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−ε) and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q > x^(5/11+ε). 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−ε) and q > x^(1/3+ε). Furthermore we show that obtaining a bound sharp up to factors of H^ε in the full range H < x^(1−ε) is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

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