  Let $\lambda$ denote the Liouville function. The Chowla conjecture, in the
two-point correlation case, asserts that $$ \sum_{n \leq x} \lambda(a_1 n +
b_1) \lambda(a_2 n+b_2) = o(x) $$ as $x \to \infty$, for any fixed natural
numbers $a_1,a_2,b_1,b_2$ with $a_1b_2-a_2b_1 \neq 0$. In this paper we
establish the logarithmically averaged version $$ \sum_{x/\omega(x) < n \leq x}
\frac{\lambda(a_1 n + b_1) \lambda(a_2 n+b_2)}{n} = o(\log \omega(x)) $$ of the
Chowla conjecture as $x \to \infty$, where $1 \leq \omega(x) \leq x$ is an
arbitrary function of $x$ that goes to infinity as $x \to \infty$, thus
breaking the "parity barrier" for this problem. Our main tools are the
multiplicativity of the Liouville function at small primes, a recent result of
Matom\"aki, Radziwi{\l}{\l}, and the author on the averages of modulated
multiplicative functions in short intervals, concentration of measure
inequalities, the Hardy-Littlewood circle method combined with a restriction
theorem for the primes, and a novel "entropy decrement argument". Most of these
ingredients are also available (in principle, at least) for the higher order
correlations, with the main missing ingredient being the need to control short
sums of multiplicative functions modulated by local nilsequences.
  Our arguments also extend to more general bounded multiplicative functions
than the Liouville function $\lambda$, leading to a logarithmically averaged
version of the Elliott conjecture in the two-point case. In a subsequent paper
we will use this version of the Elliott conjecture to affirmatively settle the
Erd\H{o}s discrepancy problem.
