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 Lindelöf Hypothesis and the Generalized Lindelöf 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 (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.

在渐进长度为\（H <x ^ {6/ 11- \ varepsilon} \）的短间隔内，我们渐近地评估x的无平方整数的数量的方差和算术级数中的x的无平方整数的数量的方差模数q与\（q> X ^ {5/11 + \ varepsilon} \）。分别在Lindelöf假设和广义Lindelöf假设的假设下，我们表明这些范围可以分别提高到\（H <x ^ {2/3-\ varepsilon} \）和\（q> x ^ {1/3 + \ varepsilon} \）。此外，我们表明，在整个范围内获得到\（H ^ {\ varepsilon} \）因子的边界锐化\（H <x ^ {1-\ varepsilon} \）等同于黎曼假说。这些结果在霍尔（Mathematika 29（1）：7–17，1982）的短时间间隔上得到了改善，而Warlimont，Vaughan，Blomer，Nunes和Le Boudec在算术级数的情况下得到了更早的结果。