The paper under review deals with asymptotic formulas for sums of the shape \[\sum_{n \in \mathbb{Z}^s \cap N \mathfrak{K}} h_1 \left( P_1 (\mathbf{n}) \right) \dotsb h_r \left( P_r (\mathbf{n}) \right)\] where \(r,s \in \mathbb{Z}_{\geqslant 2}\), \(\mathfrak{K} \subset \mathbb{R}^s\) is a fixed bounded convex subset, \(N \mathfrak{K}:= \left\lbrace N \mathbf{x} \in \mathbb{R}^s : \mathbf{x} \in \mathfrak{K} \right\rbrace\), \(P_1,\dotsc,P_r \in \mathbb{Z} \left[ X_1,\dotsc,X_s\right]\) are fixed linear polynomial satisfying the property that the non-constant parts of any two of these polynomials are pairwise linearly independent, and \(h_1,\dotsc,h_r\) belong to a large class of multiplicative functions satisfying the Ramanujan hypothesis \(h_j (n) \ll n^\varepsilon\), and containing the usual arithmetic functions \(\tau_k\), \(k^{\omega}\) with \(0 < k < 2\), \(\left| \lambda_f \right|\) where \(\lambda_f\) is the normalised Fourier coefficients of a primitive holomorphic cusp form, etc. The results are too complicated to be stated here, but it should be mentioned that the proof proceeds via Green and Tao’s nilpotent Hardy-Littlewood method [B. Green and T. Tao, Ann. Math. (2) 171, No. 3, 1753–1850 (2010; Zbl 1242.11071)]. Simplified versions of the main theorems are also given.