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arXiv · 2503.02962

Local divisor correlations in almost all short intervals

Javier Pliego·Yu-Chen Sun·Mengdi Wang
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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 $\sum_{x 0$ be a small fixed number and let $\Phi(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{\Phi(x)}$ 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 $1\leq h\leq H_2$.

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Javier Pliego, Yu-Chen Sun, Mengdi Wang. 2025-03-04. Local divisor correlations in almost all short intervals. https://arxiv.org/abs/2503.02962

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