A multiplicative function with equation \(f(p+m^3)=f(p)+f(m^3)\)

Abstract. We prove that if a multiplicative function \(f\) satisfies the conditions $$ f(p+m^3)=f(p)+f(m^3)\quad\text{and}\quad f(\pi^2)=f(\pi)^2 $$ for all primes \(p,~\pi\) and positive integers \(m\), then \(f(n)=n\) holds for all positive integers \(n\).

Key words and phrases. Multiplicative function, a Dirichlet character, the identity function, functional equation.