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 Lindelof Hypothesis and the Generalized Lindelof 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 He in the full range H < x(1-epsilon) 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.