A Note on Terence Tao's Paper "On the Number of Solutions to 4/p=1/n_1+1/n_2+1/n_3"

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 $$\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}.$$ 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 $p.$ Recently Terence Tao proved that $$\sum _{p<x}{f}_2(p)\ll x{\log }^{2}x\log \log x$$ with other results. But actually only the upper bound $x{\log }^{2}x\log {\log }^{2}x$ can be obtained in his discussion. In this note we shall use an elementary method to save a factor $\log \log x$ and recover the above estimate.