The estimate for mean values on prime numbers relative to $\frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }} $

Abstract

If n is a positive integer, let f(n) denote the number of positive integer solutions (n1, n2, n3) of the Diophantine equation $$\frac{4} {n} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }} $$ For the prime number p, f(p) can be split into f1(p) + f2(p), where fi(p) (i = 1, 2) counts those solutions with exactly i of denominators n1, n2, n3 divisible by p.