Moments of L-functions — Moments of the Riemann zeta function — Problem 41

Definitions and mathematical statement

Let $g$ be a primitive cusp form of weight $k$ for $\Gamma_0(N)$, arithmetically normalized by $a_g(1)=1$. Here $\Gamma_0(N)$ denotes the subgroup of $\mathrm{SL}_2({\mathbb{Z}})$ whose lower-left entry is divisible by $N$. Write $e(t)=\exp(2\pi i t)$. For positive Fourier indices, the normalization in the supplied record gives \[ g(z)=\sum_{n\geq 1}a_g(n)e(nz),\qquad a_g(n)=\lambda_g(n)n^{(k-1)/2},\qquad \lambda_g(1)=1. \] The record writes its Fourier expansion starting at $n=0$; the constant Fourier coefficient of a cusp form is zero, and no value of $\lambda_g(0)$ is needed here. The record does not specify the admissible weights, levels, or character conventions beyond the description above.

Let $X,Y,P\geq1$, and let $F(x,y)$ be a smooth weight function supported on \[ (X/4,4X]\times[Y/4,4Y]. \] The derivative condition is supplied in the following inconsistent notation: \[ x^iy^j\frac{\partial^i}{\partial{x^i}}\frac{\partial^j}{\partial v^j}f(u,v)\ll P^{i+j} \qquad(i,j\in{\mathbb{Z}}_{\geq0}). \] It mixes $F$ with $f$ and $(x,y)$ with $(u,v)$. A possible intended condition is \[ \left|x^iy^j\frac{\partial^{i+j}F}{\partial x^i\partial y^j}(x,y)\right| \leq C_{i,j}P^{i+j}, \] but this is an interpretation requiring confirmation; the domain on which the estimate holds and the permitted dependence of its constants are not explicitly supplied.

For coprime $\ell_1,\ell_2$, the record introduces \[ D_g(\ell_1,\ell_2;h) =\sum_{\ell_1m\pm\ell_2n=h} \lambda_g(n)\lambda_g(m)F(\ell_1m,\ell_2n). \] The natural positive-Fourier-index interpretation takes $m,n$ to be positive integers. The source does not explicitly specify their ranges, the domains or signs of $\ell_1,\ell_2$, the domain of $h$, or whether the two signs designate separate sums or a combined sum. Those conventions remain unresolved.

Supplied background and missing target. The record attributes to Harcos [Harcos] an asymptotic upper bound for this sum with implicit constants depending on $g$, and to Blomer and Harcos [BH] an asymptotic bound for an averaged version, also with implicit constants depending on $g$. It supplies neither bound nor the averaging procedure. It gives no explicit conjecture, requested estimate, or other mathematical task after this background. Consequently, a precise open assertion cannot be recovered from the supplied record alone. In particular, a request for uniformity in $g$ cannot be imposed merely from the repeated qualification about implicit constants.

Short English statement

The record describes weighted shifted convolution sums of normalized Fourier coefficients of a primitive cusp form and reports bounds whose constants depend on the form. Its derivative hypothesis is inconsistent, several summation conventions are unstated, and the actual requested estimate or conjecture is missing. The supplied status is open; this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800189$, source problem code catalogue entry, supplied complete source record. The registry attributes the problem to Hamieh, Kadiri, and Ng, with proposed year $2024$, and labels it open; this label is supplied metadata, not an independently verified assessment.
• Hamieh, Kadiri, and Ng, Moments of L-functions Problem List (2024), Problem 41, as identified in the supplied provenance: https://arxiv.org/abs/2405.17800. The registry title is "Moments of L-functions --- Moments of the Riemann zeta function --- Problem 41." The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent source consultation.
• [Harcos] Harcos, cited in the supplied record for an asymptotic upper bound with implicit constants depending on $g$. No title, publication details, precise bound, or separate link is supplied.
• [BH] Blomer and Harcos, cited in the supplied record for an asymptotic bound for an averaged version, with implicit constants depending on $g$. No title, publication details, averaging specification, precise bound, or separate link is supplied.