On a quantum Riemann Hypothesis 
Here is a revised version: On a revised quantum Riemann
hypothesis.


Robin's theorem (1984) states that
$$ \sigma(n) < e^\gamma n \log \log n$$
for all $n > 5040$ if and only if the Riemann hypothesis is true.
Recall that $γ$ is the Euler–Mascheroni constant and $σ(n)$ is the divisor function, given by
$$\sigma(n) = \sum_{d\mid n} d.$$
To formulate a quantum Riemann hypothesis, we will use Robin's theorem and the following facts:

*

*a natural number $n$ can be encoded into the cyclic group $C_n$

*a finite group $G$ can be encoded into the finite index irreducible depth $2$ subfactor $R \subseteq R \rtimes G$

*a finite index irreducible subfactor $N \subseteq M$ can be encoded into a planar algebra $\mathcal{P}$.

For a justification of the qualifier "quantum", see the following article:
Jones, Vaughan. On the origin and development of subfactors and quantum topology. Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 309--326.
Galois correspondences:

*

*the divisors $d\mid n$ are $1$-$1$ with the subgroups $H \subseteq C_n$,

*the subgroups $H \subseteq G$ are $1$-$1$ with the intermediate subfactors $R \subseteq K \subseteq R \rtimes G$,

*the intermediate subfactors $N \subseteq K \subseteq M$ are $1$-$1$ with the biprojections $b \in [e_1,id]$.

The notations match as follows:

*

*$n = |G| = [M:N] = |id : e_1|$,

*$d = |H| = [K:N] = |b : e_1|$.

The equality $|G| = |G:H| \cdot |H|$ extends to $|id : e_1| = |id:b| \cdot |b:e_1|$.
In general, $|id : e_1|$ is not necessarily an integer, but (by Jones' theorem) can be any element in
$$\{4\cos^2(\pi /n)|n=3,4,5,...\}\cup [4,+\infty).$$
Let $\mathcal{P}$ be an irreducible subfactor planar algebra. We define the analog of the set of divisors by  $$D(\mathcal{P}) := \{|b : e_1|  \text{ with } b \in [e_1,id] \},$$
(which is finite by Watatani's theorem) and the analog of the divisor function by
$$\sigma(\mathcal{P}) := \sum_{\beta \in D(\mathcal{P})} \beta.$$

Quantum Riemann Hypothesis (of depth $n$)
There is $\alpha_n>0$ such that for every irreducible depth $n$ subfactor planar algebra $\mathcal{P}$ with $\alpha:=|id : e_1|> \alpha_n$, we have
$$\sigma(\mathcal{P}) < e^\gamma \alpha \log \log \alpha.$$

Of course, a proof of this quantum Riemann Hypothesis (QRH) is not expected as an answer of this post, because it implies the usual Riemann Hypothesis (RH).
For the group case, QRH follows from RH, because for $\sigma(G):=\sigma(\mathcal{P}(R \subseteq R \rtimes G))$, we have $\sigma(G) \le \sigma(|G|)$ by Lagrange's theorem.  Idem if $|id:b|$ and $|b:e_1|$ are integers $\forall b \in [e_1,id]$, like the irreducible depth $2$ case, because then $\sigma(\mathcal{P}) \le \sigma(|id : e_1|)$.
Let's denote QRH of depth $n$ by QRH$_n$. Then, QRH$_2 \Leftrightarrow$ RH, and we can take $\alpha_2 = 5040$.
Question: Does RH imply QRH$_n \ \forall n \ge 2$? Or, do you see a counterexample for some $n$?
Bonus question: Assuming QRH$_n$ true $ \forall n \ge 2$, can the sequence $(\alpha_n)$ be bounded?

To learn more, you can watch the series Quantum Symmetries and Quantum Arithmetic.
The last video finishes on (revised) QRH.
 A: Can you clarify whether there exists a notion of direct product in this setting with the desired properties? 
If so, the asymptotics you are predicting only seem consistent with the hypothesis that there are no such objects besides groups. (I guess they do actually exist or you wouldn't ask this question.) The first version of this post suggested taking the direct sum of many factors of size less than $4$, and the OP said that this was invalid because most of them didn't have "bounded depth." Fair enough. But if you take direct products of many subfactors of small indices which have the property that they are "coprime" then it seems you can easily exceed your desired bound unless the only new objects also have rational orders. Here's a very weak explicit version of this idea using a single new object.
It's know that  $\displaystyle{\sup \frac{\sigma(n)}{e^{\gamma} n \log \log n} = 1}$. Take an $\alpha$ for which the ratio is very close to $1$. How close will be clear below.
Let $X$ be the object coming from the cyclic group $C_{\alpha}$. Now let $Y$ be any other object whose index is some number $\eta$ which is not a rational number (assuming it exists), and let $Z = X \times Y$ (assuming it exists), and let $\beta = \alpha \eta$ be the index of $Z$. Then it would seem (by considering the subgroups of $C_{\alpha}$ and those subgroups direct sum with $Y$), and using that $\eta$ is not rational, that
$$\sigma(Z) \ge \sigma(X) + \eta \sigma(X) \sim (1 + \eta) e^{\gamma} \alpha \log \log(\alpha) \sim \left(1 + \frac{1}{\eta}\right) e^{\gamma} \beta \log \log(\beta),$$
and the QRH is false.
