Moments of the Riemann zeta function (Keating–Snaith/Conrey–Ghosh)

Prove that for fixed k > 0, the 2k-th moment satisfies (1/T) ∫_0^T |ζ(1/2 + it)|^{2k} dt ∼ c_k (log T)^{k^2}, and determine c_k in accordance with the random-matrix prediction c_k = a_k f_k beyond the proven cases k = 1 and k = 2.

Background

The paper reviews the moment conjectures for ζ(1/2+it), where the main term is believed to be c_k (log T){k2}. For k = 1,2 the conjecture is proved (Hardy–Littlewood and Ingham), while higher moments lack proofs.

Random matrix theory predicts the universal part f_k, while the arithmetic factor a_k captures Euler-product contributions.

References

The asymptotic behavior of moments of the zeta function is conjectured (see ) to be of the form :

\frac 1T\int_0T |\zeta(1/2 + it)|{2k} dt \sim c_k (\log T){k2}

for some constant $c_k$.

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Moments and unitary matrices (Keating–Snaith)