On binary correlations of multiplicative functions

TL;DR

This paper investigates the binary correlations of multiplicative functions, extending previous work by Tao, and applies these results to problems in prime factorization, smooth numbers, and character sums.

Contribution

It broadens the class of multiplicative functions for which correlation estimates are known, including those uniformly distributed in arithmetic progressions, and derives multiple number-theoretic applications.

Findings

Correlation of multiplicative functions asymptotic to product of means

Independence of large prime factors of n and n+1

Small average of quadratic character over reducible polynomials

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 number of large prime factors of $n$ and $n+1$ are independent of each other with respect to the logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erd\H{o}s and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\leq x^{4-\varepsilon}$, the logarithmic average around $x$ of the real character $\chi \pmod{Q}$ over the values of a reducible quadratic polynomial is small.