Moments of L-functions — Moments of the Riemann zeta function — Problem 18

Definitions and mathematical statement

Let $f$ be a level-$1$ Hecke--Maass cusp form with spectral parameter $t_f$, and write $L(s,f)$ for the $L$-function appearing in the source. For $T>0$, put \[ M_f(T)=\int_0^T\left|L\!\left(\tfrac12+it,f\right)\right|^2\,dt. \] The supplied source attributes to Good, for fixed $t_f$, the asymptotic formula \[ M_f(T)=T P_{1,f}(\log T)+O_{t_f,\varepsilon}\!\left(T^{2/3+\varepsilon}\right) \qquad(T\to\infty), \] where $\varepsilon>0$ and $P_{1,f}$ is the degree-$1$ polynomial denoted by $P_1$ in the source. The subscript $f$ allows for dependence of the main term on the form; uniformity of the main-term coefficients is not requested. Define \[ E_f(T)=M_f(T)-T P_{1,f}(\log T). \] The problem asks for explicit dependence on $t_f$ in the bound for $E_f(T)$. In particular, it asks whether the error bound can be independent of $t_f$ when $T\geq t_f^{3/2+\delta}$ for some $\delta>0$.

On the nonnegative real spectral-parameter range implicit in that threshold, a precise uniform interpretation of this particular question is: does there exist $\delta>0$ such that, for every $\varepsilon>0$, there are constants $C_{\varepsilon,\delta}>0$ and $T_{0,\varepsilon,\delta}>0$, independent of $f$ and $t_f$, for which \[ |E_f(T)|\leq C_{\varepsilon,\delta}T^{2/3+\varepsilon} \] for every such level-$1$ form $f$ and every real \[ T\geq\max\!\left\{T_{0,\varepsilon,\delta},\,t_f^{3/2+\delta}\right\}? \] This is an explicit interpretation of the source's uniformity question, not an additional reported theorem. The source does not specify normalization conventions, treatment of nonreal spectral parameters, or whether the assertion is intended for every $\delta>0$; its wording asks about some $\delta>0$.

The source also asks for the corresponding explicit dependence on the weight $k_f$ for holomorphic Hecke eigenforms. It does not specify the level, normalization, precise remainder estimate, or a proposed relation between $T$ and $k_f$, so this part remains a request for a holomorphic analogue rather than a uniquely determined quantified assertion.

Short English statement

Determine explicitly how the error in the second moment of a level-$1$ Hecke--Maass $L$-function depends on its spectral parameter. In particular, ask whether an $O_{\varepsilon,\delta}(T^{2/3+\varepsilon})$ bound holds uniformly once $T\geq t_f^{3/2+\delta}$ for some $\delta>0$. Ask also for an analogous result with explicit weight dependence for holomorphic Hecke eigenforms, whose precise scope is unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800168$, exact source problem code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied record labels the problem open; this status is supplied metadata, not an independently verified literature assessment.
• Hamieh, Kadiri, Ng, Moments of L-functions Problem List (2024), Problem 18: https://arxiv.org/abs/2405.17800. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of access or verification in this draft.
• The problem text attributes the question to Peter Humphries, Virginia. The registry's proposer field lists Hamieh, Kadiri, Ng, with proposed year 2024.
• The fixed-spectral-parameter estimate is attributed to Good in the supplied text. No bibliographic identification of Good's underlying work is supplied.