# On binary correlations of multiplicative functions

Teräväinen, J

28 June 2018

## Journal:

Forum Math. Sigma

## Last Updated:

2021-04-07T03:50:40.987+01:00

6

## DOI:

10.1017/fms.2018.10

e10-e10

## 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.

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