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- W2022803914 abstract "An integer is called y-smooth if all of its prime factors are ⩽ y . An important problem is to show that the y -smooth integers up to x are equi-distributed among short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then all intervals of length <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msqrt> <mml:mi>x</mml:mi> </mml:msqrt> </mml:math> up to x , contain, asymptotically, the same number of y -smooth integers. We come close to this objective by proving that such y -smooth integers are so equi-distributed in intervals of length <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msqrt> <mml:mi>x</mml:mi> </mml:msqrt> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> </mml:math> , for any fixed ε < 0." @default.
- W2022803914 created "2016-06-24" @default.
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- W2022803914 date "1993-11-15" @default.
- W2022803914 modified "2023-09-23" @default.
- W2022803914 title "Smoothing ‘smooth’ numbers" @default.
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- W2022803914 doi "https://doi.org/10.1098/rsta.1993.0133" @default.
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