities, since neither assumes the conjectures being derived.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central object is a conjecture produced by the Ratios Conjecture recipe, so the real axioms are the recipe's unproved steps, plus RH and the shift conditions. No data-fitting occurs: the constants in Conjecture 5 are standard Stieltjes-Laurent coefficients or values of the Euler-product factor A from (1.2), computed over the first 1000 primes, not adjusted to match data. The independent support is external: the k=1 case reproduces the proven [14] asymptotic, the leading order matches the independently proven random-matrix Theorem 5, and the k=2 numerics leave a small residual against a computation over the first million zeros. The error-term convention is the one explicitly chosen assumption.

axioms (6)
  • domain assumption Riemann Hypothesis
    Assumed in Conjectures 1-5 and throughout (Section 1). Needed for all non-trivial zeros to lie on the critical line and for the conditional convergence of the 1/ζ Dirichlet series used in Section 7 Step 1.
  • domain assumption Ratios-recipe diagonal dominance: non-diagonal sums hm ≠ n_1...n_k oscillate away
    Section 7 Step 4: 'The ratios methodology assumes all the non-diagonal terms oscillate away.' This unproved heuristic converts the contour integral into the Euler-product main term of Conjecture 1.
  • domain assumption Shift restrictions |Re α_j| < 1/4, |Im α_j| ≪_ε T^{1-ε}, |δ| < 1/4
    Stated in Conjecture 1 and carried through the paper; they keep the zeta ratios and Euler products in their region of convergence, away from the pole at s=1.
  • standard math Holomorphy of the permutation-sum main term in the shifts
    Remark 5: 'the symmetries of the expression imply that the poles cancel to form a holomorphic function', citing Lemma 6.7 of [4]. Required for the Taylor expansion and term-by-term differentiation in Section 8.
  • standard math Approximate functional equation for ζ on the critical line, errors discarded
    Section 7 Step 2; the discarded errors are asserted to fall into the O(T^{1/2+ε}) error term. Also χ(c+it+δ)χ(1-c-it+α_j) is replaced by (t/2π)^{-δ-α_j}(1+O(1/|t|)).
  • ad hoc to paper Error-term convention O(T^{1/2+ε})
    Remark 4 and the closing remark: the authors acknowledge that O(T^{1-δ}) for 0<δ<1/2 is plausible and adopt O(T^{1/2+ε}) 'in keeping with the original statements of the recipe/Ratios Conjecture'. An explicit convention, not a derived bound.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Integer moments of the derivatives of the Riemann zeta function." pith.science (2026). https://pith.science/paper/IRL72LJZ

@misc{pith2026250907792,
  author       = {Pith},
  title        = {Pith review of: Integer moments of the derivatives of the Riemann zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRL72LJZ}},
  note         = {Machine review of arXiv:2509.07792}
}
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read the original abstract

We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a conjecture for the integer moments of mixed derivatives of the zeta function. This generalises a result of the authors where they took complex moments of the first derivative of the zeta function, evaluated at the non-trivial zeros. We approach this problem in two different ways: the first uses a random matrix theory approach, and the second by the Ratios Conjecture of Conrey, Farmer, and Zirnbauer.

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Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

  1. [1]

    J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith. Auto- correlation of random matrix polynomials.Comm. Math. Phys., 237:365–395, 2003

    work page 2003

  2. [2]

    J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith. Inte- gral moments ofL-functions.Proc. Lond. Math. Soc., 91:33–104, 2005

    work page 2005

  3. [3]

    J. B. Conrey, D. W. Farmer, and M. R. Zirnbauer. Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groupsU(N).arXiv preprint math-ph/0511024, 2005

    Pith/arXiv arXiv 2005

  4. [4]

    J. B. Conrey, D. W. Farmer, and M. R. Zirnbauer. Autocorrelation of ratios ofL- functions.Commun. Number Theory Phys., 2(3):593–636, 2008

    work page 2008

  5. [5]

    Jean Coquet

    J. B. Conrey, A. Ghosh, and S. M. Gonek. Simple zeros of zeta functions. InColloque de Théorie Analytique des Nombres “Jean Coquet” (Marseille, 1985), volume 88 of Publ. Math. Orsay, pages 77–83. Univ. Paris XI, Orsay, 1988

    work page 1985

  6. [6]

    J. B. Conrey and N. C. Snaith. Applications of theL-functions ratios conjectures. Proc. Lond. Math. Soc., 94(3):594–646, 2007

    work page 2007

  7. [7]

    D. W. Farmer. Long mollifiers of the Riemann zeta-function.Mathematika, 40:71– 87, 1993

    work page 1993

  8. [8]

    A. Fujii. On a conjecture of Shanks.Proc. Japan Acad. A, 70:109–114, 1994. INTEGER MOMENTS OF THE DERIVATIVES OF ZETA 29

    work page 1994

  9. [9]

    A. Fujii. On the distribution of values of the derivative of the Riemann zeta function at its zeros. I.Proc. Steklov Inst. Math., 276:51–76, 2012

    work page 2012

  10. [10]

    S. M. Gonek, C. P. Hughes, and J. P. Keating. A hybrid Euler-Hadamard product for the Riemann zeta function.Duke Math. J., 136:507–549, 2007

    work page 2007

  11. [11]

    C. B. Haselgrove.Tables of the Riemann Zeta Function. Royal Society mathematical tables. Published for the Royal Society at the University Press, 1960

    work page 1960

  12. [12]

    Hughes, G

    C. Hughes, G. Martin, and A. Pearce-Crump. A heuristic for discrete mean values of the derivative of the Riemann zeta function.Integers, 24:A83, 2024

    work page 2024

  13. [13]

    Hughes and A

    C. Hughes and A. Pearce-Crump. Complex moments of the derivative of the Rie- mann zeta function. (in preparation)

    work page

  14. [14]

    Hughes and A

    C. Hughes and A. Pearce-Crump. A discrete mean-value theorem for the higher derivatives of the Riemann zeta function.J. Number Theory, 241:142–164, 2022

    work page 2022

  15. [15]

    Kaptan, Y

    D. Kaptan, Y. Karabut, and C. Yildirim. Some mean value theorems for the Riemann zeta-function and DirichletL-functions.Comment. Math. Univ. St. Pauli, 60:83–87, 2011

    work page 2011

  16. [16]

    J. P. Keating and N. C. Snaith. Random matrix theory andζ(1/2+it).Comm. Math. Phys., 214:57–89, 2000

    work page 2000

  17. [17]

    Pearce-Crump.Discrete Moments of the Riemann zeta function

    A. Pearce-Crump.Discrete Moments of the Riemann zeta function. PhD thesis, University of York, 2024

    work page 2024

  18. [18]

    D. Shanks. Review of ‘Tables of the Riemann zeta function’ by C.B. Haselgrove in collaboration with J.C.P. Miller.Math. Comp., pages 84–86, 1961. DEPARTMENT OFMATHEMATICS, UNIVERSITY OFYORK, YORK, YO10 5GH, UNITEDKINGDOM Email address:christopher.hughes@york.ac.uk SCHOOL OFMATHEMATICS, FRYBUILDING, WOODLANDROAD, BRISTOL, BS8 1UG, UNITEDKING- DOM Email add...

    work page 1961

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.