Infinite Number of Changes of Sign for A Difference of Two Number-Theoretic Functions

In this paper,we present two number-theoretic functions F and G for which prF (n)−pr−1G(n) is both positive and negative infinitely often, where n has at least k distinct prime factors (k ≥ 1) and (pr−1, pr) is a couple of two consecutive primes. To be precise, we will construct infinite sequences (ni)i≥1 , (mi)i≥1 such that, F (ni) G(ni) > pr−1 pr > F (mi) G(mi) , for i = 1, 2, . . . , where each ni and mi has k distinct prime factors and F (t) and G(t) are either the Kernel or the Euler’s function of the positive integer t.