Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?

Question

Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?

Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime factorization of $n$. Combining these bounds we get the well-known inequality, $\dfrac{\phi(n)\sigma(n)}{n^2}<1$, but from the definitions

$$J_k(n)=\prod_{i=1}^t\left(p_i^{ka_i}-p_i^{k(a-1)}\right)$$

and

$$\sigma_k(n)=\prod_{i=1}^t\dfrac{p_i^{k(a_i+1)}-1}{p_i^k-1}$$

it's clear that

$$\dfrac{J_k(n)\sigma_k(n)}{n^{2k}}=\prod_{i=1}^t\left(1-p_i^{-k(a_i+1)}\right)<1$$

Are any other good pairs of upper bounds for $J_k(n)$ and $\sigma_k(n)$ known for $k\ne1$ of a form similar to those for $k=1$?

Edit: I figure we should have something like $\dfrac{n^k}{J_k(n)}<\dfrac{\sigma_k(n)}{n^k}<B_k(n)$, where $B_k(n)$ is the upper bound in question. As in the case $k=1$, it may be known for the totient function, but not for the divisor function.

Answer 1

In Tenenbaum's Introduction à la théorie analytique et probabiliste des nombres, chapter I.5, the following is proved.

A maximal order for $\sigma_k(n)$ is

• $\exp((1+o(1))\log 2 \log n / \log \log n)$, if $k=0$.
• $$ n^k \exp \left ( (1+o(1)) \frac{(\log n)^{1-k}}{(1-k)\log \log n} \right ) $$ if $0<k<1$.
• $\text{e}^\gamma n \log \log n$, if $k=1$.
• $\zeta(k) n^k$, if $k>1$.
• For $k<0$, just use $\sigma_{-k}(n)=\sigma_k(n)/n^k$.

The proof is not too hard and relies upon Mertens's theorem as usual. It is only the case $0<k<1$ that needs additional care and the full use of the PNT.

For the totient function the analysis is essentially the same, so I recommend you to read Tenenbaum's textbook for further details.

It is sensible to conjecture an easy analogue of Robin's criterion for values of $k$ other than $1$, but one should check Robin's proof to see whether the methods can be adapted (the answer is presumably "yes").

On the other hand, error terms in analogues of Grönwall's bound would of course depend on the quality of the bounds in the PNT, and here one tipically uses results from the Rosser-Schoenfeld paper or subsequent works.