A Note on Terence Tao's Paper

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 $$ \sum_{p< x}f_2(p)\ll x\log^2x\log\log x $$ with other results. But actually only the upper bound $x\log^2x\log\log^2x$ 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.