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Averages of diagonal Elliott-Halberstam problem twisted by M\"obius function with Sobolev and H\"older-Zygmund weights

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Pith's one-line read Under GRH and a weak zero-density hypothesis, weighted averages of the Möbius-twisted Elliott–Halberstam discrepancy match the expected diagonal size for all levels of distribution when the weight is Sobolev W^{2,1}.

desk verdict Solid conditional averaged bounds for a diagonal Möbius-twisted EH that still implies Goldbach, under GRH plus a weak Gonek–Hejhal input. read the letter →

arxiv 2607.09110 v1 pith:2F6QXXYZ submitted 2026-07-10 math.NT

Marco Cantarini This is my paper · ORCID
classification math.NT MSC 11N3711P3211N56
keywords Elliott–HalberstamMöbiusfunctionbinaryGoldbachexplicitformulaeSobolevweightsHölder–ZygmundGonek–Hejhalweightedaverages
topics Goldbach's Conjecture
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 studies averaged forms of the Möbius-twisted Elliott–Halberstam conjecture that appears in Huang–Li’s approach to binary Goldbach. Instead of the full maximal discrepancy, it considers smooth or Hölder–Zygmund weighted double sums of Λ(n)χ(n)μ(m) and the same sums with an extra log m. Under the generalised Riemann hypothesis and a weak form of the Gonek–Hejhal conjecture (the first moment of 1/|ζ′(ρ)|^{2} is O(T)), these averages are shown to be of the same size as the “diagonal” version of the conjecture. For weights whose second derivative is integrable the bound holds for every level of distribution up to N^{1−ε}; for Hölder–Zygmund weights of order δ the admissible level drops but never falls below N^{1/2−ε}. The same statements hold after the logarithmic weight is inserted. The author also records that the pure diagonal conjecture already implies Goldbach once classical Elliott–Halberstam is available, so the averaged results sit in a chain of implications that ends at the binary Goldbach problem.

What carries the argument

A two-dimensional Abel summation identity that rewrites the weighted double sum as a Laplace convolution of the partial-sum functions of Λχ and μ; once the truncated explicit formulae for those summatory functions are inserted, the main term becomes a double series over zeros that can be controlled by Gamma-function estimates and the weak Gonek–Hejhal hypothesis.

What would settle it

Compute or rigorously bound the first moment J_1(T)=∑_{0<γ≤T}1/|ζ′(ρ)|^{2}; if it exceeds T(log T)^c for every c>0, the error terms in the explicit formulae of Theorems 22 and 26 exceed N^{2−ε} and the averaged bounds fail.

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Extended reading notes

Core claim

Under GRH and the bound J_1(T)≪T, the character-averaged weighted sum of Λ(n)χ(n)μ(m)f((n+m)/N) (and its logarithmic counterpart) is O_ε(N^{2−ε}E(f″)) for every Sobolev weight f∈W^{2,1} supported in [0,β), for all levels of distribution θ<1; the same size holds for Hölder–Zygmund weights of order δ with a θ that depends on δ but is always at least 1/2−2ε.

Load-bearing premise

The assumption that the sum of 1/|ζ′(ρ)|^{2} over zeros up to height T grows at most linearly in T; any extra positive power of log T would make the error terms larger than the claimed main-term size.

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Editorial analysis

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

Referee Report

0 major / 5 minor

Summary. The paper introduces a diagonal variant (dEH^{μ,log}) of the Möbius-twisted Elliott–Halberstam conjecture and proves that it, together with a classical EH input of level θ, already implies the binary Goldbach conjecture (Theorem 7). It then establishes weighted averaged forms of this diagonal discrepancy. Under GRH and the weak Gonek–Hejhal bound J_1(T)≪T, for weights f with support in [0,β) belonging to the Sobolev space W^{2,1} the averaged sum over q≤N^{1-2ε} of (1/φ(q))|∑_{χ eqχ_0} χ(N)∑∑ Λ(n)χ(n)μ(m)f((n+m)/N)| is ≪_ε N^{2-ε} E(f''), and an analogous bound holds for the logarithmically weighted sum; the same statements are obtained for Hölder–Zygmund weights in C^δ with a δ-dependent range of θ that never falls below 1/2-2ε. The proofs rely on a two-dimensional Abel summation identity, truncated explicit formulae for ψ(x,χ) and M(x) (extended to all x>0), absolute convergence of double series over zeros, and careful tracking of error terms.

Significance. If the conditional results hold, they supply the first averaged evidence toward a diagonal form of the twisted Elliott–Halberstam conjecture that is already known to be strong enough for Goldbach. The two-dimensional Abel identity (Theorem 8) and its discrete counterpart cleanly decouple the arithmetic convolutions, while the absolute-convergence theorem for double zero series (Theorem 15) and the uniform explicit formulae for x>0 are reusable tools. The paper is careful to isolate the single non-standard hypothesis (Conjecture 19) and to show that the classical max_y and max_a can be removed from the Goldbach implication. These features make the work a solid, self-contained contribution to the analytic theory of Goldbach-type problems under standard hypotheses.

minor comments (5)
  1. The phrase “consistent with the bound of the diagonal versions” (abstract and §1.2) is slightly ambiguous: the proved upper bound is O(N^{2-ε}) while a true dEH would give O(N/log^A). A short remark comparing the GRH-trivial size, the size implied by dEH, and the size actually obtained would remove any possible misunderstanding.
  2. Notation for the sum over characters is declared as an “abuse” (p. 5 and after (4.1)), yet the same symbol χ is used both for the non-principal character mod q and for its primitive inducer. A single clarifying sentence at the first occurrence would help the reader.
  3. In several places (e.g., the statements of Theorems 21–28) the dependence of the implied constants on eta, δ, f is recorded only in the O-symbol; listing the parameters explicitly in the theorem statements would improve readability.
  4. Minor typographical inconsistencies appear throughout (missing spaces after commas, occasional “log (N)” versus “log(N)”, and a few duplicated words such as “a verages”). A careful copy-edit pass is recommended.
  5. The examples in §6 (Cesàro–Riesz and the Zygmund-type weight) are useful; it would be helpful to record the precise value of E(f'') or the Hölder norm for each example so that the reader can see the numerical size of the constant.

Circularity Check

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No significant circularity; the averaged bounds follow from independent explicit formulae and a general Abel identity under external hypotheses (GRH + Conjecture 19).

full rationale

The derivation chain begins from the two-dimensional Abel summation identity (Theorem 8, cited from the author's prior work [3] but stated and used as a general parameter-free identity relating weighted double sums to convolutions of summatory functions) and its discrete counterpart (Theorem 9, proved in full). These are applied to the weighted sums involving Λ(n)χ(n)μ(m)f((n+m)/N) (and the log-weighted analogue). Truncated explicit formulae for ψ(x,χ) (Theorem 17, classical under GRH, extended to x>0) and for M(x), M̃(x) (Theorems 20 and (3.14), under RH + Conjecture 19) are inserted; the resulting main terms are double series over zeros that converge absolutely by the paper's own Theorem 15 (proved here by partial summation + Stirling + the weak Gonek-Hejhal bound J1(T)≪T). Error terms are estimated by Cauchy-Schwarz, Proposition 16, and the same bound on 1/|ζ'(ρ)|, yielding the averaged estimates of Theorems 22/26 (Sobolev) and 24/28 (Hölder-Zygmund) that match the expected size of the diagonal conjecture. Section 2 shows that the diagonal form of the conjecture already implies Goldbach by a direct adaptation of Huang-Li, without appealing to the averaged results. No quantity is defined in terms of the target bound, no parameters are fitted to data and then re-predicted, no uniqueness theorem is imported from the authors to force a choice, and the self-citations supply only general analytic tools whose hypotheses do not include the Elliott-Halberstam averages. The sole load-bearing external inputs are GRH and Conjecture 19, both stated explicitly and independent of the paper's conclusions.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The central claims rest on two unproved analytic hypotheses (GRH and the weak Gonek-Hejhal bound) together with standard facts about L-functions and Sobolev/Hölder spaces. No free parameters are fitted; the only invented objects are the diagonal conjecture and the two-dimensional Abel identity, both of which are given explicit statements and proofs.

assumptions (3)
  • domain assumption Generalized Riemann Hypothesis for Dirichlet L-functions
    Used throughout §§3-5 to place all non-trivial zeros on the critical line and to obtain the square-root cancellation bounds for ψ(x,χ) and M(x).
  • domain assumption Conjecture 19: ∑_{0<γ≤T} 1/|ζ'(ρ)|^2 ≪ T
    Invoked to guarantee absolute convergence of double series over zeros (Theorem 15) and to absorb the remainder in the explicit formula for the Mertens function.
  • standard math Standard truncated explicit formulae for ψ(x,χ) and M(x) under GRH
    Taken from Montgomery-Vaughan and Ng; extended by the author to all x>0.
invented entities (2)
  • Diagonal Elliott-Halberstam conjecture twisted by Möbius (dEH^{μ,log})
    purpose: Removes the two maxima from the original EH^μ while retaining enough strength to imply binary Goldbach.
    Introduced in Conjecture 6; shown in Theorem 7 to be sufficient for Goldbach once classical EH is available.
  • Two-dimensional Abel summation identity (Theorem 8) independent evidence
    purpose: Converts a weighted double sum of arithmetic functions into a Laplace convolution of their summatory functions.
    Proved from first principles; the discrete analogue (Theorem 9) is likewise elementary.

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Pith. "Pith review of Averages of diagonal Elliott-Halberstam problem twisted by M\"obius function with Sobolev and H\"older-Zygmund weights." pith.science (2026). https://pith.science/paper/2F6QXXYZ

@misc{pith2026260709110,
  author       = {Pith},
  title        = {Pith review of: Averages of diagonal Elliott-Halberstam problem twisted by M\"obius function with Sobolev and H\"older-Zygmund weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2F6QXXYZ}},
  note         = {Machine review of arXiv:2607.09110}
}
abstract

Recalling that the so-called Elliott-Halberstam conjecture twisted by the M\"obius function $\mu(n)$ claims that \[ \sum_{q\leq N^{\theta}}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{{\scriptstyle n\equiv a\,\mod\,q}}{n\leq y}}\Lambda(n)\mu\left(N-n\right)-\frac{1}{\varphi\left(q\right)}\sum_{n\leq y}\Lambda(n)\mu\left(N-n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}} \] for every $A>0$, where $0<\theta<1$ is fixed, and also recalling that the validity of this conjecture, in combination with the validity of the classical Elliott-Halberstam for suitable $\theta$, proves the binary Goldbach conjecture, in this paper we study weighted average variants of this problem. We will show that, under Generalized Riemann Hypothesis, a weak version of the Gonek-Hejhal conjecture and working with weights belonging to the Sobolev space $W^{2,1}$ or in the H\"older-Zygmund spaces $\mathcal{C}^{\delta}$ for suitable range of $\delta$, the bound of the average is consistent with the bound of the ``diagonal versions'' of this conjecture (that is, taking $y=N$ and taking $n\equiv N\mod q)$. In particular, in the case of weights in Sobolev space, the consistent upper bound holds for the whole $0<\theta<1$ and, in the case of weights in the H\"older-Zygmund class $\mathcal{C}^{\delta}$, for $\theta$ that depends on the choice of $\delta$ but still not below the $1/2-2\varepsilon$ threshold.

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