On the estimate for a mean value relative to 4/p=1/n_1+1/n_2+1/n_3

TL;DR

This paper improves the upper bound estimate for the sum of solutions where exactly one denominator is divisible by a prime p in a specific Diophantine equation, refining previous exponential bounds to a polynomial-logarithmic bound.

Contribution

The paper provides a significantly tighter bound on the sum of solutions with a specific divisibility condition, advancing the understanding of solution distribution for the equation.

Findings

Improved the bound from exponential to polynomial-logarithmic in x.

Established a new upper bound:  x \u2212  x \u2212 1   x         x.

Enhanced the understanding of the distribution of solutions relative to prime divisibility.

Abstract

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1, n_2, n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ 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 $p.$ Recently Terence Tao proved that $$ \sum_{p< x}f_1(p)\ll x\exp({c\log x\over \log\log x}) $$ with other results. In this paper we shall improve it to $$ \sum_{p< x}f_1(p)\ll x\log^5x\log\log^2x. $$