Robin's equivalence to the Riemann hypothesis can be written as
$$\frac{\sigma(n)-n}{\gamma+\log\log\log n}<M_{\text{lm}}(\sigma(n),n)\tag{1}$$
for enough large $n$ (it is well-know this suitable choice of the integer $n$, see [1]), where $M_{\text{lm}}(a,b)$ denotes the logarithmic mean, $\sigma(n)=\sum_{1\leq d\mid n}d$ is the sum of divisors function and $\gamma$ is the Euler-Mascheroni constant. Using the relationship between the logarithmic mean and the Stolarsky mean one can to propose the problem to calculate the greatest integer $k$ for which
$$\frac{\sigma(n)-n}{\gamma+\log\log\log n}<\left(\frac{k\left(\sqrt[k]{\sigma(n)}-\sqrt[k]{n}\,\right)}{\sigma(n)-n}\right)^{-\frac{k}{k-1}}\tag{2}$$
holds.
The articles from Wikipedia for these means are Logarithmic mean and Stolarsky mean.
Question. I would like to know (approximately) what is the greatest positive integer or real number $k$ for which we can to prove that $(2)$ is true for all integer $n>n_{0}$, being $n_0$ a fixed positive constant (maybe depending of $k$) that is a suitable choice (yours) in your discussion. Many thanks.
Please the answer must be without assumption of additional conjectures, thus unconditionally. I add this requirement since I know what is the relationship between $(2)$ and $(1)$. I hope that my project (I mean the inequality $(2)$) has a good mathematical content.
I can to compare numerically $(1)$ and $(2)$ for segments of positive integers $n$'s and fixed integers $k=4$ or $k=5,6\ldots$ increasing, with my computer.
Remarks (Maybe unrelated to my post). Similar discussions (the path from $(1)$ to $(2)$) can be done from other equivalences to the Riemann hypothesis whenever you can to invok ean argument of positivity: I mean for example that $N_k>\varphi(N_k)$ where $N_k$ denotes the primorial of order $k$ and $\varphi(n)$ is the Euler's totient function or the example $\sigma(n)>\log(H_n)$ where $H_n$ denotes the harmonic number (I evoke Kaneko's claim in last paragraph of [2], I also did calculations with my computer for this last example).
References:
[1] G. Robin, Grandes Valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63, 187-213, (1984).
[2] Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, The American Mathematical Monthly, 109, No. 6 (2002), pp. 534-543.
[3] Kenneth B. Stolarsky, Generalizations of the logarithmic mean, Mathematics Magazine. 48 (1975),pp. 87–92.