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Local Moments of M\"obius Fourier Polynomials and the Riemann Hypothesis

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Pith's one-line read The Riemann hypothesis is equivalent to subpolynomial growth of high local moments of Möbius Fourier polynomials on arcs of length 1/N.

desk verdict Sound elementary equivalence: RH iff subpolynomial local moments of Möbius polynomials on the critical arc of radius c/N; new local moment-to-point inequality, no correctness gap. read the letter →

arxiv 2607.25002 v1 pith:HMHCAFML submitted 2026-07-27 math.PR math.NT

Alberto Verjovsky This is my paper · ORCID
classification math.PRmath.NT MSC 11M2611N3742A0560G50
keywords MöbiusfunctionMertensRiemannhypothesisrandomFourierserieslocalmomentstrigonometricpolynomials
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

This paper rewrites the classical Mertens form of the Riemann hypothesis as a statement about local moments. Form the normalized trigonometric polynomial whose coefficients are the Möbius values, then evaluate it at a point chosen uniformly at random inside an arc of radius c/N centered at the origin. The claim is that RH holds if and only if every finite moment of that random variable grows slower than every positive power of N. The reason the reformulation works is a quantitative inequality: a large value of the polynomial at zero cannot disappear instantly, so it remains visible to high enough local L^q averages on the critical scale set by the highest frequency. The paper does not offer a new attack on RH; it supplies an equivalent local probabilistic criterion that sits next to Denjoy's random-sign heuristic and global flatness criteria for the same polynomials.

What carries the argument

The local moment-to-point-value inequality (Proposition 3.2 / Corollary 3.3): from the crude derivative bound |S'_N|≤πN(N+1) it recovers |M(N)| from the local L^q mean of S_N on an interval of length ~1/N, at the cost of a factor N^{1/(2(q+1))} that vanishes only as q o∞.

What would settle it

Fix c>0 and check whether the local q-moments of P_N on [-c/N,c/N] remain O(N^η) for every η>0 as q is taken larger and larger; if for some fixed η the moments eventually grow faster than N^η no matter how large q is chosen, the claimed equivalence fails.

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Extended reading notes

Core claim

For any fixed c>0, the Riemann hypothesis is equivalent to the assertion that the local moments M_{q,c}(N) of the normalized Möbius polynomial P_N, sampled uniformly on the arc of radius c/N, satisfy M_{q,c}(N)=O(N^η) for every η>0 and every finite q≥1 (and also to the weaker version that only requires the bound along an unbounded set of exponents for each η). In short, subpolynomial growth of arbitrarily high finite local moments recovers the Mertens bound.

Load-bearing premise

The recovery step uses only the general bound on the derivative of a trigonometric polynomial with coefficients at most 1 in size; that forces the loss term that disappears only when moments of arbitrarily high order are allowed.

Editorial extensions

If this is right

  • RH is equivalent to a critical-scale local moment condition on deterministic Möbius polynomials, with randomness only in the evaluation point.
  • A single fixed moment yields only an exponent strictly larger than 1/2; arbitrarily high moments are required to reach the classical Mertens threshold.
  • The same local-moment bound under RH extends at once to any shrinking radius r_N with N r_N = N^{o(1)}.
  • An Orlicz/sub-Gaussian strengthening of the local moments would still imply RH, giving a deterministic analogue of random-sign Fourier behaviour.
  • The maximal radius on which RH alone forces subpolynomial local moments is left open and is conjecturally no larger than N^{-1+o(1)}.

Reading between the lines

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

  • If a sharper persistence estimate that exploits multiplicative structure of µ could replace the crude derivative bound, a single fixed local moment might already be equivalent to RH.
  • The open maximal-scale question links the criterion directly to the size of additively twisted Möbius sums beyond the untwisted Mertens bound.
  • Comparing the growth of these local moments against the corresponding moments for independent random signs would give a quantitative measure of how much dependence the true Möbius coefficients retain at scale 1/N.
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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

0 major / 6 minor

Summary. The paper studies the normalized Möbius Fourier polynomial P_N(t) = N^{-1/2} Σ_{n≤N} μ(n) e^{2πint}, evaluated at a uniform random point U_{N,c} in the arc [-c/N, c/N]. The main result, Theorem 1.3, states that for fixed c > 0 the Riemann hypothesis is equivalent to subpolynomial growth M_{q,c}(N) = O_{η,q,c}(N^η) of the local moments for every η > 0 and every finite q ≥ 1 (and to an a priori weaker version in which, for each η, the bound is only required along an unbounded set of exponents Q_η). The forward direction uses the classical Mertens form of RH and partial summation (identity (8)); the converse uses a local moment-to-point-value inequality (Proposition 3.2), based on the derivative bound |S'_N| ≤ πN(N+1), which recovers |M(N)| from local L^q-data with a loss of N^{1/(2(q+1))} (Corollary 3.3). The paper also discusses the naturality of the critical scale N^{-1}, tail/Orlicz variants, and the relation to flatness and semiflatness questions for trigonometric polynomials.

Significance. The paper is explicitly and honestly framed: it does not claim a new approach to proving RH, but an equivalent local probabilistic reformulation, together with a quantitative mechanism (the moment-to-point-value inequality (5) and its corollary (7)) showing how high local moments on the critical-scale arc recover the Mertens function. Within that scope the paper delivers what it promises. I verified the load-bearing steps: Lemma 3.1 and the persistence argument; both regimes in Proposition 3.2, including the constant C_q = (2^{q+2}π)^{1/(q+1)}; the normalization identity (6); the exponent 1/2 + 1/(2(q+1)) in (7); the partial-summation identity (8) with the correct boundary behavior; and the quantifier order in the (iii)⇒(i) argument (η fixed first, then q ∈ Q_η chosen large), which is handled correctly and is the only place where the proof could easily have gone wrong. Constants are tracked explicitly and no unproved arithmetic input is used beyond the classical Mertens equivalence. The significance is modest by design — an equivalence with RH via an elementary inequality is a reformulation, not progress toward a proof — but the critical-scale local viewpoint, the explicit loss 1/(

minor comments (6)
  1. [Sections 3, 4, 10] Cross-references are mislabeled throughout: the proof of Proposition 3.2 cites "Theorem 3.1" (should be Lemma 3.1); the proof of Corollary 3.3 cites "Theorem 3.2" (should be Proposition 3.2); Section 4 cites "the precise form of Theorem 1.4" (should be Remark 1.4); Section 10 cites "Theorem 5.3" (should be Question 5.3). Please correct all four.
  2. [Section 6] Section 6: the tail-bound criterion is stated with the hedge "together with a mild truncation or integrability condition." Since the trivial bound |X_{N,c}| ≤ √N always holds, the tail estimate P(|X_{N,c}| > λ) ≤ C_{q,η,c} N^{qη} λ^{-q} integrates directly (via E|X|^q = q∫_0^{√N} λ^{q-1} P(|X|>λ) dλ) to give M_{q,c}(N) ≪ N^η up to a logarithm, which Theorem 1.3 absorbs. Making this explicit would remove the vagueness and strengthen the section.
  3. [Section 4, proof of Theorem 1.3] Section 4, converse: the final exponent in the second term of (7) is 1/2 + 1/(2(q+1)) + ηq/(q+1); with η < ε/3 and 1/(2(q+1)) < ε/3 this is ≤ 1/2 + 2ε/3, so the conclusion O(N^{1/2+ε}) holds with room to spare. One line spelling out this arithmetic (rather than asserting it) would help the reader.
  4. [Global/notation] The math italic 'e' for the exponential (e2πint in the extracted text) and the rendering of Möbius accents appear garbled in places; please check the compiled PDF for consistent typesetting of e^{2πint}, Mq,c(N), and the name Möbius throughout.
  5. [Section 1, Remark 1.1] Remark 1.1 (the case c > N) is somewhat redundant with the sentence immediately preceding it ("The finitely many values N ≤ 2c play no role..."); the second paragraph on varying c_N is useful, but the first could be trimmed.
  6. [Sections 1 and 9] The discussion of el Abdalaoui [1] in Sections 1 and 9 would benefit from one sentence stating precisely how the global semiflatness criterion's hypothesis compares quantitatively with (ii) of Theorem 1.3 (whole-circle Haar measure vs. shrinking-arc normalized measure), since this is the closest related result in the literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.3 is a standard two-way equivalence between RH (Mertens form) and independently defined local moments, mediated by a general trigonometric inequality.

full rationale

The derivation chain is self-contained and non-circular. Local moments M_{q,c}(N) are defined from the deterministic Möbius polynomial sampled on a shrinking arc, with no reference to RH. The forward implication (RH ⇒ moment bounds) uses only classical partial summation (identity (8)) and the Mertens form of RH. The converse uses Proposition 3.2 / Corollary 3.3, a general moment-to-point-value inequality for any trigonometric polynomial with |a_n|≤1, which recovers |M(N)| from local L^q data with an explicit loss 1/(2(q+1)) that vanishes as q→∞. Self-citations [12,15] appear only in the acknowledgments as motivational background and are not inputs to any load-bearing step. No parameter is fitted to data and re-presented as a prediction; no uniqueness theorem is imported; the criterion is not a renaming of a known empirical pattern. The paper explicitly frames the result as an equivalent reformulation, not a proof of RH. Score 0 is the correct honest finding.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

Load-bearing inputs are standard: the classical Mertens equivalence for RH, partial summation, and the elementary bound |μ(n)|≤1 used to control S'_N. The sampling radius constant c>0 is arbitrary but fixed and not fitted. No empirical parameters and no new physical or arithmetic entities are postulated; local moments and U_{N,c} are definitional constructs.

free parameters (1)
  • c (arc-radius constant) = any fixed c>0
    Fixed positive constant defining the sampling arc I_{N,c}=[-c/N,c/N]. The equivalence is stated for every fixed c>0; no fitting to data. Not load-bearing as a tuned value.
assumptions (4)
  • domain assumption Classical Mertens criterion: RH ⇔ M(x)=O_ε(x^{1/2+ε}) for every ε>0
    Invoked as the target equivalent form in the Introduction and to close the proof of Thm. 1.3 (Section 4); cited to Titchmarsh, Montgomery–Vaughan, Tenenbaum.
  • standard math Partial summation / Abel summation for SN(t) in terms of M(x) (identity (8))
    Used for the RH⇒ local-moment direction in Section 4; standard real analysis.
  • standard math |μ(n)|≤1 for all n, hence |a_n|≤1 in the general trigonometric bound
    Used in Lemma 3.1 to bound |S'_N|≤πN(N+1); enters the loss exponent 1/(2(q+1)) in Cor. 3.3.
  • domain assumption Khintchine / Kahane moment comparisons for random-sign models (benchmark only)
    Cited for context in Section 2; not used in the proof of Thm. 1.3.
invented entities (1)
  • Local moments M_{q,c}(N) and random variables X_{N,c}=P_N(U_{N,c}) independent evidence
    purpose: Package the critical-scale sampling of the Möbius polynomial so that RH becomes a statement about subpolynomial moment growth.
    Definitional constructs built from standard Lebesgue measure on I_{N,c}; not postulated hidden objects. Falsifiable in the mathematical sense via the stated equivalence.

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Cite this review

Pith. "Pith review of Local Moments of M\"obius Fourier Polynomials and the Riemann Hypothesis." pith.science (2026). https://pith.science/paper/HMHCAFML

@misc{pith2026260725002,
  author       = {Pith},
  title        = {Pith review of: Local Moments of M\"obius Fourier Polynomials and the Riemann Hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMHCAFML}},
  note         = {Machine review of arXiv:2607.25002}
}
abstract

Let $$ P_N(t)=\frac1{\sqrt N}\sum_{n\leq N}\mu(n)\e^{2\pi i nt}, \qquad t\in\T=\mathbb R/\mathbb Z. $$ We give a local probabilistic reformulation of the Riemann hypothesis by evaluating the normalized M\"obius Fourier polynomial \(P_N\) at a uniform random point in an arc of radius \(c/N\). We prove that RH is equivalent to subpolynomial growth of arbitrarily high finite local moments of these random variables. The principal quantitative tool is a local moment-to-point-value inequality which recovers the value \(P_N(0)=M(N)/\sqrt N\) from local \(L^q\)-data (where $M$ denotes the Mertens function). This provides a critical-scale local criterion for RH and complements Denjoy's random-walk heuristic, Kahane's theory of random Fourier series, and global \(L^p\)-semiflatness criteria for M\"obius polynomials.

Discussion (0). Continue with ORCID to comment.

Reference graph

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