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

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 $$\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}\mathrm{.}$$ 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 $p\mathrm{.}$ Recently Terence Tao proved that $$\sum _{p<x}{f}_1(p)\ll x\exp (\frac{c\log x}{\log \log x})$$ with other results. In this paper we shall improve it to $$\sum _{p<x}{f}_1(p)\ll x{\log }^{5}x\log {\log }^{2}x\mathrm{.}$$