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:

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.

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