Shifted character sums with multiplicative coefficients

Let f(n) be a multiplicative function satisfying |f(n)|≤1, q (≤N2) be a prime number and a be an integer with (a,q)=1, χ be a non-principal Dirichlet character modulo q. In this paper, we shall prove that∑n≤Nf(n)χ(n+a)≪Nq14log⁡log⁡(6N)+q14N12log⁡(6N)+Nlog⁡log⁡(6N) and that∑n≤Nf(n)χ(n+a1)⋯χ(n+at)≪Nq14log⁡log⁡(6N)+q14N12log⁡(6N)+Nlog⁡log⁡(6N), where t≥2, a1,…,at are pairwise distinct integers modulo q.