Notes on certain arithmetic inequalities involving two consecutive primes

Let r; k be positive integers (parameters) with r ≥ 2,and let pr be the r-th prime number. let wk denote the set of positive integers n for which the number of distinct prime factors of n is greater or equal to k. by using the prime number theorem and bertrand's theorem,we will determine arithmetic functions f,g: n → nfor which f(n)-αrg(n) has infinitely many sign changes on the set wk,where αr = pr-1/pr in the framework of internal set theory (for more details,see nelson (1977)),some notions concerning nonstandard analysis as well as unlimited positive integers have been used.