On the variance of squarefree integers in short intervals and arithmetic progressions
: Gorodetsky Ofir, Matomäki Kaisa, Radziwill Maksym, Rodgers Brad
Publisher: SPRINGER BASEL AG
: 2021
: Geometric And Functional Analysis
: GEOMETRIC AND FUNCTIONAL ANALYSIS
: GEOM FUNCT ANAL
: 31
: 1
: 111
: 149
: 39
: 1016-443X
: 1420-8970
DOI: https://doi.org/10.1007/s00039-021-00557-5
: https://research.utu.fi/converis/portal/detail/Publication/54379555
We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H < x(6/11-epsilon) and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q > x(5/11+epsilon). On the assumption of respectively the Lindelof Hypothesis and the Generalized Lindelof Hypothesis we show that these ranges can be improved to respectively H < x(2/3-epsilon) and q > x(1/3+epsilon). Furthermore we show that obtaining a bound sharp up to factors of He in the full range H < x(1-epsilon) 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.
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