Shifted character sums with multiplicative coefficients, II

Let f(n) be a multiplicative function with |f(n)|≤1,q be a prime number and a be an integer with (a,q)=1, χ be a non-principal Dirichlet character modulo q. Let ε be a sufficiently small positive constant, A be a large constant, q12+ε≪N≪qA. In this paper, we shall prove that∑n≤Nf(n)χ(n+a)≪Nlog⁡log⁡qlog⁡q and that∑n≤Nf(n)χ(n+a1)⋯χ(n+at)≪Nlog⁡log⁡qlog⁡q, where t≥2,a1,…,at are distinct integers modulo q.