  Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative
functions taking values in the closed unit disc. Using an analytic approach in
the spirit of Hal\'{a}sz' mean value theorem, we compute multidimensional
averages of the shape $$x^{-l} \sum_{\mathbf{n} \in [x]^l} \prod_{1 \leq j \leq
k} f_j(L_j(\mathbf{n}))$$ as $x \rightarrow \infty$, where $[x] := [1,x]$ and
$L_1,\ldots, L_k$ are affine linear forms that satisfy some natural conditions.
Our approach gives a new proof of a result of Frantzikinakis and Host that is
distinct from theirs, with \emph{explicit} main and error terms. \\ As an
application of our formulae, we establish a \emph{local-to-global} principle
for Gowers norms of multiplicative functions. We also compute the asymptotic
densities of the sets of integers $n$ such that a given multiplicative function
$f: \mathbb{N} \rightarrow \{-1, 1\}$ yields a fixed sign pattern of length 3
or 4 on almost all 3- and 4-term arithmetic progressions, respectively, with
first term $n$.
