Correlations of multiplicative functions and applications

Abstract

We give an asymptotic formula for correlations$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$where$f,\ldots ,f_{m}$are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions$f:\mathbb{N}\rightarrow \{-1,+1\}$with bounded partial sums. This answers a question of Erdős 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$\operatorname{Re}(s)<1$or$|f(n)|$is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of$n=a+b$, where$a,b$belong to some multiplicative subsets of$\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.