A Kubilius model for sieve-theoretic sequences

Abstract

<jats:p> We bound the total variation distance in the Kubilius model for sequences with positive level of distribution. We obtain a result that we expect is qualitatively optimal. As a special case, it recovers a recent result of Ford on shifted primes, with a slightly simplified proof. In the classical case considered by Kubilius, our theorem gives a simple proof of the optimal bound discovered by Tenenbaum, up to factors of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$x^{o(1)}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$u^{o(u)}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> . </jats:p>