The divisor sigma function is defined as $$ \sigma_{\nu}(n):=\sum_{d|n}d^{\nu}\textrm{, }n\in\textbf{N} $$ It is known that under Riemann's hypothesis it holds the following inequality due to Ramanujan and Robin see here $$ \sigma_1(n)<e^{\gamma}n\log\log n\textrm{, }n>5040\tag 1 $$ ($\gamma$ is Euler-constant, $0.5772156649...$)The opposite is also true: (1) implies Riemann's hypothesis.
My question is: Are there similar inequalities in the literature for $\sigma_{\nu}(n)$, for other values of $\nu=2,3,\ldots$?
With the word similar I mean inequalities of the form $$ \sigma_{\nu}(n)<C_{\nu}n^{\nu}\log\log n\textrm{, }\forall n>n_0, $$ where $C_{\nu}$ some known constants (I need the values of these constants also).
My starting point is equation $$ 9d^2(k^2+4d)=n^2\textrm{, }(n,k,d)\in\textbf{Z}^3.\tag 2 $$ When $n$ is around $\sigma_1(n)\approx e^{\gamma}n\log\log n$ i.e. when $n$ is such that $0,85<\frac{\sigma(n)}{e^{\gamma}n\log\log n}<1$, then we have a ''jump'' in the number of solutions $L=L(n)$ of (2) from (say) $L$ to $L+6$. For example $n=10080$, $L=18$; $n=19440$, $L=24$; $n=55440$, $L=30$; $n=443520$, $L=36$; $\ldots$.
Equation (2) is equivalent (have the same number of solutions) to $$ x^3+y^3+z^3=n\textrm{, }x+y+z=0 $$ For the case $$ x^3+y^3+z^3=n\textrm{, }x+y+z=t, $$ we have the general equation $$ \left(\frac{n-t^3}{3d}+2t\right)^2-4d=k^2, $$ which I suspect to have solutions $n$ around $\sigma_{\nu}(n)\approx C_{\nu}n^{\nu}\log\log n$, $\nu=\nu(t)$.