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arXiv · 2504.14511

Variance of square-full integers in short intervals and arithmetic progressions

Yotsanan Meemark·Watcharakiete Wongcharoenbhorn
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Abstract

Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected order, under the assumption of a certain quasi-Riemann hypothesis. We also prove an asymptotic formula for the variance in arithmetic progressions, averaging over a quadratic residue and a nonresidue by a half, which is of smaller order of magnitude than the aforementioned bound for all primes $q\gg x^{51/114+\varepsilon}$.

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Yotsanan Meemark, Watcharakiete Wongcharoenbhorn. 2025-04-20. Variance of square-full integers in short intervals and arithmetic progressions. https://arxiv.org/abs/2504.14511

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