This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the justification of the term quantum.
Let first recall that for $n \in \mathbb{N}$, $\sigma$ the divisor function, and $\gamma$ the Euler–Mascheroni constant, then $$\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma}.$$ Next Robin's theorem states that the Riemann hypothesis is true if and only if for $n$ large enough $$\sigma(n) < e^\gamma n \log \log n.$$
Let $\mathfrak{F}$ be the set of irreducible finite index subfactor planar algebras (up to equivalence), and let $\mathfrak{F}_r$ be those of depth $r \ge 2$.
Some reminders:
- Watatani's theorem states that for $\mathcal{P} \in \mathfrak{F}$, its biprojection lattice $[e_1, id]$ is finite.
- The index of a biprojection $b$ is noted $|b : e_1|$, and $|id:e_1|$ is the index of $\mathcal{P}$, also noted $|\mathcal{P}|$.
- The index is multiplicative, i.e. for $b,b'$ biprojections then $$b \le b' \Rightarrow |b':e_1| = |b':b| \cdot |b:e_1|.$$
- The index $|b:e_1|$ is a cyclotomic integer, and restricted by Jones index theorem.
Consider the divisor set $$D(\mathcal{P}) = \{ |b : e_1| \ \text{ with } \ b \in [e_1,id] \},$$ which then is a finite subset of the (generically infinite) number theoretic divisor set $D(|\mathcal{P}|)$ in the ring of cyclotomic integers (of appropriate degree), see this post.
Consider the divisor function
$$\sigma(\mathcal{P}) = \sum_{d \in D(\mathcal{P})} d.$$
Question 1: Is there a (finite) constant $\gamma_r$ such that the following equality holds?
$$\limsup_{\mathcal{P} \in \mathfrak{F}_r, \\ |\mathcal{P}| \to \infty} \frac{\sigma(\mathcal{P})}{|\mathcal{P}| \log \log |\mathcal{P}|} = e^{\gamma_r}$$
Note that $\gamma_2 = \gamma$ because the index of a biprojection of $\mathcal{P} \in \mathfrak{F}_2$ is a rational integer, and for $G$ a finite group (in particular $C_n$), the subfactor planar algebras $\mathcal{P}(G)$ is in $\mathfrak{F}_2$, and there is a 1-1 (Galois) correspondance between $H\le G$ and the biprojection of $\mathcal{P}(G)$, whose indices are precisely given by $|H|$.
FYI, $\mathfrak{F}_2$ contains exactly the finite dimensional Hopf C*-algebra subfactor planar algebras.
Assume that Question 1 admits a positive answer (otherwise the asymptotic should be improved), we can then formulate a (revised) depth $r$ quantum Riemann hypothesis (QRH$_r$) as follows:
(QRH$_r$) For $\mathcal{P} \in \mathfrak{F}_r$ and $|\mathcal{P}|$ large enough then $\sigma(\mathcal{P}) < e^{\gamma_r} |\mathcal{P}| \log \log |\mathcal{P}|$.
Note that QRH$_2$ is (obviously) equivalent to RH.
Question 2: What is about QRH$_r$ when $r>2$?
We can provide a variation, replacing $\mathfrak{F}_r$ by the set of $\mathcal{P} \in \mathfrak{F}$ for which the indices of the biprojections are in $\mathbb{Z}[\zeta_r]$ with $\zeta_r = e^{2\pi i/r}$ and $r$ minimal. Let us call this variation QRH$^r$. If relevant, we can combine both variation as QRH$_{r'}^{r}$.
Note that it would be inappropriate to replace $\sigma(\mathcal{P})$ by $$\tilde{\sigma}(\mathcal{P})=\sum_{b \in [e_1,id]} |b :e_1|,$$ because then above asymptotic would be incorrect, but mostly because there would be no more relation with RH, because if $\mathcal{P}_G$ is the group subfactor planar algebra for the finite group $G$ then $|\mathcal{P}_G| = |G|$ and $\tilde{\sigma}(\mathcal{P}_G)$ is the sum of the order of the subgroups of $G$ (let us call it $\tilde{\sigma}(G)$), which explodes for the abelian $p$-groups as $G = C_2^n$:
$$ \begin{array}{c|c} n&1&2&3&4&5&6&7 \newline \hline |C_2^n|&2&4&8&16&32&64&128 \newline \hline \tilde{\sigma}(C_2^n)&3&11&51&307&2451&26387&387987 \end{array} $$
To learn more, you can watch the series Quantum Symmetries and Quantum Arithmetic.
The last video finishes on QRH.
