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 $$H < x^{6/11 - \varepsilon }$$ 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.

A variance fork-free numbers in arithmetic progressions of given modulus

“…We should mention a very recent work [3] of Ofir Gorodetsky, Kaisa Matomäki, Maksym Radziwiłł and Brad Rodgers, in which the range of validity for the asymptotic formula extends to q as small as x 5/11 , which far surpasses what we have here. On the other hand their results are proven only for prime q and their error term which corresponds to our first error term, not being their prime concern but being our main concern, is far weaker than what we have.…”

“…On the other hand their results are proven only for prime q and their error term which corresponds to our first error term, not being their prime concern but being our main concern, is far weaker than what we have. We should in particular mention that we found our argument for Lemma 3.6 after looking over [3].…”

“…The question of whether one averages over a complete set of residues, as here, or over a reduced one, as in [3] and [9], is, most likely harmless but still remains to be seen. The squarefree numbers are, in contrast to the primes, not better understood outside the reduced residue classes, and therefore it isn't clear, at least to the author, neither why averaging over a reduced system is guaranteed to be just as insightful as averaging over a complete system, nor that averaging over a complete system is no harder.…”

Squarefrees are Gaussian in short intervals

“…Keating and Rudnick [23] studied this problem in a function field setting, connecting it with Random Matrix Theory, and suggested based on this work that (1.1) will hold for H ≤ X 1−ε . The best known result is [12], where it is shown that (1.1) holds for H ≤ X 6/11−ε unconditionally and H ≤ X 2/3−ε on the Lindelöf Hypothesis. In fact in [12] it is shown that even an upper bound of order √ H for H ≤ X 1−ε for all ε > 0 would already imply the Riemann Hypothesis.…”

“…The best known result is [12], where it is shown that (1.1) holds for H ≤ X 6/11−ε unconditionally and H ≤ X 2/3−ε on the Lindelöf Hypothesis. In fact in [12] it is shown that even an upper bound of order √ H for H ≤ X 1−ε for all ε > 0 would already imply the Riemann Hypothesis.…”