AbstractWe evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$
H
<
x
6
/
11
-
ε
and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$
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 $$H < x^{2/3 - \varepsilon }$$
H
<
x
2
/
3
-
ε
and $$q > x^{1/3 + \varepsilon }$$
q
>
x
1
/
3
+
ε
. Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$
H
ε
in the full range $$H < x^{1 - \varepsilon }$$
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.
On the distribution of squarefree integers in arithmetic progressions
