On the variance of squarefree integers in short intervals and arithmetic progressions

TL;DR

This paper analyzes the variance of squarefree integers in short intervals and arithmetic progressions, providing asymptotic results and improvements under hypotheses like Lindelöf, and linking bounds to the Riemann Hypothesis.

Contribution

It extends previous results by establishing new variance bounds for squarefree integers in short intervals and progressions, under various hypotheses, and connects these bounds to the Riemann Hypothesis.

Findings

Variance bounds for squarefree integers in short intervals up to $x^{6/11 - \\varepsilon}$

Variance bounds in arithmetic progressions for moduli up to $x^{5/11 + \\varepsilon}$

Improved ranges under Lindelöf and Generalized Lindelöf hypotheses

Abstract

We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H < x^{6/11 - \varepsilon}$ and the variance of the number of squarefree integers up to $x$ in arithmetic progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On the assumption of respectively the Lindel\"of Hypothesis and the Generalized Lindel\"of Hypothesis we show that these ranges can be improved to respectively $H < x^{2/3 - \varepsilon}$ and $q > x^{1/3 + \varepsilon}$. Furthermore we show that obtaining a bound sharp up to factors of $H^{\varepsilon}$ in the full range $H < x^{1 - \varepsilon}$ is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.