Counting primes in the interval (n^2,(n+1)^2)

Abstract

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