Moments of L-functions — Moments of the Riemann zeta function — Problem 9

Definitions and mathematical statement

The supplied excerpt does not define $I_k(T)$ or $a_3$. The following formulation is conditional on their customary normalization for moments of the Riemann zeta function; this normalization requires confirmation against the complete source.

Let $\zeta(s)$ denote the meromorphic continuation of $\sum_{n=1}^{\infty}n^{-s}$, initially defined for $\operatorname{Re}(s)>1$. For a positive integer $k$ and a real number $T>0$, set \[ I_k(T)=\int_0^T\left|\zeta\!\left(\tfrac12+it\right)\right|^{2k}\,{\,\mathrm{d}} t. \] Under the customary arithmetic-factor normalization, set \[ a_3=\prod_{p\ \mathrm{prime}}\left(1-\frac1p\right)^9 \sum_{r=0}^{\infty}\binom{r+2}{2}^{\!2}p^{-r}. \] The excerpt attributes to Soundararajan, reference [S], the lower bound \[ I_3(T)\geq \frac{20.26\,a_3}{9!}(1+o(1))T(\log T)^9 \qquad(T\to\infty). \] The principal problem is to exhibit a real constant $C>20.26$ and prove that, for every real $\varepsilon>0$, there exists $T_0>1$ such that every real $T\geq T_0$ satisfies \[ I_3(T)\geq \frac{(C-\varepsilon)a_3}{9!}\,T(\log T)^9. \] Equivalently, prove \[ \liminf_{T\to\infty}\frac{9!\,I_3(T)}{a_3T(\log T)^9}\geq C>20.26. \] The source describes $42$ as the best possible coefficient. This is retained as the source's stated target, without asserting an independently verified upper bound or asymptotic formula.

The source also asks for improvements to lower bounds for $I_k(T)$ for $k>3$. Under the interpretation above, $k$ ranges over integers at least $4$; the supplied excerpt gives neither the relevant baseline bounds nor their constants, so this additional request remains unspecified quantitatively.

Short English statement

Under the explicitly provisional standard normalization, improve the coefficient $20.26$ in the lower bound for the sixth moment of the Riemann zeta function. The source gives $42$ as the optimal target and also requests improvements for higher moments, without specifying their baseline bounds.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $800159$, exact source problem code catalogue entry, upstream ID $800159$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry status is open; it is not an independently verified mathematical assessment.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 9, under moments of the Riemann zeta function: https://arxiv.org/abs/2405.17800. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The problem text attributes the question to Nathan Ng, Lethbridge. The registry separately lists Hamieh, Kadiri, Ng as proposers and 2024 as the proposed year.
• The displayed lower bound is attributed in the supplied excerpt to Soundararajan, reference [S]. Full bibliographic details for [S] are absent from the supplied record.
• The excerpt identifies works [BBLR], [BCR], and [HY] as available and potentially useful. Their full bibliographic details and individual links are absent from the supplied record.