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.