The structure of logarithmically averaged correlations of multiplicative
functions, with applications to the Chowla and Elliott conjectures

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative
functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider
correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):=
\widetilde{\lim}_{m \to \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq
n \leq x_m} \frac{g_0(n+ah_0) \dots g_k(n+ah_k)}{n} $$ where $1 \leq \omega_m
\leq x_m$ are numbers going to infinity as $m \to \infty$, and
$\widetilde{\lim}$ is a generalised limit functional extending the usual limit
functional. We show a structural theorem for these sequences, namely that these
sequences $f$ are the uniform limit of periodic sequences $f_i$. Furthermore,
if the multiplicative function $g_0 \dots g_k$ "weakly pretends" to be a
Dirichlet character $\chi$, the periodic functions $f_i$ can be chosen to be
$\chi$-isotypic in the sense that $f_i(ab) = f_i(a) \chi(b)$ whenever $b$ is
coprime to the periods of $f_i$ and $\chi$, while if $g_0 \dots g_k$ does not
weakly pretend to be any Dirichlet character, then $f$ must vanish identically.
As a consequence, we obtain several new cases of the logarithmically averaged
Elliott conjecture, including the logarithmically averaged Chowla conjecture
for odd order correlations. We give a number of applications of these special
cases, including the conjectured logarithmic density of all sign patterns of
the Liouville function of length up to three, and of the M\"obius function of
length up to four.