On binary correlations of multiplicative functions - Teräväinen, J (Cambridge University Press, 2018)
We study logarithmically averaged binary correlations of bounded multiplicative functions g1 and g2. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever g1 or g2 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 gj, 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 g1 and g2 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 ≤ x4−𝜀, the logarithmic average around x of the real character 𝜒 (mod Q) over the values of a reducible quadratic polynomial is small.