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.
Categories: math.NT