Moments of L-functions — Moments of the Riemann zeta function — Problem 31

Definitions and mathematical statement

The supplied record reports a result attributed to Topacogullari [T1]: for \[ 1\le h\ll x^{2/3}, \qquad D_{3,2}(x,h)=xP_{3,h}(\log x)+O\!\left(x^{8/9+\varepsilon}\right), \] where \[ P_{3,h}(t)=\sum_{j=0}^{3}a_j(h)t^j, \qquad a_3(h)\ne0. \] The notation $P_{3,h}$ makes explicit the supplied assertion that the coefficients of the degree-three polynomial depend on $h$; the excerpt writes this polynomial as $P_3$. It also states that a result was obtained for $D_{2,3}(x,h)$, without giving that result.

The functions $D_{3,2}$ and $D_{2,3}$ are not defined in the supplied record. In particular, no summation formula, shift convention, or endpoint convention is supplied. The domains of $x$ and $h$, the quantification of $\varepsilon$, and the dependence and uniformity of the implied constants in $\ll$ and $O$ are also unspecified. Reading the display as an asymptotic estimate would conventionally involve positive $x$ tending to infinity, but its complete quantifiers cannot be recovered from this excerpt.

The excerpt contains a reported established result rather than an explicit conjecture or requested improvement. Consequently, it does not determine a precise open mathematical problem. The registry's supplied open label is retained as source metadata and is not an independently verified mathematical assessment.

Short English statement

The excerpt attributes to Topacogullari [T1] an estimate for the undefined quantity $D_{3,2}(x,h)$, with a degree-three polynomial main term depending on $h$, and mentions an unspecified result for $D_{2,3}(x,h)$. Definitions, full quantifiers, and the intended open question are missing, so this is an incomplete statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $800180$, exact source problem code catalogue entry; title: "Moments of L-functions --- Moments of the Riemann zeta function --- Problem 31." The record attributes the problem to Hamieh, Kadiri, and Ng, with proposed year 2024, and supplies the status open.
• Supplied source list: Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 31. Supplied URL: https://arxiv.org/abs/2405.17800. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not independently verified access or source verification.
• The excerpt attributes the displayed estimate to Topacogullari [T1]. No bibliographic entry identifying [T1] is supplied, so that attribution and the underlying theorem cannot be independently resolved from this record.