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.