Ramanujan's $\pi(x)^{2}<{\frac {ex}{\log x}}\pi {\bigg (}{\frac {x}{e}}{\bigg )}$

Question

Quote from https://en.wikipedia.org/wiki/Prime_Counting_Function#Inequalities

"In his well-known notebooks, Ramanujan[23] proves that the inequality

$$\pi(x)^{2}<{\frac {ex}{\log x}}\pi {\bigg (}{\frac {x}{e}}{\bigg )}$$

holds for all sufficiently large values of $x$."

It is the "holds for all sufficiently large values of $x$" part that I need some help. I do not have the book to verify the proof (Is is correct?), and I wonder if someone has computed the minimum value of x?

Answer 1

This is discussed by Axler in arXiv:1703.02407; see Theorems 1.3, 1.4 and surrounding discussion. In particular, if the Riemann hypothesis is true, then $$ \pi(x)^{2}<{\frac {ex}{\log x}}\pi {\bigg (}{\frac {x}{e}}{\bigg )} \qquad\mbox{ for all }x\ge 38 358 837 683. $$ Regardless of the Riemann hypothesis, Axler proves that $$ \pi(x)^{2}<{\frac {ex}{\log x}}\pi {\bigg (}{\frac {x}{e}}{\bigg )} \qquad\mbox{ for all }x\ge e^{9032}. $$ The largest known integer $x$ for which the inequality fails is $$ x=38 358 837 682; $$ we have $$ \pi(x)^2=2704950040057325824, \qquad {\frac {ex}{\log x}}\pi {\bigg (}{\frac {x}{e}}{\bigg )}= 2704950040042588896.1001\ldots $$