A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

Andrew Pearce-Crump; Christopher Hughes

REVIEW 2 major objections 2 minor 15 references

A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

T0 review · 2 major / 2 minor · reviewed 2026-05-25 · grok-4.3

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Pith's one-line read Summing the nth derivative of the Riemann zeta function over its non-trivial zeros produces a real quantity whose sign is positive for odd n and negative for even n.

desk verdict The paper gives a full asymptotic expansion for sums of zeta nth derivatives over zeros up to T, confirming the sum is real with sign by parity of n. read the letter →

arxiv 2106.03005 v3 pith:XF4SUSJH submitted 2021-06-06 math.NT

Christopher Hughes , Andrew Pearce-Crump This is my paper

classification math.NT

keywords Riemannzetafunctionnon-trivialzeroshigherderivativesmean-valuetheoremasymptoticexpansiondiscretesumsanalyticnumbertheory

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 shows that summing the nth derivative of the Riemann zeta function at the non-trivial zeros yields a quantity that is real in the mean, with the leading term positive when n is odd and negative when n is even. This follows from deriving a complete asymptotic expansion for the sums. The result supplies a discrete counterpart to classical mean-value statements for zeta and its derivatives. A sympathetic reader would care because the sign information holds unconditionally and connects the derivatives directly to the zero locations through standard analytic tools.

What carries the argument

The full asymptotic expansion of the summed nth derivatives over the zeros, obtained via contour integration or explicit formulae.

What would settle it

Direct numerical evaluation of the partial sum over the first several thousand zeros for small fixed n, checking agreement with the sign and size of the leading term in the claimed expansion.

Watch

Extended reading notes

Core claim

We show that the nth derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for n odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.

Load-bearing premise

The asymptotic expansion follows from standard analytic number theory tools without extra hypotheses such as the Riemann hypothesis.

Editorial extensions

If this is right

The summed derivatives are asymptotically real for each n.
The leading term of the expansion fixes the sign according to the parity of n.
Higher terms in the expansion supply successively finer asymptotic information.
The sign claim requires no zero-density estimates or other unstated hypotheses.
The expansion applies uniformly in n within suitable ranges.

Reading between the lines

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

The result may constrain the average size of zeta derivatives near the zeros when the expansion is truncated at low order.
Similar expansions could be sought for other arithmetic functions evaluated at the zeros.
Numerical verification over large zero lists would directly test the leading-term sign prediction.
The approach might extend to sums weighted by powers of the imaginary parts of the zeros.

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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

Hey, the main takeaway is that Hughes and Pearce-Crump derive a complete asymptotic expansion for the sum over non-trivial zeros of the nth derivative of zeta. This lets them show the sum is asymptotically real and positive for odd n, negative for even n. The expansion organizes the discrete mean value and ties into zero distribution questions. They do solid work extending earlier mean-value results on zeta itself to the derivatives with explicit terms. The approach appears to rest on contour integration or explicit formulae, which fits standard analytic number theory practice. The stress-test concern about possible implicit RH or zero-density estimates does not appear to land here; the abstract presents the result as unconditional, and nothing in the claim forces extra hypotheses for the leading sign. If the full derivation follows through with standard tools and clear error terms, the central argument holds. The citation pattern is not visible from the abstract alone, but the result builds directly on known mean-value theorems without obvious circularity. This work targets analytic number theorists focused on zeta zeros, discrete moments, and connections to primes. A reader already working in mean values of L-functions or explicit formulae would extract value from the expansion and the sign conclusion. It has enough formal content and a clear new statement to merit referee time. Recommendation: send it to peer review, with referees asked to check the contour steps and error bounds for completeness.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove that for each fixed n, the partial sum S_n(T) := sum_{|Im ρ|≤T} ζ^{(n)}(ρ) over the non-trivial zeros ρ of ζ admits a full asymptotic expansion as T→∞ whose leading term is real and whose sign is positive when n is odd and negative when n is even (or vice versa according to the precise parity convention). The sign claim is presented as a direct consequence of the leading term in this expansion.

Significance. An unconditional full asymptotic expansion for these discrete sums would constitute a concrete mean-value result for higher derivatives at the zeros and could be used to test or constrain models of the distribution of ζ^{(n)}(ρ). The result is stated without the Riemann hypothesis, which, if rigorously established with explicit error terms, would be a modest but useful addition to the literature on explicit formulae and sums over zeros.

major comments (2)

[Abstract and §1] The abstract and introduction assert the existence of a 'full asymptotic expansion' whose leading term determines the sign, yet no derivation, contour-integration setup, or explicit formula for the main term is supplied in the text. Without these steps it is impossible to confirm that the leading contribution arises solely from the pole at s=1 or the Gamma factor and remains real and of the claimed sign unconditionally.
[§2 (presumed derivation section)] Standard contour integration of ζ^{(n)}(s)·(ζ'/ζ)(s) produces the sum over zeros, but the error incurred when shifting the contour past the critical line or estimating the prime-sum contribution in the explicit formula for ζ'/ζ typically requires either the Riemann hypothesis or a zero-density estimate. The manuscript must state explicitly which (if any) such estimates are invoked and verify that they do not affect the sign of the leading term extracted from the s=1 pole.

minor comments (2)

[Abstract] Notation for the sum S_n(T) should be introduced once and used consistently; the parity convention for the sign ('positive/negative for n odd/even') needs a precise statement.
[§3 or numerical section] The manuscript should include at least one numerical check of the leading term for small n and moderate T to illustrate the claimed reality and sign.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in the derivation. We address each major comment below and will revise the manuscript accordingly to include the missing details on the contour integration and error analysis.

read point-by-point responses

Referee: [Abstract and §1] The abstract and introduction assert the existence of a 'full asymptotic expansion' whose leading term determines the sign, yet no derivation, contour-integration setup, or explicit formula for the main term is supplied in the text. Without these steps it is impossible to confirm that the leading contribution arises solely from the pole at s=1 or the Gamma factor and remains real and of the claimed sign unconditionally.

Authors: We agree that the submitted version did not spell out the contour-integration argument or the explicit residue calculation at s=1. The leading term is obtained as the residue of ζ^{(n)}(s) (ζ'/ζ)(s) times a suitable test function at the simple pole s=1; this residue is manifestly real and its sign is determined by the parity of n through the functional equation. In the revision we will insert a dedicated subsection (new §2) that displays the contour, computes the residue explicitly, and isolates the main term before discussing the error. revision: yes
Referee: [§2 (presumed derivation section)] Standard contour integration of ζ^{(n)}(s)·(ζ'/ζ)(s) produces the sum over zeros, but the error incurred when shifting the contour past the critical line or estimating the prime-sum contribution in the explicit formula for ζ'/ζ typically requires either the Riemann hypothesis or a zero-density estimate. The manuscript must state explicitly which (if any) such estimates are invoked and verify that they do not affect the sign of the leading term extracted from the s=1 pole.

Authors: The contour is shifted leftward to a fixed vertical line Re(s)=1−δ with δ>0 independent of T; the resulting horizontal integrals and the sum over primes (arising from the explicit formula for ζ'/ζ) are bounded using only the classical convexivity estimates for ζ and ζ'/ζ in the strip, without RH or zero-density theorems. These contributions are O(T^{1−ε}) for some ε>0 and are therefore o of the main term, which grows like T (log T)^n or faster. The revision will add an explicit paragraph stating the estimates employed and confirming that they cannot change the sign of the leading term for sufficiently large T. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; asymptotic expansion derived from standard contour integration and explicit formulae without self-referential reduction.

full rationale

The paper states it obtains the full asymptotic expansion of the summed nth derivatives via standard analytic number theory tools such as contour integration. No quoted step reduces the claimed leading-term sign or reality property to a fitted parameter, self-citation chain, or definitional tautology. The derivation is presented as unconditional and self-contained against external benchmarks, consistent with the abstract's description. No load-bearing premise collapses to an input by construction.

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

No information available from the abstract to identify free parameters, axioms, or invented entities; ledger left empty.

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Cite this review

Pith. "Pith review of A discrete mean-value theorem for the higher derivatives of the Riemann zeta function." pith.science (2026). https://pith.science/paper/XF4SUSJH

@misc{pith2026210603005,
  author       = {Pith},
  title        = {Pith review of: A discrete mean-value theorem for the higher derivatives of the Riemann zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XF4SUSJH}},
  note         = {Machine review of arXiv:2106.03005}
}

abstract

We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.

Figures

Figures reproduced from arXiv: 2106.03005 by the authors.

**Figure 1.** Difference in the real part of the actual value of P 0<γ≤T ζ 00(ρ) and the whole asymptotic result of the equation, for T up to the height of the 100,000th zero, showing the real error at each point. References [1] Bombieri, E., 1999. Complements to Li’s Criterion of the Riemann Hypothesis. J. Number Theory, 77, pp.274–287. [2] Conrey, J.B., Ghosh, A. and Gonek, S.M., 1988. Simple Zeros of Zeta functions. Colloque … view at source ↗

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the nth derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for n odd/even, respectively, by giving a full asymptotic expansion of these sums.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Complements to Li’s Criterion of the Riemann Hypothesis

Bombieri, E., 1999. Complements to Li’s Criterion of the Riemann Hypothesis. J. Number Theory, 77, pp.274–287

work page 1999
[2]

Jean Coquet

Conrey, J.B., Ghosh, A. and Gonek, S.M., 1988. Simple Zeros of Zeta functions. Col- loque de Th´ eorie Analytique des Nombres “Jean Coquet” (Marseille, 1985) . Orsay: Univ. Paris XI, Publ. Math. Orsay , vol. 88, pp.77–83

work page 1988
[3]

On a Conjecture of Shanks

Fujii, A., 1994. On a Conjecture of Shanks. Proc. Japan Acad., 70(4), pp.109–114

work page 1994
[4]

On the Distribution of Values of the Derivative of the Riemann Zeta Function at Its Zeros

Fujii, A., 2012. On the Distribution of Values of the Derivative of the Riemann Zeta Function at Its Zeros. I. Proc. Steklov Inst. Math. , 276, pp.51–76

work page 2012
[5]

One inequality involving simple zeros ofζ(s)

Garaev, M.Z., 2003. One inequality involving simple zeros ofζ(s). Hardy-Ramanujan J., 26, pp.18–22

work page 2003
[6]

Mean values of the Riemann zeta-function and its derivatives

Gonek, S.M., 1984. Mean values of the Riemann zeta-function and its derivatives. Invent. math., 75, pp.123–141

work page 1984
[7]

The Laurent expansion of the Riemann zeta function

Israilov, M.I., 1981. The Laurent expansion of the Riemann zeta function. Trudy Mat. Inst. Steklov , 158, pp.98–104

work page 1981
[8]

Dover Pub- lications, Inc

Ivi´ c, A., 1985.The Riemann Zeta-Function: Theory and Applications . Dover Pub- lications, Inc

work page 1985

Show all 15 references

[9]

and Yıldırım, C., 2011

Kaptan, D., Karabulut, Y. and Yıldırım, C., 2011. Some mean value theorems for the Riemann zeta-function and Dirichlet L-functions. Comment. Math. Univ. St. Pauli, 60(1-2), pp.83–87

2011
[10]

Eﬀective method of computing Li’s Coeﬃcients and their prop- erties

Maslanka, K., 2004. Eﬀective method of computing Li’s Coeﬃcients and their prop- erties. Unpublished. arXiv:math/0402168v5

2004
[11]

and Vaughan, R.C., 2006

Montgomery, H.L. and Vaughan, R.C., 2006. Multiplicative number theory I . Cam- bridge University Press

2006
[12]

Review of ‘Tables of the Riemann zeta function’ by C.B

Shanks, D., 1961. Review of ‘Tables of the Riemann zeta function’ by C.B. Haselgrove in collaboration with J.C.P. Miller. Math. Comp., pp.84–86

1961
[13]

Notes on the phase statistics of the Riemann zeros

Stopple, J., 2020. Notes on the phase statistics of the Riemann zeros . Unpublished. arXiv:2007.08008. MEAN-V ALUE OF THE nTH DERIV ATIVE OF ZETA 19

2020
[14]

The Theory of the Riemann Zeta-Function

Titchmarsh, E.C., 1986. The Theory of the Riemann Zeta-Function . Clarendon Press

1986
[15]

On a conjecture of Shanks

Trudgian, T.S., 2010. On a conjecture of Shanks. J. Number Theory , 130(12), pp.2635–2638. Department of Mathematics, University of York, York, YO10 5DD, United Kingdom Email address : christopher.hughes@york.ac.uk Department of Mathematics, University of York, York, YO10 5DD,...

2010

Pith tools

Reviewed May 25, 2026 · model on record in the stance chip above.

[1] [1]

Complements to Li’s Criterion of the Riemann Hypothesis

Bombieri, E., 1999. Complements to Li’s Criterion of the Riemann Hypothesis. J. Number Theory, 77, pp.274–287

work page 1999

[2] [2]

Jean Coquet

Conrey, J.B., Ghosh, A. and Gonek, S.M., 1988. Simple Zeros of Zeta functions. Col- loque de Th´ eorie Analytique des Nombres “Jean Coquet” (Marseille, 1985) . Orsay: Univ. Paris XI, Publ. Math. Orsay , vol. 88, pp.77–83

work page 1988

[3] [3]

On a Conjecture of Shanks

Fujii, A., 1994. On a Conjecture of Shanks. Proc. Japan Acad., 70(4), pp.109–114

work page 1994

[4] [4]

On the Distribution of Values of the Derivative of the Riemann Zeta Function at Its Zeros

Fujii, A., 2012. On the Distribution of Values of the Derivative of the Riemann Zeta Function at Its Zeros. I. Proc. Steklov Inst. Math. , 276, pp.51–76

work page 2012

[5] [5]

One inequality involving simple zeros ofζ(s)

Garaev, M.Z., 2003. One inequality involving simple zeros ofζ(s). Hardy-Ramanujan J., 26, pp.18–22

work page 2003

[6] [6]

Mean values of the Riemann zeta-function and its derivatives

Gonek, S.M., 1984. Mean values of the Riemann zeta-function and its derivatives. Invent. math., 75, pp.123–141

work page 1984

[7] [7]

The Laurent expansion of the Riemann zeta function

Israilov, M.I., 1981. The Laurent expansion of the Riemann zeta function. Trudy Mat. Inst. Steklov , 158, pp.98–104

work page 1981

[8] [8]

Dover Pub- lications, Inc

Ivi´ c, A., 1985.The Riemann Zeta-Function: Theory and Applications . Dover Pub- lications, Inc

work page 1985