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ON BINARY CORRELATIONS OF MULTIPLICATIVE FUNCTIONS

Published online by Cambridge University Press:  28 June 2018

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 Erdős and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\leqslant x^{4-\unicode[STIX]{x1D700}}$ , the logarithmic average around $x$ of the real character $\unicode[STIX]{x1D712}\hspace{0.6em}({\rm mod}\hspace{0.2em}Q)$ over the values of a reducible quadratic polynomial is small.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2018

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