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REVIEW 3 major objections 5 minor 14 references

Asymptotic properties of zeros of Riemann zeta function

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Pith's one-line read The paper proves the zeta zeros' asymptotic expansion defines a whole family of 'Riemann sequences', including complex ones whose zeros lie off the critical line.

desk verdict The main existence theorem for Riemann sequences is analytically solid, but the paper's concrete complex example rests on a numerical x-ray, not a proof. read the letter →

arxiv 2507.07253 v1 pith:WDBVFHYG submitted 2025-07-09 math.NT

Juan Arias de Reyna , Yves Meyer This is my paper · ORCID
classification math.NT MSC 11M2652C2330D99
keywords RiemannzetafunctionzerosofsequencescrystallinemeasuresfunctionalequationHurwitzasymptoticexpansionhypothesis
topics The Riemann Hypothesis
open problems 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 paper tries to characterize the sequence of nontrivial zeros of the Riemann zeta function by an intrinsic property, without assuming the Riemann hypothesis. Its candidate is the 'Riemann sequence' property: an asymptotic expansion of $\sum_k 2z/(z^2+\alpha_k^2)$ with fixed coefficients $a_n$ built from Euler and Bernoulli numbers. The paper proves the zeta zeros form one such sequence, then constructs infinitely many others, and exhibits a concrete complex Riemann sequence whose computed zeros are not all on the critical line. If these constructions are right, the zeta zero sequence is not singled out by its asymptotic behaviour alone; the remaining conjecture is that it is the only real Riemann sequence. That would give a characterization of zeta's zeros that does not settle, and does not presuppose, the Riemann hypothesis.

What carries the argument

The load-bearing object is the Riemann sequence (Definition 4): a sequence of complex numbers with positive real parts, bounded imaginary parts, and conjugate symmetry, satisfying the asymptotic expansion (16) with coefficients $a_{2n+1}=2^{-2n-2}(8-E_{2n})$ and $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n)$. The proof that the zeta zeros form such a sequence goes through the Weierstrass product for $\Xi(t)$ and the Stirling expansion for $\log\Gamma$ (Theorems 1 and 3). For the new examples, the machinery is a construction of crystalline measures—tempered distributions whose support and Fourier support are both locally finite—via the finite Fourier transform on $\mathbb{Z}/N^2\mathbb{Z}$ (Propositions 8 and 9); applying the equivalence from [9] turns such a measure into a Dirichlet series $g_N(s)$ satisfying zeta's functional equation, and adding a small multiple of it to $\zeta(s)$ yields $\zeta_N(s)$ whose zeros are controlled by Proposition 12 and have the product representation (26). The explicit example $\zeta_M(s)$ is a linear combination of Hurwitz zeta functions chosen from an explicit self-dual measure.

What would settle it

Run a rigorous contour count around the rectangle $R=(-21,22)\times(-10,80)$ for $\zeta_M$: a count other than 31 zeros plus one pole, or any zero with real part above $\sigma_0=10.564029176912431172$, would disprove the concrete example; for the general theorem, a zero of some $\zeta_N$ with $|\operatorname{Im} t|\ge 3/2$ for a small allowed $\delta$ would break Proposition 12.

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Extended reading notes

Core claim

On its own terms, the central discovery is that the zeta zeros satisfy the asymptotic law $$\sum_{k\in\mathbb{N}}\frac{2z}{$z^{2}$+\$tau_k^{2}$} \simeq \frac12\log\frac{z}{2\pi}+\sum_{n=1}^\infty \frac{a_n}{z^n},$$ with $a_{2n+1}=2^{-2n-2}(8-E_{2n})$ and $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n)$, and that this law is not exclusive to them. Theorem 14 states that for every odd $N$ and integer $T\ge 2$ with $N^2>4NT+1$, a small perturbation $\zeta_N(s)=\zeta(s)+\delta g_N(s)$—where $g_N$ is an entire Dirichlet series satisfying the same functional equation as zeta—has a zero sequence that is again a Riemann sequence. Section 6 writes down an explicit $\zeta_M(s)$ built from Hurwitz zeta functions whose computed zeros include points off the critical line, which the paper presents as a concrete complex Riemann sequence. The paper's conclusion, stated as a conjecture, is that the zeta zeros may be the only real Riemann sequence; if that uniqueness holds, it characterizes the zeta zeros intrinsically without settling the Riemann hypothesis.

Load-bearing premise

For the infinite family, the proof depends on the product formula (26) holding exactly for the zeros of the constructed functions; for the concrete example, it depends on the numerical search having found every zero and on the claimed zero-free region being exact rather than approximate.

Editorial extensions

If this is right

  • Every admissible pair $(N,T)$ with $N$ odd, $T\ge 2$, and $N^2>4NT+1$ yields a meromorphic function $\zeta_N$ sharing zeta's functional equation, with a unique simple pole at $s=1$, whose zeros with positive real part form a Riemann sequence; these functions are not expected to have Euler products.
  • The explicit $\zeta_M$ provides a complex Riemann sequence, so the asymptotic property (16) alone does not force zeros onto the critical line; the Riemann hypothesis is not a consequence of this intrinsic zero law.
  • If the conjecture that only one real Riemann sequence exists is correct, then the zeta zero sequence is characterized among all Riemann sequences by being real—an intrinsic statement that bypasses the ordinary formulation of the Riemann hypothesis.
  • Every Riemann sequence gives a secondary zeta function $Z_\alpha(s)$ with a double pole at $s=1$, simple poles at the odd negative integers, and prescribed values at the even negative integers (Theorems 26–27), so the family of sequences carries a whole class of zeta-like functions.
  • A real Riemann sequence different from $(\tau_n)$ would be a concrete way to disprove the Riemann hypothesis without exhibiting a zero off the critical line; the paper explicitly points to this as a possible route.

Reading between the lines

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

  • Beyond the paper, the existence of complex Riemann sequences suggests that any characterization of zeta's zeros must add an arithmetic input—such as an Euler product—beyond analytic self-duality.
  • The construction makes a precise numerical prediction: a rigorous contour integral over the rectangle $R=(-21,22)\times(-10,80)$, together with a verified zero-free bound $\sigma\ge\sigma_0=10.564029176912431172$ for $\zeta_M$, would turn the complex Riemann sequence example from numerical evidence into a theorem; the paper itself leaves that verification open.
  • Because the construction passes through crystalline measures, Riemann sequences can be viewed as the zeta-side of self-dual combs; a classification of self-dual measures with a gap at the origin would likely translate into a classification of all Riemann sequences, not just the real ones.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces the notion of a Riemann sequence: a sequence of complex numbers satisfying the same asymptotic expansion as the nontrivial zeros of the Riemann zeta function, namely sum_k 2z/(z^2+alpha_k^2) ~ (1/2)log(z/2pi) + sum_n a_n/z^n with the coefficients a_n given by Euler and Bernoulli numbers, together with structural axioms on ordering, symmetry, imaginary parts, and a sector condition. The authors prove that the zeta zero parameters tau_n form a Riemann sequence (Theorem 3), construct infinitely many Riemann sequences via crystalline measures and finite Fourier analysis (Theorem 14), present an explicit candidate complex Riemann sequence attached to a combination of Hurwitz zeta functions (Section 6), and derive common analytic properties of the associated Xi, zeta, and secondary zeta functions (Sections 7 and 8). The advertised program is to characterize the zeta zeros as the unique real Riemann sequence.

Significance. If the existence of complex Riemann sequences were rigorously established, this would give a new intrinsic characterization of the zeros of the Riemann zeta function, independent of the Riemann hypothesis, and would connect this problem to the theory of crystalline measures. The analytic core of the paper is sound: Theorem 1 and the derivation of the asymptotic relation from the classical product formula, the product-representation argument behind Theorem 14, and the structural study of Riemann sequences in Sections 7 and 8 are coherent and clearly presented. The paper is honest that the zeta zeros themselves are used as a source of properties rather than as an assumed conclusion, and it gives explicit formulas for residues and special values of the secondary zeta function Z_alpha. The main weakness is Section 6: the sole evidence for the existence of a complex Riemann sequence is a numerical computation whose completeness and error control are not certified, and the paper's central advertised claim therefore currently exceeds what is proved.

major comments (3)
  1. [Section 6] The zero-free half-plane sigma >= sigma_0 for the function zeta_M is not proved. The displayed equation 'This happens for sigma_0 = 10.564029176912431172' is preceded by an expression ending with '... where the dots represents all the other terms in zeta_M(s) taken with coefficients in absolute value'. An ellipsis with absolute values is not a tail bound. A rigorous argument requires an explicit dominating convergent series, interval arithmetic, or an effective bound on the remaining terms of the Dirichlet expansion. This matters because condition (c) of Definition 4 demands a uniform bound on |Im(alpha_n)| for all zeros and condition (d) demands the sector condition |arg(alpha_n)| < pi/4; without the zero-free region, neither can be asserted for all zeros of zeta_M.
  2. [Section 6, Figure 1 and zero table] The count 'In our case 31 zeros and a pole' and the subsequent computation of the zeros are described heuristically through an x-ray. This is not a certified argument-principle computation: there is no rigorous enclosure of the boundary integral, no error bound on the listed zeros, and no proof that the table contains every zero in the rectangle R=(-21,22)x(-10,80). A missed zero with small positive ordinate and large |Re(s)-1/2|, or a zero outside R with positive ordinate, would violate condition (a) or (d) of Definition 4. To support the claim that zeta_M gives a complex Riemann sequence, the authors must either provide a rigorous certificate of the zero count (for example interval arithmetic on the argument variation) or explicitly state that the example is only numerical evidence.
  3. [Remark 15 and Section 6] The paper's overarching claim that complex Riemann sequences exist is not established by Theorem 14 alone, because Theorem 14 constructs Riemann sequences without proving that their zeros are off the critical line. Remark 15 states 'This proves that there are complex Riemann sequences' after citing the Section 6 numerics. Since the Section 6 verification is presently non-rigorous, that sentence overreaches the proof. The manuscript should either upgrade the numerical verification to a rigorous one or weaken the claim to a conjecture or numerical demonstration, while Theorem 14 would then still establish the existence of Riemann sequences, but not their complexity.
minor comments (5)
  1. [Abstract and Section 2] The abstract writes the displayed asymptotic relation for real x, while Theorem 3 states it for complex z in the sector |arg z| <= pi/2 - epsilon; please reconcile the notation so that the sector condition is visible from the outset.
  2. [Section 6, definition of zeta_M] In the displayed formula for zeta_M(s), the last term reads '(zeta(s,5/12)+zeta(7/12))'; the second Hurwitz zeta argument is missing an 's' and should be zeta(s,7/12).
  3. [Section 6, zero table] The two-column presentation 'beta gamma beta gamma' is visually confusing; please use a proper table with column headers and clear row separators, and state explicitly the convention beta=Re(s), gamma=Im(s).
  4. [Section 6, Figure 1] The x-ray uses thick and thin lines to mark real and purely imaginary values; in grayscale or small print these may be hard to distinguish. Please add a legend and possibly labels for the axes.
  5. [Proposition 12] The statement says the function does not vanish 'nor on the square [-3/2,3/2] x [-3/2,3/2]'; this is a rectangle, not a square, and the wording could be clarified.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Riemann-sequence expansion is derived from product representations, not assumed; the self-citation to [12] is not load-bearing, though Section 6's numerical verification is non-rigorous.

full rationale

The paper's central theorem (Theorem 14) does not assume the target asymptotic (16). It starts from the product representation (26) for Ξ_N from Proposition 13, the zero-free region control of Proposition 12, and the fact that ζ_N(s)≃1; the expansion of ∑2z/(z^2+α_n^2) is then obtained by the same Gamma-factor calculation as in Section 2. The coefficients a_n come from the Stirling expansion of log Γ, not from the zeta zeros, so no input is renamed as a prediction. The construction of crystalline measures is proved self-containedly in Propositions 7–9 via finite Fourier-transform dimension counting; the reference to the second author's [12] is described as 'follows the line of' but the existence theorem is proved in the paper, so it is not load-bearing. The concrete example in Section 6 rests on a numerical x-ray and a bare zero-free value σ0=10.564029176912431172 without rigorous error bounds; this is a correctness/rigor gap, not circularity, because the structural expansion would follow from the same derivation once those numerical facts are certified. Overall, no equation reduces to its own input; the self-citation is minor and non-essential.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central asymptotic expansion uses standard entire-function theory and Stirling expansions. The examples rely on the crystalline measure method from the second author's prior work and on Hamburger's theorem. No new physical entities are postulated. The main non-standard input is the numerical verification in Section 6, which is better treated as a red flag than as a stated axiom.

free parameters (2)
  • delta = sufficiently small, |delta| < delta_0
    In Proposition 12, zeta_N = zeta + delta g_N requires delta small enough to keep the function zero-free in a fixed rectangle. The existence of delta_0 is shown, but no explicit value is used.
  • N and T = odd N, integer T >= 2, N^2 > 4NT + 1
    Construction parameters for the crystalline measure and for the initial gap T in the Dirichlet series. The existence proof works for any such pair, so these are design choices rather than fitted constants.
assumptions (7)
  • standard math Hadamard-Weierstrass product representation for zeta and Xi, with convergence of the zero series
    Used in Section 2 and Proposition 13 to pass from zeros to the logarithmic sum and to the asymptotic expansion (14).
  • standard math Hamburger's equivalence between crystalline measures with hat mu = mu and Dirichlet series with functional equation (Theorem 5)
    Cited from [9] and used as the bridge from crystalline measure constructions to zeta-like functions in Sections 4 and 5.
  • standard math Stirling expansion for log Gamma and the Bateman/Gradshteyn integral identities used in Theorem 25
    The asymptotic constants and the Mellin transform computations depend on these standard identities; the paper notes one typo in the tables.
  • standard math Jensen's theorem for counting zeros of entire functions of finite order
    Used in Proposition 13 to obtain the bound on the number of zeros and the convergence of sum |alpha|^{-2}.
  • standard math Eigenvalue dimension table for the finite Fourier transform on Z/N^2Z
    Used in Proposition 8 to prove the existence of a nonzero real symmetric Fourier-invariant function vanishing on an interval.
  • domain assumption Growth bound for the Hurwitz zeta function, |(s-1)zeta(s,a)| <= C e^{c|s| log(2+|s|)}
    Invoked in Proposition 13 to control the growth of Xi_N and to justify the product representation; this is a standard analytic number theory input.
  • ad hoc to paper Numerical completeness of the Section 6 x-ray and zero table
    The claim that zeta_M yields a complex Riemann sequence rests on the computed zero-free region and the x-ray showing all relevant zeros. This is not proved with interval arithmetic and is the paper's main non-rigorous input.

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Pith. "Pith review of Asymptotic properties of zeros of Riemann zeta function." pith.science (2026). https://pith.science/paper/WDBVFHYG

@misc{pith2026250707253,
  author       = {Pith},
  title        = {Pith review of: Asymptotic properties of zeros of Riemann zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDBVFHYG}},
  note         = {Machine review of arXiv:2507.07253}
}
abstract

We try to define the sequence of zeros of the Riemann zeta function by an intrinsic property. Let $(z_k)_{k\in \mathbb{N}}$ be the sequence of nontrivial zeros of $\zeta(s)$ with positive imaginary part. We write $z_k= 1/2+i\tau_k$ (RH says that these $\tau_k$ are all real). Then the sequence $(\tau_k)_{k\in \mathbb{N}},$ satisfies the following asymptotic relation \[\sum_{k\in\mathbb{N}}\frac{2x}{x^2+\tau_k^2}\simeq \frac12\log\frac{x}{2\pi}+\sum_{n=1}^\infty \frac{a_n}{x^n},\,\,x\to +\infty\] where $a_{2n+1}=2^{-2n-2}(8-E_{2n})$, $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n).$ Are there other sequences $(\alpha_k)_{k\in \mathbb{N}},$ of real or complex numbers enjoying this property? These problems are addressed in this note.

Figures

Figures reproduced from arXiv: 2507.07253 by the authors.

Figure 1
Figure 1. x-ray of ζM(s) in R = (−21, 22) × (−10, 80) [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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