On the variance of squarefree integers in short intervals and arithmetic progressions - Gorodetsky, O; Matomäki, K; Radziwiłł, M; Rodgers, B (Springer, 2021)
We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H< x6/11-ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q> x5/11+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< x2/3-ε and q> x1/3+ε. Furthermore we show that obtaining a bound sharp up to factors of Hε in the full range H< x1-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.