Moments of L-functions — Moments of the Riemann zeta function — Problem 33

Definitions and mathematical statement

The supplied source presupposes quantities $D_{3,2}(x,h)$ and $P_3$, but does not define either of them. Wherever these quantities are defined and $x>0$, denote the residual by \[ E(x,h):=D_{3,2}(x,h)-xP_3(\log x). \] The problem attributed to Nathan Ng (Lethbridge) asks for exact, simple formulae for the coefficients of $P_3$, and for numerical investigation of \[ |E(x,h)|=\left|D_{3,2}(x,h)-xP_3(\log x)\right|. \] It further asks whether this error is of size $\sqrt{x}$.

The supplied record does not specify the definition or normalization of $D_{3,2}$, the degree or defining characterization of $P_3$, whether its coefficients depend on $h$, or the admissible values and relationship of $x$ and $h$. The subscript in $P_3$ alone does not determine its degree. The phrase "of size $\sqrt{x}$" does not specify an upper bound, a two-sided order of magnitude, a numerical heuristic, or an averaged assertion. Thus the displayed residual records the requested investigation, but the supplied information does not determine a self-contained asymptotic conjecture with precise quantifiers.

Short English statement

Find exact, simple expressions for the coefficients of the source's $P_3$, numerically examine the residual $D_{3,2}(x,h)-xP_3(\log x)$, and investigate its proposed $\sqrt{x}$ scale. Definitions, parameter ranges, and the intended meaning of that scale are missing from the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800182$, source code catalogue entry, title: "Moments of L-functions --- Moments of the Riemann zeta function --- Problem 33." The supplied registry labels the record open; this status is not independently verified here.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 33. Supplied source URL: https://arxiv.org/abs/2405.17800. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are source metadata, not a claim of access or verification in preparing this draft.
• The problem text attributes the question to Nathan Ng (Lethbridge); the registry's proposer field lists Hamieh, Kadiri, Ng, with proposed year 2024.
• The source refers to a "recent related article of Nguyen [Nguyen]." No title, year, URL, or complete bibliographic entry for that reference is supplied.