Moments of the Riemann zeta function on the critical line

The asymptotic behaviour of the kth moment of the Riemann zeta function is conjectured to be 1/T∫_0^T |ζ(1/2 + it)|^2kdt ∼ c_k log(T)^k^2, for some constant c_k.

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Fact	Value	Source
Formal Conjectures: research statements in the file	4 open, 2 solved	Formal Conjectures

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Cite

@misc{cairn-paper-riemann-zeta-moments,
  title        = {Moments of the Riemann zeta function on the critical line},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-riemann-zeta-moments}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

moment_isEquiv. The asymptotic behaviour of the $k$ th moment of the Riemann zeta function is conjectured to be $\frac{1}{T} \int_{0}^{T} ∣ ζ (1/2 + i t) ∣^{2 k} d t \sim c_{k} lo g (T)^{k^{2}},$ for some constant $c_{k}$ .

moment_isEquiv₃. The constant in the asymptotic formula for the 3rd zeta moment is conjectured to be $42 a_{3} /9!$ [CG98].

moment_isEquiv₄. The constant in the asymptotic formula for the 4th zeta moment is conjectured to be $24024 a_{4} /16!$ [CGo01].

moment_isEquivₖ. The Keating–Snaith conjecture [KS00] for the $k$ th moment of the Riemann zeta function, provides the value of $c_{k}$ in the asymptotic formula [Co26, §3.4.5].

The moments $\frac{1}{T} \int_{0}^{T} ∣ ζ (1/2 + i t) ∣^{2 k} d t$ are conjectured to be asymptotic to $c_{k} (lo g T)^{k^{2}}$ for some constants $c_{k}$ . This is a theorem for $k = 1$ (Hardy–Littlewood, $c_{1} = 1$ ) and $k = 2$ (Ingham, $c_{2} = 1/ (2 π^{2})$ ), and these are the only proven cases. Conrey–Ghosh and Conrey–Gonek conjectured $c_{3} = 42 a_{3} /9!$ and $c_{4} = 24024 a_{4} /16!$ , with $a_{k}$ the arithmetic factor $a_{k} = \prod_{p} (1 - 1/ p)^{k^{2}} \sum_{m \geq 0} (m m + k - 1)^{2} p^{- m}$ ; Keating–Snaith conjectured $c_{k} = a_{k} f_{k}$ with $f_{k} = \prod_{j = 0}^{k - 1} j! / (j + k)!$ from random matrix theory, recovering $g_{k} = (k^{2})! f_{k} = 1, 2, 42, 24024$ .

The statements and constants below are taken from §3.4.5 of [Co26].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.RiemannZetaMoments (4 statements).

theorem moment_isEquiv (k : ℕ) : ∃ c > (0 : ℝ),
    moment k ~[atTop] fun T ↦ c * T.log ^ (k ^ 2)

theorem moment_isEquiv₃ : moment 3 ~[atTop] fun T ↦ 42 * arithmeticFactor 3 / 9 ! * T.log ^ 9

theorem moment_isEquiv₄ :
    moment 4 ~[atTop] fun T ↦ 24024 * arithmeticFactor 4 / 16 ! * T.log ^ 16

theorem moment_isEquivₖ (k : ℕ) : moment k ~[atTop]
    fun T ↦ (arithmeticFactor k * ∏ j ∈ Finset.range k, (j ! : ℝ) / (j + k) !) *
      T.log ^ (k ^ 2)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Co26] A. Connes, The Riemann Hypothesis: Past, Present and a Letter Through Time, arXiv:2602.04022, §3.4.5.
[In26] A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. 27 (1926), 273–300.
[CG98] J. B. Conrey, A. Ghosh, *A conjecture for the sixth power moment of the Riemann zeta-function*, Internat. Math. Res. Notices 1998, no. 15, 775–780.
[CGo01] J. B. Conrey, S. Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), no. 3, 577–604.
[KS00] J. P. Keating, N. C. Snaith, Random matrix theory and $ζ (1/2 + i t)$ , Commun. Math. Phys. 214 (2000), no. 1, 57–89.

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.