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

Abstract

We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H<x^{6/11-\epsilon }$ and the variance of the number of squarefree integers up to $x$ in arithmetic progressions modulo $q$ with $q>x^{5/11+\epsilon }$. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $H<x^{2/3-\epsilon }$ and $q>x^{1/3+\epsilon }$. Furthermore we show that obtaining a bound sharp up to factors of $H^{\epsilon }$ in the full range $H<x^{1-\epsilon }$ 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.

40 pages, 2 figures. Incorporated referees' comments. Accepted for publication in GAFA