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I'm trying to understand Ramanujan's proof of Bertrand's postulate, but I don't get the step in which it says

But is easy to see that

$\log\Gamma(x) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2}) \le \log[x]! - 2\log[\frac{1}{2}x]! \le \log\Gamma(x+1) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2})$

Then, applying Stirling's approximation, Ramanujan gets to:

$\log[x]! - 2\log[\frac{1}{2}x]! < \frac{3}{4}x$ if $x > 0$

and

$\log[x]! - 2\log[\frac{1}{2}x]! > \frac{2}{3}x$ if $x > 300$

How does he use Stirling formula to get those inequalities?

I know this have been asked before, but I'm trying to find another explanation because I'm not familiar with the method used by the user who answered.

I hope someone can help me understand.

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  • $\begingroup$ I think you can do it with just the error estimate of Stirling $$0\leq\log\frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^x}\leq\frac1{12x}$$ for all $x>0$ $\endgroup$
    – user10354138
    Dec 6, 2018 at 8:20

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