New extensions of some classical theorems in number theory

In this paper we show that for k∈N∪{0}, under natural assumptions on the functions g and h, for a large class of Riemann integrable functions f:[0,1]k+1→R (not all, for k∈N; and all, for k=0), the following equality holdslimx→∞1h(x)∑n⩽xf(nx,ln1nln1x,…,lnknlnkx)g(n)=∫01f(x,1,…,1︸k-times)dx. Using these results for prime numbers, we obtain some new extensions of the classical version from 1917 Polyaʼs theorem in number theory.