  In this note, we show that there are many infinity positive integer values of
$n$ in which, the following inequality holds $$
\left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2
n}{\log\log n}\right\rfloor\leq\pi\big((n+1)^2\big)-\pi(n^2). $$
