Multiplicative functions satisfying the functional equation κf(m^2+n^2) = f(κm^2) + f(κn^2)

Abstract

For a fixed positive integer \kappa, the functional equation \kappa f(m^2 + n^2) = f(\kappa m^2) + \kappa f(n^2), (m,n\in\mathbb{N}) is solved for multiplicative functions f. This complements a 1996 result of Chung [2] which deals with the case \kappa=1. The method used relies on the sum of two squares theorem in number theory.