Robin's inequality for odd numbers

Question

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?

Answer 1

Choie et. al. had more concerns than just odd $n;$ if we ask when their argument kicks in, it is simpler in appearance. Small prime $p,$

$$s(n)=s(pn)/s(p)< \frac{p}{1+p}\left(e^{\gamma}\log(\log(pn))+\frac{0.64821365}{\log(\log(pn))}\right) \; ?< ? \; e^{\gamma}\log(\log(n))$$

or $$ \frac{p}{1+p}\left( \frac{\log(\log(pn))}{ \log \log n}+\frac{0.363945701}{\log(\log(pn))\log \log n}\right) \; ?< 1 ? $$

This decreases as $n$ increases, using simple $$ \log \log n < \log \log pn < \log \log n + \frac{\log p}{\log n} $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

For odd $n \geq 17,$ we find $s(n) < e^\gamma \log \log n.$

For $n \neq 0 \pmod 3$ and $n \geq 56, \; \;$ $s(n) < e^\gamma \log \log n.$

For $n \neq 0 \pmod 5$ and $n \geq 898, \; \; \;$ $s(n) < e^\gamma \log \log n.$

For $n \neq 0 \pmod 7$ and $n \geq 19479, \; \; \;$ $s(n) < e^\gamma \log \log n.$

For $n \neq 0 \pmod {11}$ and $n \geq 19913559, \; \; \;$ $s(n) < e^\gamma \log \log n.$