Correlations of multiplicative functions and applications

We give an asymptotic formula for correlations Xn6x f1(P1(n))f2(P2(n)) · · · · · fm(Pm(n)) where f . . . , fm are bounded “pretentious” multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions f : N → {−1, +1} with bounded partial sums. This answers a question of Erdös from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either f(n) = n s for Re(s) < 1 or |f(n)| is small on average. This settles an old conjecture of K´atai. Third, we apply our theorem to count the number of representations of n = a+b where a, b belong to some multiplicative subsets of N. This gives a new ”circle method-free” proof of the result of Brüdern.