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A1 Journal article – refereed

ON BINARY CORRELATIONS OF MULTIPLICATIVE FUNCTIONS

List of Authors: Joni Teräväinen

Publisher: CAMBRIDGE UNIV PRESS

Publication year: 2018

Journal: Forum of Mathematics, Sigma

Journal name in source: FORUM OF MATHEMATICS SIGMA

Journal acronym: FORUM MATH SIGMA

Volume number: 6

ISSN: 2050-5094

Abstract

We study logarithmically averaged binary correlations of bounded multiplicative functions g(1) and g(2). A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever g(1) or g(2) does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions g(j), namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of g(1) and g(2) is asymptotic to the product of their mean values. We derive several applications, first showing that the numbers of large prime factors of n and n + 1 are independent of each other with respect to logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erdos and Pomerance on two consecutive smooth numbers. Thirdly, we show that if Q is cube-free and belongs to the Burgess regime Q <= x(4-epsilon), the logarithmic average around x of the real character chi (mod Q) over the values of a reducible quadratic polynomial is small.

We study logarithmically averaged binary correlations of bounded multiplicative functions g(1) and g(2). A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever g(1) or g(2) does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions g(j), namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of g(1) and g(2) is asymptotic to the product of their mean values. We derive several applications, first showing that the numbers of large prime factors of n and n + 1 are independent of each other with respect to logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erdos and Pomerance on two consecutive smooth numbers. Thirdly, we show that if Q is cube-free and belongs to the Burgess regime Q <= x(4-epsilon), the logarithmic average around x of the real character chi (mod Q) over the values of a reducible quadratic polynomial is small.

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