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

Abstract

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