  We show the following bounds on the prime counting function $\pi(x)$ using
principles from analytic number theory, giving an estimate: $$2 \log 2 \geq
\limsup_{x \rightarrow \infty} \frac{\pi(x)}{x / \log x} \geq \liminf_{x
\rightarrow \infty} \frac{\pi(x)}{x / \log x} \geq \log 2$$ 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: $$ \left
\lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log
x}\right)-\frac{\left(\log x\right)^{2}}{\log\left(\log x\right)}\right \rfloor
\leq \pi((x+1)^{2}) - \pi(x^{2}) \leq $$ $$ \left
\lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log
x}\right) + \log^{2}x\log\log x \right \rfloor$$
