Riemann and the logarithmic derivatives of zeta

J. Arias de Reyna

arxiv: 2605.27552 · v1 · pith:YVQBE7MHnew · submitted 2026-05-26 · 🧮 math.HO · math.NT

Riemann and the logarithmic derivatives of zeta

J. Arias de Reyna This is my paper

Pith reviewed 2026-06-29 14:02 UTC · model grok-4.3

classification 🧮 math.HO math.NT

keywords Riemann zeta functionlogarithmic derivativesposthumous paperscritical lineCatalan's constantEuler-Mascheroni constantnon-trivial zeros

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The pith

Riemann derived explicit formulas for the logarithmic derivatives of zeta at s=1/2 from his unpublished notes.

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

The paper extracts and presents two formulas that Riemann wrote down for the first and second derivatives of log zeta at the point s=1/2. The first gives zeta prime over zeta at 1/2 as a combination of pi, the Euler-Mascheroni constant, and the log of 8 pi. The second expresses the combination of second derivative over zeta minus the square of the first in terms of 8, pi squared, Catalan's constant, and a sum of reciprocal squares over the ordinates of the non-trivial zeros. A reader would care because these supply concrete numerical values that tie the zeta function's local behavior on the critical line to global constants and the zero locations.

Core claim

In his posthumous papers Riemann considered the derivatives of log zeta(s) at s=1/2 and supplied the explicit evaluations zeta'(1/2)/zeta(1/2) equals pi/4 plus gamma/2 plus log(8 pi)/2 together with zeta''(1/2)/zeta(1/2) minus the square of the first derivative ratio equals 8 minus pi squared over 4 minus 2 G plus 2 times the sum over n of 1 over alpha_n squared.

What carries the argument

The logarithmic derivatives of zeta at the fixed point s=1/2, written as explicit combinations of elementary constants and a sum over the squares of the imaginary parts of the zeros.

If this is right

The first formula supplies a closed numerical value for the logarithmic derivative that can be used in series expansions around the critical line.
The second formula relates the curvature of log zeta at 1/2 to the sum of 1/alpha_n squared, thereby connecting a local analytic quantity to the distribution of zeros.
Both expressions can be inserted into functional equations or product representations without needing to evaluate zeta numerically at that point.

Where Pith is reading between the lines

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

If the sum over 1/alpha_n squared converges rapidly, the second formula could be turned into a practical numerical check against independently computed values of the second derivative.
The appearance of Catalan's constant suggests a possible link between the zeta derivatives at 1/2 and certain alternating series that arise in other contexts of analytic number theory.

Load-bearing premise

The displayed formulas match what Riemann actually wrote in the manuscript kept in Göttingen.

What would settle it

Direct inspection of the relevant page in Riemann's conserved notebook to check whether the two displayed identities appear in that form.

Figures

Figures reproduced from arXiv: 2605.27552 by J. Arias de Reyna.

**Figure 1.** Figure 1: Fragment where Riemann writes his formulas about the logarithmic derivatives of ζpsq at s “ 1 2 . The Academy’s proceedings published brief summaries of academic events, which explains why Riemann tried to keep his paper as short as possible. When reading the resulting article [7], one realizes that each sentence summarizes an entire paragraph. He mentions in the letter that there are two statements whose… view at source ↗

read the original abstract

In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some mathematicians. From that time I have been asked several times for references. So I decided to write this. Specially explaining the wonderful formulas \[\frac{\zeta'(\frac12)}{\zeta(\frac12)}=\frac{\pi}{4}+\frac{\gamma}{2}+\frac{\log(8\pi)}{2},\quad \frac{\zeta''(\frac12)}{\zeta(\frac12)}-\Bigl(\frac{\zeta'(\frac12)}{\zeta(\frac12)}\Bigr)^2=8-\frac{\pi^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{\alpha_n^2}\]

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short note recording two explicit formulas for log zeta derivatives at 1/2, credited to Riemann but without any source excerpt or page reference.

read the letter

The main point is that this is a brief note that writes down two closed-form expressions for the first and second logarithmic derivatives of zeta at s=1/2, attributing them directly to one of Riemann's posthumous papers held in Göttingen. The formulas are given with the usual constants (gamma, log(8 pi), Catalan's G, and the sum over 1/alpha_n squared) and the author mentions having shared the second one informally around 2010 before deciding to publish for reference purposes.

The paper does what it sets out to do by putting the expressions into print in a compact, readable way. Anyone who has needed these particular values now has a citable source instead of hunting through old notes or computing them from scratch.

The soft spot is exactly the one the stress-test flags: the attribution is stated as fact but the text contains no quotation from the manuscript, no shelf-mark, no page number, and no indication of how the constants were extracted from Riemann's notation. For a history-of-mathematics note that is a noticeable gap; readers cannot verify the claim from the paper itself. There is also no derivation or numerical check supplied, which keeps the work purely expository.

This is aimed at analytic number theorists who work with explicit values on the critical line or who track historical expressions for the zeta function. It has too little original content or analysis to justify sending it to referees. I would not bring it to a reading group or cite it in new work, and I would not recommend peer review.

Referee Report

2 major / 0 minor

Summary. The manuscript asserts that in one of his posthumous papers conserved in Göttingen, Riemann computed and recorded explicit values for the first and second logarithmic derivatives of ζ(s) at s = 1/2. It displays the two formulas ζ'(1/2)/ζ(1/2) = π/4 + γ/2 + log(8π)/2 and ζ''(1/2)/ζ(1/2) − (ζ'(1/2)/ζ(1/2))² = 8 − π²/4 − 2G + 2 ∑_{n=1}^∞ 1/α_n², states that these were shared with mathematicians around 2010, and presents the note as an explanation of those formulas.

Significance. If the attribution to Riemann's manuscript is accurate and the formulas are faithfully transcribed, the note would record a previously undocumented calculation by Riemann on the zeta function at the critical line, of potential interest to historians of analytic number theory. The formulas combine classical constants (π, γ, Catalan's G) with a sum that appears to involve the ordinates of zeta zeros, but the manuscript supplies neither a derivation nor any verification that these expressions match Riemann's own notation or calculations.

major comments (2)

[Abstract, first paragraph] Abstract, first paragraph: the central historical claim—that the displayed formulas appear in Riemann's Göttingen manuscript—is advanced solely by assertion, with no quotation, shelf-mark, page reference, or excerpt from the document. This absence makes the attribution impossible to verify and is load-bearing for the paper's thesis.
[Abstract] The second displayed formula invokes the sum ∑ 1/α_n² without defining α_n or indicating how the constant terms (8, −π²/4, −2G) arise from Riemann's calculations; no derivation or cross-reference to the manuscript is supplied to support these explicit values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our short note. We address each major comment below and indicate planned revisions.

read point-by-point responses

Referee: [Abstract, first paragraph] Abstract, first paragraph: the central historical claim—that the displayed formulas appear in Riemann's Göttingen manuscript—is advanced solely by assertion, with no quotation, shelf-mark, page reference, or excerpt from the document. This absence makes the attribution impossible to verify and is load-bearing for the paper's thesis.

Authors: The note is a brief record of formulas shared from the Göttingen collection rather than a critical edition or full transcription. The attribution rests on the author's examination of the relevant posthumous papers. We will revise the abstract and main text to include the specific shelf-mark and page reference for the document in question. revision: yes
Referee: [Abstract] The second displayed formula invokes the sum ∑ 1/α_n² without defining α_n or indicating how the constant terms (8, −π²/4, −2G) arise from Riemann's calculations; no derivation or cross-reference to the manuscript is supplied to support these explicit values.

Authors: The purpose of the note is to present the explicit values recorded by Riemann, not to re-derive them. We will add a parenthetical definition of α_n as the positive ordinates of the non-trivial zeros. The constants are those appearing directly in the manuscript entry; a cross-reference to the relevant page will be supplied in revision. No derivation is provided because none appears in the source document itself. revision: partial

Circularity Check

0 steps flagged

No circularity: paper reports historical attribution of formulas to Riemann manuscript without any derivation or self-referential reduction.

full rationale

The paper states that the displayed identities for the logarithmic derivatives of zeta at s=1/2 appear in one of Riemann's posthumous Göttingen papers and simply reproduces the two explicit formulas. No derivation chain, parameter fitting, ansatz, or uniqueness theorem is supplied inside the paper itself; the text contains only the bare historical claim plus the formulas. Because nothing is derived from inputs within the document, none of the enumerated circularity patterns (self-definitional, fitted-input-called-prediction, self-citation load-bearing, etc.) can be exhibited by quoting equations that reduce to their own premises. The attribution may be unverifiable from the given text alone, but that is an evidentiary rather than a circularity issue.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is an expository historical note; it introduces no free parameters, axioms, or invented entities of its own.

pith-pipeline@v0.9.1-grok · 5679 in / 1227 out tokens · 50480 ms · 2026-06-29T14:02:14.495135+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 3 canonical work pages · 1 internal anchor

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work page arXiv 2006
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work page internal anchor Pith review Pith/arXiv arXiv 2019
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[1] [1]

Arias de Reyna Computation of the secondary zeta function https://arxiv.org/abs/2006.04869, arXiv:2006.04869, (2020)

J. Arias de Reyna Computation of the secondary zeta function https://arxiv.org/abs/2006.04869, arXiv:2006.04869, (2020)

work page arXiv 2006

[2] [2]

R. P. Brent, D. J. Platt, T. S. Trudgian , Accurate estimation of sums over zeros of the Riemann zeta-function https://arxiv.org/abs/2009.13791, arXiv:2009.13791, (2009)

work page arXiv 2009

[3] [3]

H. M. Edwards , Riemann's Zeta Function, Dover Publ. Inc., Mineola, New York, (2001) reprinted from the Academic Press, Inc. edition in 1974

2001

[4] [4]

S. R. Finch , Mathematical constants, Cambridge University Press, 2003

2003

[5] [5]

The Mangoldt function and the non-trivial zeros of the Riemann zeta function

J. Guillera , The Mangoldt function and the non-trivial zeros of the Riemann zeta function https://arxiv.org/abs/1705.06367, arXiv:1705.06367, (2019)

work page internal anchor Pith review Pith/arXiv arXiv 2019

[6] [6]

E. Landau Nouvelle démonstration pour la formule de Riemann sur le nombre des nombres premiers inférieurs à une limite donnée et démonstration d’une formule plus générale pour le cas des nombres premiers d’une progression arithmétique https://www.numdam.org/item/?id=ASENS_1908_3_25__399_0, Ann. Sci. Éc. Norm. Supér.(3) 25 (1908) 399--442

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Riemann , Uber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsb

B. Riemann , Uber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsb. Preuss. Akad. Wiss. (1859) 671--680 (English translation, in the book by Edwards E )

[8] [8]

Riemann Gesammelte Mathematische Werke, Wissenschaftlicher Nachlass und Nachträge - Collected Papers, ed

B. Riemann Gesammelte Mathematische Werke, Wissenschaftlicher Nachlass und Nachträge - Collected Papers, ed. Raghavan Narasimhan, Springer-Verlag, Berlin, Heidelberg 1990

1990

[9] [9]

E. C. Titchmarsh , The theory of the Riemann zeta-function, 2nd ed. with a preface by D. R. Heath-Brown, Oxford University Press, New York , 1986

1986

[10] [10]

Voros , Zeta Functions over Zeros of Zeta Functions, Lecture Notes of the Unione Mathematica Italiana, Springer-Verlag, Berlin, Heidelberg 2010

A. Voros , Zeta Functions over Zeros of Zeta Functions, Lecture Notes of the Unione Mathematica Italiana, Springer-Verlag, Berlin, Heidelberg 2010

2010

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http://mpmath.org/

The mpmath development team , mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.4.0), 2026. http://mpmath.org/

2026