Robin's inequality
$$\sigma\_1(n)<e^\gamma n\log\log n$$
at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma\_1(n)=\sum\_{d|n}d$ is sum of divisors function applied to integer $n$.


Is there a necessary and sufficient condition involving sum of squares of divisors function $\sigma\_2(n)=\sum\_{d|n}d^2$?



Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?
I would expect no: the big difference between $\sigma\_1$ and $\sigma\_k$ for
$k\ge2$ is that in the latter case $\sigma\_k(n)/n^k$ is trivially bounded by
$\zeta(k)$, which doesn't leave much room for a Robin type criterion.


