Effective Asymptotic Formulae for Multilinear Averages of Multiplicative Functions

Let $f_1,\dots ,f_k:\mathbb{N}\to \mathbb{C}$ 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 $$x^{-l}\sum _{\mathbf{n}\in [x{]}^l}\prod _{1\le j\le k}{f}_j(L_j(\mathbf{n}))$$ as $x\to \mathrm{\infty }$, where $[x]:=[1,x]$ and $L_1,\dots ,L_k$ 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 $f:\mathbb{N}\to \{-1,1\}$ yields a fixed sign pattern of length 3 or 4 on almost all 3- and 4-term arithmetic progressions, respectively, with first term $n$.