Estimates of the bounds of $\pi (x)$ and $\pi ((x+1{)}^2)-\pi (x^2)$

Abstract

We show the following bounds on the prime counting function $\pi (x)$ using principles from analytic number theory, giving an estimate: $$2\log 2\ge \underset{x\to \infty }{\mathrm{limsup}}\frac{\pi (x)}{x/\log x}\ge \underset{x\to \infty }{\mathrm{liminf}}\frac{\pi (x)}{x/\log x}\ge \log 2$$ for all $x$ sufficiently large. We also conjecture about the bounding of $\pi ((x+1{)}^2)-\pi (x^2)$, as is relevant to Legendre's conjecture about the number of primes in the aforementioned interval such that: $$\lfloor \frac{1}{2}\left(\frac{{\left(x+1\right)}^2}{\log \left(x+1\right)}-\frac{x^2}{\log x}\right)-\frac{{\left(\log x\right)}^2}{\log \left(\log x\right)}\rfloor \le \pi ((x+1{)}^2)-\pi (x^2)\le$$ $$\lfloor \frac{1}{2}\left(\frac{{\left(x+1\right)}^2}{\log \left(x+1\right)}-\frac{x^2}{\log x}\right)+{\log }^{2}x\log \log x\rfloor$$

8 pages, 0 figures