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 (n ₁, n ₂, n ₃) 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 f ₁(p) + f ₂(p), where f _i (p) (i = 1, 2) counts those solutions with exactly i of denominators n ₁, n ₂, n ₃ divisible by p.

In this paper, we shall study the estimate for mean values

$\sum\limits_{p < x} {f_i (p),i = 1,2,} $

, where p denotes the prime number.