Guy Robin proved
$$
\frac{\sigma(n)}{n} < \exp(\gamma)ln(ln(n))+ \frac{0.6482}{ln(ln(n))},
$$
for all integers $n>2,$


where $\gamma$ is Euler's constant, and $ln$ is the logarithm to the base $exp(1).$


Question: There are analogue formulas for the sum of the $k$-th powers of the positive
divisors of $n.$



Analogue of Guy Robin's formula for upper bound of $\sigma(n)/n$
Oh, yes, I gave a fairly complete answer at:


[A hierarchy of k-highly composite numbers](https://mathoverflow.net/questions/41802/a-hierarchy-of-k-highly-composite-numbers/41818#41818)


There is a procedure iniated by Ramanujan that gives a sequence of particularly large values of $$\frac{\sigma\_k(n)}{n^k},$$ essentially by taking any $\delta > 0$ and finding the $n = n\_\delta$ giving the maximum of $$\frac{\sigma\_k(n)}{n^{k + \delta}},$$ and choosing the largest if more than one $n$ gives the maximum. The recipe gives a recipe for the prime factorization of $n = n\_\delta.$ Once one has a value, it follows that the original $\frac{\sigma\_k(n)}{n^k}$ is larger than for any $m < n.$


There is a fair amount of work involved in interpolating these values into a bound of the type you quote. So, as far as I know, it has only been carried out in entirety for $k=0,1,$ where the former refers to the raw count of divisors. 


EDIT: I found the complete answer for $k=0,$


[What is the lower bound for highly composite numbers?](https://mathoverflow.net/questions/43103/what-is-the-lower-bound-for-highly-composite-numbers/43105#43105)


