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 } 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.
Highlights
Generalized Lindelof Hypothesis we show that these ranges can be improved to respectively H < x2/3−ε and q > x1/3+ε
By analogy with questions about prime numbers, a basic problem in analytic number theory is to understand the distribution of squarefree numbers in arithmetic progressions and in short intervals
Squarefree numbers ought to be a simpler, more regular sequence than primes, and yet they present distinct challenges; for instance we can determine whether n is prime in polynomial time [AKS04], but there is no known polynomial time algorithm to determine whether n is squarefree
Summary
For large H, say H = x, estimating (2) asymptotically is straightforward, but obtaining an error term Oε(x1/4+ε) is an open problem, even conditionally on the Riemann Hypothesis (see [Liu16] for the best result in this direction). N≤x with C(h) a constant depending only on h Summing this conjectural estimate over h recovers Theorem 1 but only in the range H < X1/2−ε. Keating and Rudnick [KR16] obtained Theorems 1 and 2 in the context of function fields in the limit of a large field size Their results hold in the (analogues of) the ranges Xε ≤ H ≤ X1−ε and xε ≤ q ≤ x1−ε. A formulation of this perspective seems to have been first made in [GH91] This is in contrast to the analogous process generated by prime-counting, where one may conjecture the appearance of Hurst parameter 1/2—that is, usual Brownian motion. The notation n ∼ N in the subscript of a sum will mean that N ≤ n < 2N
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