Moments of L-functions — Moments of the Riemann zeta function — Problem 39

Definitions and mathematical statement

For a positive integer $k$ and a positive integer $m$, let the generalized divisor function be \[ d_k(m)=\#\{(a_1,\ldots,a_k)\in{\mathbb{Z}}_{\ge1}^{\,k}:a_1\cdots a_k=m\}. \] Let $r\ge3$ be an integer, let $k_1,\ldots,k_r$ be positive integers, and let $h_1,\ldots,h_r$ be pairwise distinct positive integers. For real $x\ge1$, define \[ S_{\mathbf{k},\mathbf{h}}(x)=\sum_{\substack{n\in{\mathbb{Z}}\,;\ 1\le n\le x}}\prod_{j=1}^{r}d_{k_j}(n+h_j). \] The problem is to formulate an explicit conjectural asymptotic formula for $S_{\mathbf{k},\mathbf{h}}(x)$ as $x\to\infty$. In particular, seek an explicit leading expression $A_{\mathbf{k},\mathbf{h}}(x)$ such that one conjectures \[ \frac{S_{\mathbf{k},\mathbf{h}}(x)}{A_{\mathbf{k},\mathbf{h}}(x)}\longrightarrow1. \] Here the parameters are held fixed when taking the limit; the supplied source does not specify uniformity in them, the desired error term, or the extent of any lower-order expansion. The source writes $(k_1,\ldots,k_r)\in{\mathbb{N}}^r$; the formulation above uses the positive-integer convention. If zero is intended to belong to ${\mathbb{N}}$, the source does not specify the meaning of $d_0$.

The source motivates this problem by analogy with conjectural Hardy--Littlewood asymptotics for \[ \sum_{\substack{n\in{\mathbb{Z}}\,;\ 1\le n\le x}}\Lambda(n+h_1)\cdots\Lambda(n+h_r), \] where $\Lambda(m)=\log p$ if $m=p^a$ for a prime $p$ and an integer $a\ge1$, and $\Lambda(m)=0$ otherwise. This analogy does not supply a formula for the divisor correlation. The source also states that Ng--Thom and Tao used different probabilistic arguments to derive conjectures for $r=2$, but supplies no specific references for those works.

Short English statement

For every fixed choice of at least three pairwise distinct positive shifts and positive divisor-function orders, conjecture an explicit asymptotic formula for the sum of the corresponding shifted divisor-function products. The source compares this task with Hardy--Littlewood prime correlations and mentions probabilistic conjectures for two shifts. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800187$, source code catalogue entry, in the AMR Open Problem Lists collection. The supplied record attributes the problem to Hamieh, Kadiri, and Ng, with proposed year 2024, and labels it open. This label is supplied registry metadata, not an independently verified mathematical or literature assessment. The record was created and updated on 2026-10-16; upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Hamieh, Kadiri, and Ng, Moments of L-functions Problem List (2024), Problem 39, under Moments of the Riemann zeta function. Supplied source URL: https://arxiv.org/abs/2405.17800. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not claims of access or verification for this draft.
• The supplied statement invokes the Hardy--Littlewood conjectures as an analogy and attributes probabilistic conjectures for $r=2$ to Ng--Thom and Tao. It provides no bibliographic details or links for these contextual references.