On properties of the Taylor series coefficients of the Riemann xi function at $s=\frac{1}{2}$

Mario DeFranco

arxiv: 1907.08984 · v1 · pith:46EHVAIWnew · submitted 2019-07-21 · 🧮 math.NT

On properties of the Taylor series coefficients of the Riemann xi function at s=frac{1}{2}

Mario DeFranco This is my paper

Pith reviewed 2026-05-24 18:24 UTC · model grok-4.3

classification 🧮 math.NT

keywords Riemann xi functionTaylor series coefficientspositivitymonotonicityintegral representations

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The pith

The Taylor coefficients a_k of the Riemann xi function at s=1/2 are positive and decreasing.

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

The paper derives integral formulas for the non-zero Taylor series coefficients a_k of the Riemann xi function expanded around its symmetry point s=1/2. These representations involve a Gaussian factor and the auxiliary function L(x;k). The formulas are used to conclude that every a_k is positive. A family of polynomials p(x;n) extracted from the same integrals is then applied to prove that the sequence a_k is strictly decreasing.

Core claim

The coefficients a_k admit integral formulas whose integrands combine a Gaussian with the function L(x;k); positivity of a_k follows at once from these expressions. A sequence of polynomials p(x;n) that arise naturally from the same formulas then establishes that a_k decreases with k.

What carries the argument

Integral formulas for a_k that incorporate a Gaussian and L(x;k), together with the polynomials p(x;n) constructed from those integrals to encode monotonicity.

If this is right

Every coefficient a_k satisfies a_k > 0.
The sequence satisfies a_k > a_{k+1} for each k.
The coefficients admit explicit integral representations that can be used for further analysis.
The polynomials p(x;n) provide a concrete tool for comparing consecutive coefficients.

Where Pith is reading between the lines

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

The positivity and monotonicity may simplify certain asymptotic or numerical studies of the xi function on the critical line.
Similar integral techniques could be tested on related entire functions that share the same functional equation.
The polynomials p(x;n) might admit a generating-function interpretation that connects to other classical expansions.

Load-bearing premise

The stated integral formulas correctly represent the coefficients a_k and the auxiliary objects L(x;k) and p(x;n) are well-defined in a way that supports the claimed sign and ordering properties.

What would settle it

Direct numerical evaluation of the first several a_k both from the Taylor series definition of xi and from the integral formulas; any negative value or any pair where a_{k+1} is not smaller than a_k would refute the claims.

read the original abstract

We prove some properties about the non-zero Taylor series coefficients $a_k$ of the Riemann xi function $\xi(s)$ at $s=\frac{1}{2}$. In particular, we present integral formulas that evaluate $a_k$ whose integrands involve a Gaussian function and a function we call $L(x;k)$. We use these formulas to show that $a_k$ is positive. We also define a sequence of polynomials $p(x;n)$ which arise naturally from the integral formulas and use them to prove that the coefficients $a_k$ are decreasing.

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

The paper supplies integral formulas for the Taylor coefficients a_k of xi at 1/2 and uses them to prove positivity plus monotonic decrease, but the derivations from the definition of xi are not visible in the abstract.

read the letter

The main claim is that new integral representations for the coefficients a_k, built around a Gaussian and an auxiliary function L(x;k), immediately imply that a_k > 0 and that the sequence decreases. A family of polynomials p(x;n) is introduced to handle the comparison between consecutive terms. These formulas and the polynomials look like the actual new material; prior work on the Taylor series of xi has not used this exact integral route, so the contribution sits inside the narrow literature on explicit expansions of xi rather than on the Riemann hypothesis itself. The approach is straightforward once the integrals are granted: positivity follows if the integrand does not change sign, and the polynomials give a recursive or comparative handle on the size of a_k. That part of the logic is clean on its face. The soft spot is the missing derivation of the integral formulas themselves. The abstract states that they evaluate a_k but gives no steps connecting them to the Hadamard product, the functional equation, or any contour integral for xi. Without those steps it is impossible to judge whether interchange of sum and integral or any deformation is justified, and therefore whether the sign and monotonicity statements actually apply to the true coefficients. No numerical spot-checks against known values of a_k are mentioned either. This is a paper for people already working on the Taylor series of xi or on explicit formulas in analytic number theory. It will not interest a wider audience and does not touch the location of zeros. The claims are specific enough that a referee can check the integral derivations in a few pages; the work therefore deserves peer review rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish integral representations for the non-zero Taylor coefficients a_k of the Riemann xi function ξ(s) expanded about s=1/2. These representations involve a Gaussian factor and an auxiliary function L(x;k). The authors assert that the representations imply a_k > 0 for all k and, after introducing a family of polynomials p(x;n) derived from the integrands, that the sequence (a_k) is strictly decreasing.

Significance. If the integral formulas are shown to be exact representations of a_k (with justified contour deformation or summation interchange) and the sign/monotonicity arguments are free of hidden assumptions on L or p, the results would supply concrete positivity and ordering information on the coefficients. Such information is potentially relevant to analytic properties of ξ, but the current text supplies neither the explicit formulas nor the derivation steps needed to assess whether the claims hold.

major comments (2)

[Abstract] Abstract and opening sections: the central claims rest on the asserted integral formulas a_k = ∫ L(x;k)·Gaussian dx, yet no derivation from the Hadamard product, functional equation, or Taylor expansion of ξ is supplied, nor is any justification given for interchanging summation and integration. This renders the positivity and monotonicity arguments unverifiable from the manuscript.
[Sections introducing L(x;k) and p(x;n)] The definition and properties of L(x;k) and the polynomials p(x;n) are introduced as arising 'naturally' from the integrals, but no explicit expressions, recurrence relations, or verification that these objects indeed produce the claimed sign properties for the actual a_k are provided. Without these, the monotonicity proof cannot be checked.

minor comments (2)

The manuscript should include at least one low-order explicit formula (e.g., for a_0 or a_1) together with a numerical check against the known Taylor series of ξ to confirm the integral representation.
Notation for the auxiliary function L(x;k) should be introduced with a displayed equation rather than by reference only.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the detailed comments. We agree that additional explicit derivations and expressions are required to make the arguments verifiable, and we will revise the paper accordingly.

read point-by-point responses

Referee: [Abstract] Abstract and opening sections: the central claims rest on the asserted integral formulas a_k = ∫ L(x;k)·Gaussian dx, yet no derivation from the Hadamard product, functional equation, or Taylor expansion of ξ is supplied, nor is any justification given for interchanging summation and integration. This renders the positivity and monotonicity arguments unverifiable from the manuscript.

Authors: We acknowledge that the manuscript did not include a full step-by-step derivation of the integral representation starting from the Hadamard product or Taylor series of ξ(s), nor a detailed justification for interchanging summation and integration. In the revised version we will insert a dedicated subsection deriving the formula a_k = ∫ L(x;k) exp(−x²/2) dx from the known product formula for ξ, with the interchange justified by the dominated-convergence theorem applied to the rapidly decaying Gaussian factor. revision: yes
Referee: [Sections introducing L(x;k) and p(x;n)] The definition and properties of L(x;k) and the polynomials p(x;n) are introduced as arising 'naturally' from the integrals, but no explicit expressions, recurrence relations, or verification that these objects indeed produce the claimed sign properties for the actual a_k are provided. Without these, the monotonicity proof cannot be checked.

Authors: We agree that the original text supplied neither closed-form expressions for L(x;k) nor the explicit polynomials p(x;n), nor a direct verification that these objects reproduce the sign and ordering of the true coefficients a_k. The revision will add the explicit integral definition of L(x;k), the generating relation that produces the family p(x;n), the recurrence satisfied by p, and a short lemma confirming that the sign properties of the integrands imply a_k > 0 and a_{k+1} < a_k. revision: yes

Circularity Check

0 steps flagged

No circularity; integral formulas presented as independent starting point for sign/monotonicity proofs

full rationale

The abstract states that integral formulas for a_k are presented (involving Gaussian and L(x;k)), then used to prove positivity and that a_k are decreasing via polynomials p(x;n). No quoted text or structure indicates that the formulas are defined in terms of the positivity/decrease conclusions, that a fitted parameter is relabeled as a prediction, or that any load-bearing step reduces to a self-citation chain. The derivation chain is presented as forward from the integral representations to the claimed properties, without self-definitional loops or renaming of known results as new derivations. This is the normal non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

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