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I am reviewing Ramanujan's proof of Bertrand's Postulate which can be found here.

At step #7, he writes: "But it is easy to see that..."

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

As $\Gamma(x+1) = [x]!$, this seems reasonable enough to me.

My question is: why did Ramanujan choose $\frac{1}{2}$. It seems like this would be true for any $n \ge 0$ where:

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

Is that correct?



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Having cracked a book and taken out my magnifying glass, I feel that you may have mis-stated (7). I would be the last to down-vote for a typo... –  daniel Mar 23 '13 at 23:33
Thanks for the tip. I'll check it and update my question. –  Larry Freeman Mar 23 '13 at 23:58
@daniel: I checked it and it's possible that 1/3 may be 1/2. It does look like 1/3 to me. Is there any other area where you see a mistake? That's the only ambiguity that I see. –  Larry Freeman Mar 24 '13 at 0:02
Thanks! I've change the detail to 1/2. Thanks very much for the correction! :-) –  Larry Freeman Mar 24 '13 at 0:08
For those interested, here is an easier-to-read html version of Ramanujan's proof of Betrand's Postulate. –  Larry Freeman Mar 24 '13 at 0:54
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1 Answer

up vote 5 down vote accepted

Inequality (7) is delicate. $\log x!-\log\Gamma(x)$ is a little bigger than $2\log (x/2)!-2\log\Gamma(x/2+1/2),$ which makes the lower bound tight. $\log\Gamma(x+1)$ is no different from $\log x!$ so he relies on $2\log\Gamma(x/2+1/2)$ being slightly less than $2\log(x/2)!$ to achieve the upper bound.

In (8) he refers to the Stirling approximation for the bounds in (7), so there may have been an element of computational convenience in the choice 1/2. Once he establishes the numerical bound in (8),(9) he has no further use for the $\log\Gamma$ functions used in (7).

There may be room for a constant other than 1/2 but anything you do will have an effect at both ends. $2\log\Gamma(x/2+3/4)$ might be closer to and still less than $2\log(x/2)!$ but will the left inequality hold? It appears not.

Whether 1/2 is optimal I am not sure but I would guess so. When you examine the asymptotic behavior of $\log\Gamma$ the derivative becomes important. Question 94582, though I can't link to it at the moment, concerned this property. The factor 1/2 in argument of $\log\Gamma$ (and 2 outside it) make it something to check as an explanation of the choice of 1/2.

Edit: In light of this question it appears that $1/2$ is needed to give desired asymptotic behavior to the l.h.s. and r.h.s. For completeness:

$PG(x) :=$ Polygamma(x). From (7):

$$\ln\Gamma(x) -2\ln\Gamma\left(\frac{x+1}{2}\right)\leq \cdot \leq \ln\Gamma\left(x+1\right)-2\ln\Gamma\left(\frac{x+1}{2}\right) $$

$$\frac{d}{dx}\left(\ln\Gamma(x)-2\ln\Gamma\left(\frac{x+1}{2}\right)\right)= PG(x)-PG\left(\frac{x+1}{2}\right)\sim \ln 2^-$$

$$\frac{d}{dx}\left(\ln\Gamma\left(x+1\right)-2\ln\Gamma\left(\frac{x+1}{2}\right)\right)= PG\left(x+1\right)-PG\left(\frac{x+1}{2}\right)\sim \ln 2^+$$

So $\log[x]! -2\log[x/2]!$ is trapped between two functions which approach $\log 2$ from above and below. The 1/2 is from $\frac{x+1}{2}$ and is essential to the asymptotic limits.

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