I am working on a proof to a problem, and stumbled upon related work by Guy Robin, who created this function as an upper bound of $\sigma_1(x)$--the sum of divisors:$$e^{\gamma}\ln \ln n+\frac{0.6483n}{\ln \ln n}$$
Using that,I made another upper bound:$$e^{\gamma}\ln(\gamma+\ln n)+\frac{n}{\ln\ln n}$$
I am comparing it to another function, again related: $$H(n)+e^{H(n)}\ln H(n)$$ where $H(n)$ is the harmonic series.
So, to compare their ratio of convergence (or divergence), I created this function:$$f(n)=\frac{\sigma_1(n)}{e^{\gamma}n\ln (\gamma+\ln n)+\frac{n}{\ln \ln n}}$$ where $\sigma_1(n)$ is the sum of divisors and $\gamma$ is Euler's constant. What I want to know is an upper bound (as tight as possible) of $f(n)$.
EDIT: I know that $y=1.0$ as an absolute upper bound, and it seems that $y=0.9$ is a better one, but I can't prove it, neither do I know any better upper bounds.
