Moments of L-functions — Moments of the Riemann zeta function — Problem 21

Definitions and mathematical statement

Let $g$ be the fixed form in the source, and let $\mathcal H_k^{\mathrm{prim}}(p^\nu)$ denote the source's intended family of primitive forms $f$ of weight $k$ and level $p^\nu$. Write $L(s,f\otimes g)$ for the associated Rankin--Selberg $L$-function, with central point $s=\tfrac12$ as stipulated in the source.

Source-level problem. Establish upper bounds, as $\nu\to\infty$, for the third and fourth moments of the central values \[ \left\{L(\tfrac12,f\otimes g):f\in\mathcal H_k^{\mathrm{prim}}(p^\nu)\right\}, \] following the works identified in the source as [ChL1] and [ChL2].

Unresolved specifications. The source does not define its moment convention. For example, unweighted absolute moments would be \[ A_r(\nu)=\sum_{f\in\mathcal H_k^{\mathrm{prim}}(p^\nu)}\left|L(\tfrac12,f\otimes g)\right|^r, \qquad r\in\{3,4\}, \] whereas unweighted power moments would be \[ S_r(\nu)=\sum_{f\in\mathcal H_k^{\mathrm{prim}}(p^\nu)}L(\tfrac12,f\otimes g)^r, \qquad r\in\{3,4\}. \] These are alternative interpretations, not an assertion that both are requested. The record does not specify whether harmonic weights or division by the family size are intended, and it supplies no reality or positivity hypothesis permitting these alternatives to be identified. It also gives no target exponent, explicit bounding function, or required degree of sharpness.

The precise class, normalization, and character of the primitive forms, and the corresponding specifications of $g$, are not supplied. The record does not explicitly state that $p$ is prime or that $p$ and $k$ remain fixed as $\nu\to\infty$; these intended parameter conditions require clarification. Thus the displayed notation preserves the source's formulation without resolving its missing hypotheses.

As background, the source reports that Balkanova studied the fourth moment of $L(\tfrac12,f)$ with $f$ varying as above, citing [balkanova]. This is a supplied contextual claim about the untwisted family, not a bound asserted here.

Short English statement

For a fixed form $g$, seek upper bounds for third and fourth moments of Rankin--Selberg central values as the level $p^\nu$ of the varying primitive form grows. The source leaves the family specifications, moment weights, parameter conditions, and quantitative targets unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $800171$, exact source code catalogue entry, upstream ID $800171$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry labels this problem open; that label is not an independently verified mathematical or literature assessment.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 21, https://arxiv.org/abs/2405.17800. The supplied background records access on 2026-10-16 and extraction by arxiv_latex; these provenance details are supplied, not independently verified here. The registry title is "Moments of L-functions --- Moments of the Riemann zeta function --- Problem 21," while the supplied statement concerns Rankin--Selberg families.
• The registry attributes the proposal to Hamieh, Kadiri, Ng (2024); the extracted problem carries the attribution "Alia Hamieh, UNBC."
• [ChL1] and [ChL2]: references invoked by the supplied statement as models for the requested moment bounds. Their bibliographic entries are absent from the supplied record.
• [balkanova]: reference invoked for Balkanova's study of the fourth moment of the untwisted family $L(\tfrac12,f)$. Its bibliographic entry is absent from the supplied record.