On the variance of squarefree integers in short intervals and arithmetic progressions

Abstract

Abstract We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H &lt; x^{6/11 - \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo>&lt;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>6</mml:mn><mml:mo>/</mml:mo><mml:mn>11</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q &gt; x^{5/11 + \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>&gt;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>5</mml:mn><mml:mo>/</mml:mo><mml:mn>11</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H &lt; x^{2/3 - \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo>&lt;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> and $$q &gt; x^{1/3 + \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>&gt;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mi>ε</mml:mi></mml:msup></mml:math> in the full range $$H &lt; x^{1 - \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo>&lt;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> 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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