Correlations of multiplicative functions and applications

We give an asymptotic formula for correlations Sigma(n <= x)f(1)(P-1(n))f(2)(P-2(n)) . . . f(m)(P-m(n)), where f, . . . ,f(m) 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 Erdos from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either f(n) = n(s) for Re(s) < 1 or vertical bar f(n)vertical bar is small on average. This settles an old conjecture of Katai. 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 a result of Brudern.