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The third moment of the logarithm of zeta and a twisted pair correlation conjecture

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Pith's one-line read Assuming RH and three correlation conjectures, the third moment of the real part of log zeta is an explicit constant plus O(1/log T), matching random-matrix predictions.

desk verdict The third moment result is new and the proof is careful, but the twisted pair correlation conjecture is visibly designed to make the constant come out, and the ramp region only has smoothed conditional support. read the letter →

arxiv 2412.20099 v1 pith:4ZLDKSZN submitted 2024-12-28 math.NT

Alessandro Fazzari , Maxim Gerspach This is my paper · ORCID
classification math.NT MSC 11M0611M26
keywords RiemannzetafunctionlogarithmofthirdmomentSelbergcentrallimittheorempaircorrelationtripletwistedrandommatrixtheory
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 paper establishes conditional estimates for the third moment of the logarithm of the Riemann zeta function on the critical line, refining what the Selberg central limit theorem alone predicts. Assuming the Riemann Hypothesis, the pair correlation conjecture, the triple correlation conjecture, and a new 'twisted' pair correlation conjecture, it proves that the average of $(\Re\log\zeta(1/2+it))^3$ over $t\in[T,2T]$ equals an explicit constant $c_P+c_Z$ with an error $O(1/\log T)$, while the corresponding average for the imaginary part is $O(1/\log T)$. This matches the finer random-matrix prediction [KS00], which high-precision numerical data had suggested. The interest is that the third moment is the first quantity that distinguishes the real and imaginary parts of $\log\zeta$ beyond their shared leading Gaussian behaviour.

What carries the argument

The load-bearing object is the twisted pair correlation function $F_n(\alpha)$, a normalized sum over pairs of zeros $\gamma,\gamma'$ of $n^{i\gamma}T^{i\alpha(\gamma-\gamma')}\omega(\gamma-\gamma')$, together with the conjectured piecewise-linear density $m_n(\alpha)$ of Conjecture 1.5. For primes the paper notes the identity $\frac{\Lambda(n)}{\log T}m_n(\alpha)=H_*\bigl(\alpha,\frac{\Lambda(n)}{\log T}\bigr)$, where $H_*$ is the non-discrete part of the triple correlation kernel; this identity is what makes the $P(t)Z(t)^2$ contribution cancel against the $Z(t)^3$ contribution in the third moment. The rest of the machinery is the decomposition of $\log\zeta(1/2+it)$ as a sum $P(t)+Z(t)$ of a prime-power sum and a zero sum (Proposition 2.4), which converts the third moment into integrals of $P^3$, $P^2Z$, $PZ^2$, and $Z^3$ that are then evaluated with the correlation conjectures.

What would settle it

Numerically evaluate the twisted pair correlation sum $F_n(\alpha)$ for a small prime power, say $n=2$, at a height where millions of zeros are known, and compare with Conjecture 1.5's piecewise-linear $m_n(\alpha)$ across the interval $[1-\log n/\log T,\,1]$; a systematic deviation would falsify the cancellation mechanism. Alternatively, compute $M^\Re_3(T)$ and $M^\Im_3(T)$ on the same zero data and check that the real part approaches $c_P+c_Z$ at the $1/\log T$ scale while the imaginary part tends to zero.

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Extended reading notes

Core claim

The central discovery is Theorem 1.1: under RH, the pair correlation conjecture (Conjecture 1.2), the triple correlation conjecture (Conjecture 1.3), and the twisted pair correlation conjecture (Conjecture 1.5), the real-part third moment satisfies $M^\Re_3(T)=c_P+c_Z+O(1/\log T)$ with $c_Z=-\pi^2/4$ and $c_P=\frac{3}{4}\sum_{p}\sum_{m\ge2}\frac{1}{m\,p^m}\sum_{k+\ell=m}\frac{1}{k\ell}$, while the imaginary-part third moment is $O(1/\log T)$. The constant splits naturally: $c_Z$ comes from the triple correlation of zeros and $c_P$ from prime-power sums, and the twisted pair correlation conjecture is exactly the mechanism that forces the cross term between one prime power and two zeros to cancel against the pure zero contribution, leaving $c_Z$ intact. The paper argues that this twisted conjecture is of independent interest and supports it by proving it in a bounded range of $\alpha$ under RH (Proposition 1.7) and in a larger smoothed range under a uniform Hardy-Littlewood conjecture (Theorem 1.9).

Load-bearing premise

The load-bearing premise is the exact claimed shape of the twisted pair correlation function, and in particular the identity for primes that makes the prime-zero cross term cancel the pure zero term; if the true twisted correlation differs, the constant $c_Z=-\pi^2/4$ would change.

Editorial extensions

If this is right

  • The real part of $\log\zeta$ has a non-Gaussian constant-order third moment, so its logarithmic moments are not pure Gaussian at the constant scale.
  • The imaginary part's third moment vanishes at order $1/\log T$, matching the random-matrix prediction that odd moments of the imaginary part are negligible.
  • The twisted pair correlation conjecture becomes a new structural statement linking prime powers to zero-pair statistics, and the paper proves it in restricted ranges.
  • A uniform Hardy-Littlewood conjecture implies the full smoothed twisted pair correlation on a large range, connecting a prime-counting heuristic to zero-correlation statistics.
  • The constants $c_P$ and $c_Z$ are explicit and computable, so the third-moment prediction can be verified numerically at finite $T$.

Reading between the lines

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

  • The identity linking $m_n(\alpha)$ to $H_*$ likely reflects a deeper reciprocity between prime-twisted pair correlations and triple zero correlations; if made structural, it could supply a new route to the triple correlation conjecture.
  • One can test the twisted pair correlation numerically for small $n$ (for example $n=2$) with existing high-zero data; deviations from the piecewise-linear $m_n$ would change the third-moment constant and show up long before any failure of RH.
  • The dependence on the uniform Hardy-Littlewood conjecture may be loose: a weaker error term might still prove the twisted conjecture on a shorter $\alpha$-range, leaving the final constant unchanged.
  • The same twisted device should transfer to other $L$-functions with the same random-matrix statistics, giving the third-moment constant for $\log L(1/2+it)$.
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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 / 4 minor

Summary. The paper proves conditional estimates for the third moment of the real and imaginary parts of log ζ(1/2+it), refining Selberg's central limit theorem. Theorem 1.1 states that, assuming RH, Montgomery's pair correlation conjecture, Hejhal's triple correlation conjecture, and a new "twisted pair correlation" conjecture (Conjecture 1.5), one has M_3^Re(T) = c_P + c_Z + O(1/log T) and M_3^Im(T) = O(1/log T), with c_P an explicit prime sum and c_Z = -π^2/4. The proof decomposes the third moment into prime and zero contributions: P^3, P^2 Z, P Z^2, and Z^3. The Z^3 term is evaluated using the triple correlation conjecture, and the P Z^2 term using the twisted pair correlation conjecture; the two contributions cancel in a way that produces c_Z. The paper also gives partial support for the twisted pair conjecture: Proposition 1.7 proves part of it unconditionally on RH in a restricted α-range, and Theorem 1.9 proves a smoothed version on a larger range conditional on a uniform Hardy-Littlewood conjecture.

Significance. If the conditional theorem is correct, it is a genuine refinement of Selberg's central limit theorem and provides the first third-moment analogue matching Keating-Snaith predictions, with the imaginary part vanishing at the predicted order. The paper contains a substantial amount of careful analytic-number-theoretic calculation: the decomposition into prime and zero contributions is carried out in detail, Proposition 1.7 gives real unconditional support for the twisted pair correlation in the small-α range, and Theorem 1.9 is a nontrivial smoothed conditional result. The main caveat is that the central constant depends on the exact shape of a new conjecture that is supported only weakly in the range that matters for the cancellation; this limits the strength of the claim that the paper fully explains the third moment.

major comments (3)
  1. [Section 1, Conjecture 1.5 and following discussion; Section 5, Proposition 5.1; Section 9, Theorem 1.9] The value of c_Z in Theorem 1.1 is fixed by the exact cancellation, in Propositions 5.1 and 6.1, of the β-dependent integrals, and that cancellation requires the precise piecewise-linear ramp of m_n(α) on the intervals [-1-Λ(n)/log T, -1] and [1-logn/log T, 1-(logn-Λ(n))/log T]. The paper's own evidence for Conjecture 1.5 is weakest precisely there: Proposition 1.7 covers only |α| < 1 - logn/logT - δ_T, and Theorem 1.9 is a smoothed statement whose error term O(log log T / Λ(n)) is not small for high prime powers such as n = 2^a, since then Λ(n) = log 2. The introduction explicitly states that the transition from F_n(α) to F_n(α;ψ_U) seems to incur an extra factor of size √n in the error term, which obstructs removing the smoothing. Since the identity Λ(n)/log T · m_n(α) = H_*(α, Λ(n)/log T) is described only as "what is needed" for the cancellation, the main constant is currently hostage to an unverified sharp form of the twisted pair conjecture. Please either supply a proof (or a much sharper verification) of the ramp region of Conjecture 1.5, or explicitly state in Theorem 1.1 that the constant is conditional on this sharp form and discuss how the constant would change under plausible deviations of m_n.
  2. [Section 3, Proposition 3.2] The proof of Proposition 3.2, which establishes the O(1/log x) bound for the P(t)^3 contribution to the imaginary part, is omitted entirely with the text "We will not give the proof details". This is one of the four terms needed for the imaginary-part estimate in Theorem 1.1. Even if the adaptation from Proposition 3.1 is straightforward, a published proof should include the details or at least display the cancellation of the six terms explicitly; as written, this component of the main theorem is unsupported.
  3. [Section 5, around Eq. (5.5)] The application of Conjecture 1.5 in Proposition 5.1 is made with a specific test function r(u) = k_hat(2πβu), and the error term E_n from the conjecture is asserted to be absorbed "on average over n" without a displayed calculation. Because the sums over n run up to x ≤ T^{1/3}, the average of E_n is not immediate from its definition for n = q^a with a > 1; please provide the short calculation that justifies this absorption, or replace the assertion by a bound with the relevant averaging over prime powers displayed.
minor comments (4)
  1. [Section 1, Conjecture 1.5] In the displayed statement of Conjecture 1.5, the summation range "T ≤ γ,γ′,2T" is missing the condition γ′ ≤ 2T; it should read "T ≤ γ,γ′ ≤ 2T".
  2. [Section 1, after Proposition 1.7] The text refers to "Theorem 1.7" when it should refer to "Proposition 1.7" in the sentence "Due to the symmetry of F_n(α) discussed above, Theorem 1.7 provides...".
  3. [References] The reference [Odl] is incomplete: it lacks a year and publication venue. Please provide a complete bibliographic entry for the preprint.
  4. [Section 2, Lemma 2.6] The proof of Lemma 2.6 is presented as a sketch; in particular, the contour-integral bounds leading to Eq. (2.26) are stated without all intermediate steps. Given that this lemma is used later in Lemma 2.8 and in Section 9, a slightly fuller derivation would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the conditional derivation rests on explicitly stated correlation conjectures; the admitted 'what is needed' identity is a transparency caveat, not a definitional reduction.

full rationale

Theorem 1.1 is a fully conditional statement: it assumes RH, Montgomery's pair correlation conjecture, Hejhal's triple correlation conjecture, and the twisted pair correlation conjecture (Conjecture 1.5). The constant c_Z = -pi^2/4 emerges from the cancellation of the beta-dependent integrals in Propositions 5.1 and 6.1. The authors explicitly note that for primes, Lambda(n)/log T * m_n(alpha) = H_*(alpha, Lambda(n)/log T), and they state that this identity is 'what is needed' for the twisted-pair contribution to cancel the triple-correlation contribution. This is an honest admission that the shape of m_n is motivated by the desired cancellation; however, Conjecture 1.5 is not merely a restatement of the third-moment constant. It is a concrete, stronger statement about a twisted pair correlation function of zeros, it is implied by the strong Conjecture 1.4, and it receives partial independent support from Proposition 1.7 (unconditional on a range) and Theorem 1.9 (smoothed, under a uniform Hardy-Littlewood conjecture). The proof of Theorem 1.1 is self-contained given these assumptions; no fitted parameter is renamed as a prediction, and no load-bearing self-citation or imported uniqueness theorem is used. The paper's own caveat about the 'sqrt(n)' obstruction in the transition from smoothed to sharp F_n affects the evidence for Conjecture 1.5, but that is a correctness/confidence concern, not a circularity of the derivation chain. Overall, the central claim retains independent content and the assumptions are transparent, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central theorem rests on RH, two established correlation conjectures, and a new twisted pair correlation conjecture whose exact form is partly engineered to produce the target constant. Conjecture 1.8 is an additional uniform Hardy-Littlewood input used only for partial support.

assumptions (5)
  • domain assumption Riemann Hypothesis
    Used throughout, e.g., Lemma 2.2 and Propositions 2.4 and 2.5, to approximate log zeta by prime and zero sums.
  • domain assumption Montgomery pair correlation conjecture (Conjecture 1.2)
    Used in Proposition 6.1 to evaluate the three-zero contribution Z(t)^3.
  • domain assumption Hejhal triple correlation conjecture (Conjecture 1.3)
    Used in Propositions 6.1 and 6.2 to evaluate Z(t)^3.
  • ad hoc to paper Twisted pair correlation conjecture (Conjecture 1.5)
    New conjecture needed for the P(t)Z(t)^2 contribution; its form is chosen so that it cancels with the triple correlation contribution.
  • domain assumption Uniform Hardy-Littlewood conjecture (Conjecture 1.8)
    Used to prove partial support for the twisted pair conjecture for larger alpha, as in Theorem 1.9.
invented entities (1)
  • Twisted pair correlation function F_n(alpha) and Conjecture 1.5 independent evidence
    purpose: Controls the interaction of a prime power n^{i gamma} with Montgomery's pair correlation function, needed for the one-prime-power and two-zero contribution.
    New mathematical object in this paper. It yields specific asymptotic formulas that are partially proven (Propositions 1.7 and 1.9) and are in principle checkable numerically.

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Pith. "Pith review of The third moment of the logarithm of zeta and a twisted pair correlation conjecture." pith.science (2026). https://pith.science/paper/4ZLDKSZN

@misc{pith2026241220099,
  author       = {Pith},
  title        = {Pith review of: The third moment of the logarithm of zeta and a twisted pair correlation conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZLDKSZN}},
  note         = {Machine review of arXiv:2412.20099}
}
read the original abstract

We prove precise conditional estimates for the third moment of the logarithm of the Riemann zeta function, refining what is implied by the Selberg central limit theorem, both for the real and imaginary parts. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which explains the interaction of a prime power with Montgomery's pair correlation function. We believe this to be of independent interest, and devote substantial effort to its justification. Namely, we prove this conjecture on a certain range unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.

Figures

Figures reproduced from arXiv: 2412.20099 by the authors.

Figure 1
Figure 1. The function mn(α) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros

    math.NT 2025-07 conditional novelty 7.0 of 10

    The joint moments of log ζ and a smoothed zero-counting function factorize into Gaussian moments, showing asymptotic independence.

Reference graph

Works this paper leans on

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