Local divisor correlations in almost all short intervals
Javier Pliego, Yu-Chen Sun, Mengdi Wang
TL;DR
The paper addresses local divisor correlations in very short intervals by studying sums of the form $\sum_{x<n\le x+H_1} d_k(n)d_l(n+h)$ and proving that, for almost all $x$ in $[X,2X]$ and almost all $h$ with $|h|\le H_2$, these sums admit the expected main-term asymptotic $c_{h,k,l}(\log X)^{k+l-2}H_1$ with an error $o(H_1(\log X)^{k+l-2})$, under the scales $H_1\ge(\log X)^{\Phi(X)}$ and $(\log X)^{1000k\log k}\le H_2\le H_1^{1-\varepsilon}$. The authors develop a circle-method framework that combines major-arc and minor-arc analyses with a long–short interval comparison and Dirichlet-character twists, leveraging Vinogradov–Korobov zero-free regions to obtain robust mean-value bounds for Dirichlet polynomials. A key innovation is the reduction to a bounded-majorant framework via the $f_k$-construction, which allows precise control of fluctuations and the use of large-value estimates to handle exceptional cases. The resulting local asymptotics extend the understanding of divisor correlations in the short-interval regime and connect to prior big-picture results on divisor sums and circle-method techniques in additive problems.
Abstract
Let $ k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x<n\leq x+H_1}d_k(n)d_l(n+h)$. More precisely, let $\varepsilon>0$ be a small fixed number and let $Φ(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{Φ(x)}$ and $(\log x)^{1000k\log k}\leq H_2\leq H_1^{1-\varepsilon }$, the expected asymptotic formula holds for almost all $x\in[X,2X]$ and almost all $1\leq h\leq H_2$.