Creating an inequality for the numerator significance

Question

An open mathematical problem is if the Euler-Mascheroni constant is irrational and if so transcendental. There has been some progress in (dis)proving this. It is known that:

$\gamma\in\mathbb{Q}\implies q\ge10^{244663}$ where $q$ is the denominator of $\gamma$

This is a statement proved by Papanikolaou in his paper here. So I wonder if an inequality for the numerator, say $p$, is of any significance. What I am asking is, if there is an inequality for the denominator, does having one for the numerator help in proving (ir)rationality? I am asking this question since I found this inequality for the numerator: $$p\ge-\ln\left(\Gamma(10^{244663}+1)\right)+\sum_{k=1}^\infty\left(\frac{10^{244663}}{k}-\ln\left(1+\frac{10^{244663}}{k}\right)\right)$$And I wonder if it is of any significance.

Answer 1

I agree with Calvin. Here are some more thoughts/details:

• If $q \geq B$ for some $B>0$, then $p = \frac{p}{q}\, q \geq \gamma B \approx 0.5572\, B$.
• If $p \geq C$ for some $C>0$, then $q = \frac{q}{p}\, p \geq \gamma^{-1} C \approx 1.732\, C$.
• You should verify that your bound is a positive number, otherwise it is worthless.
• Once this is done, you could check whether your bound is larger than $\gamma B = 0.5772 \cdot 10^{244633}$, only then have you improved the bound found by Papanikolaou.

I have serious doubts that your bound is nonnegative, since $$ C_{n} = -\ln\left(\Gamma(n+1)\right)+\sum_{k=1}^\infty\left(\frac{n}{k}-\ln\left(1+\frac{n}{k}\right)\right) \xrightarrow{n\to\infty} -\infty, $$ where I used Wolfram Alpha, and your $n = 10^{244633}$ is rather large..