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- W4311441454 abstract "We show that, for almost all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the interval <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis x comma x plus left-parenthesis log x right-parenthesis Superscript 2.1 Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2.1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(x, x+(log x)^{2.1}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains products of exactly two primes. This improves on a work of the second author that had <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3.51> <mml:semantics> <mml:mn>3.51</mml:mn> <mml:annotation encoding=application/x-tex>3.51</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in place of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2.1> <mml:semantics> <mml:mn>2.1</mml:mn> <mml:annotation encoding=application/x-tex>2.1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown’s mean value theorem for sparse Dirichlet polynomials." @default.
- W4311441454 created "2022-12-26" @default.
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- W4311441454 date "2023-05-09" @default.
- W4311441454 modified "2023-10-17" @default.
- W4311441454 title "Almost primes in almost all short intervals II" @default.
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- W4311441454 doi "https://doi.org/10.1090/tran/8869" @default.
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