On the series expansion of the secondary zeta function about $s=1$ and its coefficients

Artur Kawalec

On the series expansion of the secondary zeta function about $s=1$ and its coefficients

T0 review · 0 major / 6 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read The regular Laurent coefficients of the secondary zeta function at its double pole s=1 are given by an explicit Stieltjes-style limit over the ordinates of the Riemann zeros.

desk verdict Clean, expected generalization of Hassani’s n=0 limit to all Cn under RH, with a useful BPT acceleration and solid numerical checks; incremental but correctly done. read the letter →

arxiv 2603.21555 v2 pith:JPT5OOZK submitted 2026-03-23 math.NT

Artur Kawalec This is my paper · ORCID

classification math.NT MSC 11M2611M06

keywords secondaryzetafunctionLaurentseriesStieltjesconstantsRiemannzeroshypothesiszerosumsBrenttheorem

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 secondary zeta function is the Dirichlet series formed from the positive imaginary parts of the non-trivial zeros of the Riemann zeta function, assuming the Riemann hypothesis. It possesses a double pole at s=1 whose Laurent expansion contains an infinite sequence of regular coefficients Cn. This paper proves that each Cn equals a concrete limit: the difference between a partial sum of log^n(γ)/γ over zeros up to height T and an elementary main term coming from the Riemann–von Mangoldt formula, taken as T tends to infinity. The expression is the precise analogue of the classical formula for the Stieltjes constants of the ordinary zeta function. Direct numerical checks with millions of zeros, further sharpened by Brent’s error-reduction theorem, confirm the formula to many decimal places and yield high-precision values of the coefficients.

What carries the argument

Stieltjes integration of the weight log^m(t)/t against the zero-counting function N(T)=L(T)+Q(T), which isolates an elementary antiderivative A(T) whose subtraction leaves a remainder that converges to (-1)^m Cm.

What would settle it

Evaluate the partial-sum expression for C0 with the first 10^10 ordinates and compare the result, plus the explicit Brent error bound, against the independently known 19-digit value of C0; a discrepancy larger than the bound would refute the claimed limit.

Watch

Extended reading notes

Core claim

For every integer n greater than or equal to zero the regular coefficient Cn in the Laurent series of the secondary zeta function about s=1 is recovered by the limit formula Cn = lim (T→∞) (-1)^n {sum_{γ<T} log^n(γ)/γ - [1/(2π(n+1)(n+2))] log^{n+1}(T) log(T^{n+1}/(2π)^{n+2})}.

Load-bearing premise

Every non-trivial zero is assumed to lie exactly on the critical line, so that its imaginary part is a real positive number that can be summed directly.

Editorial extensions

If this is right

The formula supplies an independent computational path to the coefficients Cn that does not rely on the Arias-de-Reyna algorithm.
Inserting the Brent–Platt–Trudgian correction improves the truncation error from O(log^{m+1}T/T) to O(log^{m+1}T/T^2), recovering many extra correct digits from a fixed zero database.
The same analysis yields an explicit integral representation of every Cn in terms of the oscillatory remainder Q(t).
High-precision tables of Cn for arbitrary n become available once sufficiently many ordinates are known.

Reading between the lines

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

The identical limit construction can be repeated at the simple poles of Z(s) that sit at the negative odd integers, producing analogous regular coefficients there.
The rapid growth of |Cn| visible in the computed table is consistent with a radius of convergence exactly equal to 2 and suggests factorial-type asymptotics.
Because the formula needs the Riemann hypothesis only up to height T, systematic comparison of the limit against independently computed Cn offers a practical numerical probe of the hypothesis itself.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing worth knowing is that Theorem 1 is exactly what it claims: the natural Stieltjes-integration extension of Hassani’s n=0 formula to every regular Laurent coefficient Cn of the secondary zeta function about s=1. The derivation is the standard one already used by Ivić and by Bondarenko–Ivić–Saksman–Seip; the author simply inserts the log^m factor, identifies the constant term with the known series, and writes the limit explicitly. That is new in the literature, even if it is the expected next step rather than a conceptual leap.

What the paper does well is keep the argument transparent and then check it. With 2 million zeros the raw sums recover the Arias-de-Reyna reference values to the number of digits predicted by the O(log^{n+1}T/T) error; adding the Brent–Platt–Trudgian correction improves the error by an extra 1/T factor and immediately yields 10–12 correct digits. The author also supplies a short table of high-precision Cn values obtained independently by the ADR algorithm, so the numerical side is reproducible and non-circular.

Soft spots are minor and proportionate. Everything assumes RH from the first sentence, but that is definitional for Z(s) itself, not a hidden gap. The radius-of-convergence remark and a few lower-limit conventions are slightly pedantic but harmless. Self-citation of the author’s earlier ADR tables is legitimate because those tables are the only high-precision source available. No load-bearing flaw appears.

This is a short, solid note for the small community that actually computes constants attached to zeta zeros. It will not change anyone’s view of the secondary zeta function, but it gives a practical formula and a clean error analysis. I would send it to a referee; the math is standard, the checks are honest, and the result is useful inside its niche. Accept after light polishing.

Referee Report

0 major / 6 minor

Summary. The paper studies the secondary zeta function Z(s)=sum_gamma gamma^{-s} (sum over positive imaginary parts of non-trivial zeros, under RH). It recalls the known Laurent expansion of Z(s) about the double pole at s=1, with regular coefficients C_n, and proves a limit formula (Theorem 1) expressing each C_n as the T->infty limit of the partial sum sum_{gamma<T} log^n(gamma)/gamma minus an explicit main term built from the Riemann-von Mangoldt asymptotic. The proof proceeds by Stieltjes integration of N(T)=L(T)+Q(T), extraction of the main term A(T) by repeated integration by parts, and identification of the resulting constant with the regular part of the Laurent series via the exp-log expansion of the remainder integral already used by Ivic and by Bondarenko-Ivic-Saksman-Seip. Numerical checks with 2e6 zeros recover the first few C_n to the expected number of digits; an application of Brent-Platt-Trudgian (BPT) error bounds improves the remainder by an extra 1/T factor (Theorem 2) and is verified numerically.

Significance. The main result is a clean, expected generalization of Hassani's n=0 limit (and of Brent's high-precision evaluation of C_0) to all regular Laurent coefficients C_n. The derivation is classical Stieltjes integration under RH and correctly matches the integral representation already present in the literature for the regular part of Z(s). The numerical verification against independent high-precision values obtained by the Arias de Reyna algorithm, together with the concrete BPT improvement, supplies a practical computational tool. The contribution is incremental rather than foundational, but it is self-contained, correctly executed, and of clear interest to specialists working on secondary zeta functions and sums over zeros.

minor comments (6)

Throughout: the manuscript repeatedly writes 'Brent's (BPT) Theorem' and 'the (BPT) method'. BPT is the joint work of Brent-Platt-Trudgian; the attribution should be corrected for accuracy and consistency with the references.
Section 2, display (11)-(12): the lower-limit constant B_m is defined with a special case for m=0 that relies on the convention 0^0=1. A short clarifying sentence would remove any ambiguity for the reader.
Section 2, (16)-(17) and (21): several typographical slips appear (missing closing parentheses, 'Qt)' for Q(t), and an incomplete integral sign). These should be cleaned before publication.
Section 3, numerical checks: the text states that C_0 computed via ADR was 'offset by log^2(2pi)/(4pi)'. A one-line explanation of the origin of that offset (or a pointer to the earlier paper) would help the reader reconcile the two values.
Table 1 caption and surrounding text: the table is said to list coefficients 'to 50 digits' while the displayed entries for large n are given in scientific notation with fewer significant figures; a brief remark on the actual precision claimed for each entry would be useful.
References: several arXiv identifiers and journal citations are incomplete or slightly inconsistent in format; a uniform bibliographic style should be applied.

Circularity Check

0 steps flagged · score 1.5 of 10

No significant circularity: Theorem 1 is a standard Stieltjes-integral derivation under RH; self-citations supply only independent numerical benchmarks.

full rationale

The central claim (Theorem 1) is obtained by inserting the log^m factor into the Stieltjes integral of N(T)=L(T)+Q(T), evaluating the main term A(T) by repeated integration by parts, and identifying the constant remainder with the regular Laurent coefficients already present in Ivić and Bondarenko–Ivić–Saksman–Seip. The RH hypothesis is stated from the first sentence and is definitional for Z(s) itself, not a hidden circular premise. High-precision reference values of Cn used for verification come from the independent Arias-de-Reyna algorithm (cited as [2] and the author’s earlier arXiv:2403.15741); they are not fitted parameters that are then re-predicted. The BPT correction is an external error-bound improvement applied after the main formula is already derived. Consequently the derivation chain does not reduce by construction to its own inputs, and the only self-citations are non-load-bearing numerical checks. Score 1.5 reflects that minor self-citation presence without any circular reduction of the claimed limit formula.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper works entirely inside classical analytic number theory under RH. No free parameters are fitted; the only non-standard constants that appear (A0,A1,A2) are taken from Brent's published error bounds. The secondary zeta function itself and the coefficients Cn are not new entities but objects already studied in the cited literature.

assumptions (3)

domain assumption Riemann Hypothesis: every non-trivial zero of ζ(s) has real part 1/2.
Used from the first sentence to define γ_n and the secondary zeta function Z(s).
standard math Riemann-von Mangoldt formula N(T)=L(T)+Q(T) with the stated bounds on S(T) and f(T).
Invoked in equation (5) and the subsequent integration-by-parts argument.
standard math Brent-Platt-Trudgian error bounds with numerical constants A0=2.067, A1=0.059, A2=0.007.
Cited from [3] and inserted into Theorem 2 to improve the remainder.

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Pith. "Pith review of On the series expansion of the secondary zeta function about $s=1$ and its coefficients." pith.science (2026). https://pith.science/paper/JPT5OOZK

@misc{pith2026260321555,
  author       = {Pith},
  title        = {Pith review of: On the series expansion of the secondary zeta function about $s=1$ and its coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPT5OOZK}},
  note         = {Machine review of arXiv:2603.21555}
}

abstract

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

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