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)^{{k^2}.} 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_0^T |\zeta(1/2 + it)|^{2k} dt \sim c_k (\log T)^{{k^2}}

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)

For the $2k^{\text{th}}$ moment, it is conjectured that as $T\rightarrow\infty$ M_k(T)\sim a_kg_k(\log T)^{{k^2}} where $a_k$ is an arithmetic factor in terms of an Euler product and $g_k$ is an geometric factor.

— The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors (2608.25721 - MacMillan, 26 Aug 2026) in Section 1, Introduction

It is conjectured in that

I_k(T)=TP_{k^2}(\log T)+O(T^{{\frac12+\varepsilon}),}

where $P_{k^2}(x)$ is a polynomial of degree $k^2$.

— The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point (2608.25812 - Chen et al., 26 Aug 2026) in Section 1, Introduction

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

Background

References

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