Sums of multiplicative functions with shifted arguments (Q1311138)

Let \(d(n)\), \(d_ k(n)\) denote the usual divisor functions, \(r(n)\) the number of distinct representations of \(n\) in the form of a sum of two squares. If \(f(n)\) is a multiplicative function with the only restriction \(| f(n)|\leq 1\), asymptotic representations of the sums \[ \sum_{n\leq x} f(n) d_ k(n) d(n-1), \qquad \sum_{n\leq x} f(n) d_ k(n) r(n-1) \] are presented. For the first sum no proof and for the second sum only the main steps of proof are given.