Zeta Moments from Shifted Divisor Correlations

Definitions and mathematical statement

Let $\zeta(s)$ denote the meromorphic continuation of the Riemann zeta function, initially defined by $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ for $\operatorname{Re}s>1$. For a positive integer $k$ and a real number $T>0$, define the critical-line moment \[ M_k(T)=\int_0^T\left|\zeta\!\left(\tfrac12+it\right)\right|^{2k}\,{\,\mathrm{d}} t. \] The critical line, integration interval, and positive-integer range of $k$ are conventional interpretations used here to display the question; the supplied record does not explicitly specify them.

For such $k$, the ordinary $k$-fold divisor function is \[ d_k(n)=\#\{(n_1,\ldots,n_k)\in{\mathbb{Z}}_{\geq1}^{\,k}:n_1\cdots n_k=n\},\qquad n\in{\mathbb{Z}}_{\geq1}. \] A standard example of a shifted-divisor correlation is \[ D_k(X;h)=\sum_{\substack{n\in{\mathbb{Z}}_{\geq1}\\n\leq X,\ n+h\geq1}}d_k(n)d_k(n+h), \qquad X\geq1,\quad h\in{\mathbb{Z}}. \] This example fixes notation only: the source does not identify the precise shifted divisor sums required, and it does not specify whether ordinary divisor coefficients or coefficients carrying additional complex shifts are intended.

Requested derivation. Obtain an arithmetic derivation of the conjectured $2k$th moments of the Riemann zeta function from shifted-divisor problems, including the required $k$-fold convolutions of shifted divisor sums. Under the conventional moment interpretation above, this asks for a derivation of the intended conjectural asymptotics of $M_k(T)$ as $T\to\infty$.

Unresolved specifications. The supplied record gives neither the exact moment asymptotic nor its constants, lower-order terms, or error term. It also gives no definition of the required $k$-fold convolution, no shift ranges, and no uniformity or error estimates for the shifted-divisor problems. Consequently, the requested derivation cannot be made into a uniquely specified theorem from this record alone. In particular, the displayed example $D_k(X;h)$ must not be treated as a complete specification of the required arithmetic input.

Short English statement

The problem asks for an arithmetic derivation of conjectured zeta-function moments using shifted divisor sums and their required $k$-fold convolutions. The exact moment conjecture, convolution operation, parameter ranges, and necessary estimates are unspecified. The supplied open status is dated curation, and this statement-only draft requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003272$, title Zeta Moments from Shifted Divisor Correlations, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2016 and no named proposer.
• Supplied report citation: Analytic Number Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/53, pp. 2975--3030. https://doi.org/10.4171/owr/2016/53. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, "OWR 53 report text": https://ems.press/content/serial-article-files/46657?nt=1.
• Supplied literature source, "Conrey and Keating": https://doi.org/10.1098/rsta.2014.0313.
• Supplied literature source, "Moments survey": https://arxiv.org/abs/1506.06842.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the shifted-divisor approach provides substantial moment heuristics and finite-order results, while a derivation of the full Riemann-zeta moment conjectures remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This records the supplied editorial assessment, not independent mathematical verification of this draft.