Moments of L-functions — Moments of the Riemann zeta function — Problem 43

Definitions and mathematical statement

Let $\Gamma=\mathrm{SL}_2({\mathbb{Z}})$ be the full modular group. The supplied problem concerns a primitive cusp form $g$ of weight $k$ for $\Gamma$, with the weight allowed to vary through the admissible weights.

Under the usual holomorphic interpretation, the ambient space is $S_k(\Gamma)$: the space of holomorphic functions $g\colon\mathbb H\to{\mathbb{C}}$, where $\mathbb H=\{z\in{\mathbb{C}}:\operatorname{Im}z>0\}$, satisfying \[ g\!\left(\frac{az+b}{cz+d}\right)=(cz+d)^k g(z) \qquad \left(\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma\right) \] and vanishing at the cusp. This describes a conventional interpretation of the terminology; the supplied statement does not explicitly specify holomorphicity or the normalization of the primitive form.

Requested statement, with unresolved target. Establish an asymptotic upper bound that is uniform in the weight aspect for primitive cusp forms $g$ of weight $k$ for $\Gamma$. The supplied record does not identify the quantity to be bounded, the proposed bound, any auxiliary parameters, or their ranges. Consequently, it does not determine an explicit inequality or a complete quantified mathematical assertion. In particular, uniformity in the weight would require the relevant implied constants to be independent of $k$ over a specified range, but the remaining permitted dependencies are not supplied.

The source motivates the request by applications to moment computations for Rankin--Selberg $L$-functions associated with primitive forms of varying large weights. This motivation does not specify which moment, $L$-function, or other expression is the intended target of the upper bound.

Short English statement

The source asks for an asymptotic upper bound uniform in the weight of a primitive cusp form for the full modular group, motivated by moments of Rankin--Selberg $L$-functions. The quantity to be bounded is missing, so a precise inequality cannot be recovered from the supplied record.

Sources

• Local UnsolvedMath registry: permanent ID $800191$, exact source problem code catalogue entry; supplied title: "Moments of L-functions --- Moments of the Riemann zeta function --- Problem 43." The registry labels the problem open; this is supplied metadata, not an independently verified mathematical status.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 43. Supplied source URL: https://arxiv.org/abs/2405.17800. The record attributes the item to A. Hamieh, UNBC, and lists Hamieh, Kadiri, Ng in its proposer field.
• Supplied provenance: extraction method arxiv_latex; source access date 2026-10-16. This date and provenance are supplied metadata; no independent source access or verification is asserted. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.