In this article (Theorem 1.2) there is a proof for Robin's inequality for odd numbers, $\sigma(n)/n< e^{\gamma}\log(\log(n))$ where $\gamma$ is the Euler-Mascheroni constant and $\sigma(n)$ is the divisor function. The proof is not hard but uses Euler's Totient function and other considerations.

In this other preprint (Theorem 3.4), the same theorem is stated but a much simpler proof is presented. I think there must be some kind of mistake in the proof because it is too easy compared to the proof already published but I'm not sure.

The author uses the bound

$\sigma(n)/n<e^{\gamma}\log(\log(n))+\frac{0.6483}{\log(\log(n))}$

and the fact that the divisor function is multiplicative. In the following way:

$\sigma(n)/n=s(n)=s(2n)/s(2)<2/3(e^{\gamma}\log(\log(2n))+\frac{0.6483}{\log(\log(2n))})< e^{\gamma}\log(\log(n))$

Is there any mistake in this proof?