Odd order cases of the logarithmically averaged Chowla conjecture

A famous conjecture of Chowla states that the Liouville function $\lambda(n)$
has negligible correlations with its shifts. Recently, the authors established
a weak form of the logarithmically averaged Elliott conjecture on correlations
of multiplicative functions, which in turn implied all the odd order cases of
the logarithmically averaged Chowla conjecture. In this note, we give a new and
shorter proof of the odd order cases of the logarithmically averaged Chowla
conjecture. In particular, this proof avoids all mention of ergodic theory,
which had an important role in the previous proof.