Moment bound $\frac1T\int_0^T|\zeta(\tfrac12+it)|^{2k}dt=O(T^{\varepsilon})$ for $k=1,2$

Proved

LindelofHypothesis.moment_bound_of_k_le_two

by Lucas · Oct 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

analytic-number-theorylindelof-hypothesisriemann-zeta

For $k\in\{1,2\}$ and every real $\varepsilon>0$ ,

\frac1T\int_0^T\left|\zeta\!\left(\tfrac12+it\right)\right|^{2k}dt=O\!\left(T^{\varepsilon}\right)\qquad(T\to\infty).

These are the two cases of the moment formulation of the Lindelöf hypothesis that are known; the case $k=3$ is open.

Preamble

import Mathlib
import Definitions.Def_RiemannVonMangoldt_ZeroCounts
import Definitions.Def_LindelofHypothesis_GrowthAndZeroCounts

open Complex Filter Asymptotics
open scoped ENNReal

Formal statement

namespace LindelofHypothesis

theorem moment_bound_of_k_le_two (k : ℕ) (hk : k = 1 ∨ k = 2) (ε : ℝ) (hε : 0 < ε) :
    (fun T : ℝ => (1 / T) * ∫ t in (0 : ℝ)..T, ‖riemannZeta (1 / 2 + t * I)‖ ^ (2 * k))
      =O[atTop] (fun T : ℝ => T ^ ε) := by sorry

end LindelofHypothesis

Source

Wikipedia, "Lindelöf hypothesis" (snapshot supplied as Lindelöf_hypothesis.pdf, printed 12 Oct 2026), https://en.wikipedia.org/wiki/Lindel%C3%B6f_hypothesis; section "Means of powers (or moments) of the zeta function", sentence "This has been proved for k = 1 or 2".

Human review

Endorsed by Shuze Chen · Oct 13, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 13, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Moment bound 1T∫0T∣ζ(12+it)∣2kdt=O(Tε)\frac1T\int_0^T|\zeta(\tfrac12+it)|^{2k}dt=O(T^{\varepsilon})T1​∫0T​∣ζ(21​+it)∣2kdt=O(Tε) for k=1,2k=1,2k=1,2

Moment bound $\frac1T\int_0^T|\zeta(\tfrac12+it)|^{2k}dt=O(T^{\varepsilon})$ for $k=1,2$