New extensions of some classical theorems in number theory

Abstract

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 holds lim x → ∞ 1 h ( x ) ∑ n ⩽ x f ( n x , ln 1 n ln 1 x , … , ln k n ln k x ) g ( n ) = ∫ 0 1 f ( x , 1 , … , 1 ︸ k - times ) d x . Using these results for prime numbers, we obtain some new extensions of the classical version from 1917 Polyaʼs theorem in number theory.