一类恒等函数问题的研究

A Note on the Identity Function

设f:N→R+∪{0},g:N→C是完全积性函数,若f(p+1)=g(p)+1和f(p^2+q^3)=g(p^2)+g(q^3)对所有素数p,q均成立,则对所有素数p,q,π,f(p+1)=f(p^2+q^3)=0,g(π)=-1,或者对所有正整数n,f(n)=g(n)=n.

In this paper,we prove that if f:N →R+∪{0},g:N →Care completely multiplicative functions such that f(p+1)=g(p)+1and f(p^2+q^3)=g(p^2)+g(q^3)holds for all primes pand q,then either f(p+1)=f(p^2+q^3)=0,g(π)=-1for all primes p,q,πor f(n)=g(n)=nfor all positive integers n.