### Riemann Returns

Alain Connes is probably best known for playing with connections between physics and the Riemann hypothesis. Actually, there is an easy physical way of looking at the zeta function, using the free Riemann gas. Imagine a system with eigenstates labelled by the prime numbers, and energy eigenvalues $E_p = E_0 \textrm{log} p$. Multiparticle states are given by ordinals $n$ which decompose into prime factors, $n = p_1 p_2 p_3 \cdots p_i$. Setting $s = \frac{E_0}{k T}$ one can write the partition function

$Z(T) = \sum_n e^{- \frac{E_n}{kT} } = \sum_n e^{- s \textrm{log} n} $

$= \sum_n \frac{1}{n^s} = \zeta (s)$

The Hagedorn temperature corresponds to the $\zeta$ pole at $s = 1$. What would happen if we considered knot zeta functions instead? States would be labelled not by integers, but by knots, which similarly factor into primes. Instead of prime factors $p$ there are polynomials $J(K)$, and logarithms of $J(K)$ are associated with energy levels.

As it happens, such logarithms (for the normalised colored Jones polynomial) occur in the well known volume conjecture, which states that the hyperbolic volume of a knot complement is given by

$V(K) = 2 \pi \textrm{lim}_{N \rightarrow \infty} \frac{1}{N} \textrm{log} |J(K)|$

where $J(K)$ is evaluated at the root of unity $q = e^{\frac{2 \pi i}{N}}$. These volumes can be associated to singular spaces appearing in 4d spin foam models, where there is in fact a partition function interpretation. Note that as $N \rightarrow \infty$ the so-called cosmological constant often associated with a quantum symmetry should disappear, as one expects (and observes) for classical spacetimes.

$Z(T) = \sum_n e^{- \frac{E_n}{kT} } = \sum_n e^{- s \textrm{log} n} $

$= \sum_n \frac{1}{n^s} = \zeta (s)$

The Hagedorn temperature corresponds to the $\zeta$ pole at $s = 1$. What would happen if we considered knot zeta functions instead? States would be labelled not by integers, but by knots, which similarly factor into primes. Instead of prime factors $p$ there are polynomials $J(K)$, and logarithms of $J(K)$ are associated with energy levels.

As it happens, such logarithms (for the normalised colored Jones polynomial) occur in the well known volume conjecture, which states that the hyperbolic volume of a knot complement is given by

$V(K) = 2 \pi \textrm{lim}_{N \rightarrow \infty} \frac{1}{N} \textrm{log} |J(K)|$

where $J(K)$ is evaluated at the root of unity $q = e^{\frac{2 \pi i}{N}}$. These volumes can be associated to singular spaces appearing in 4d spin foam models, where there is in fact a partition function interpretation. Note that as $N \rightarrow \infty$ the so-called cosmological constant often associated with a quantum symmetry should disappear, as one expects (and observes) for classical spacetimes.

## 2 Comments:

Dear Kea,

a nice coincidence.

I happen to be working with a TGD inspired idea about what conditions most general zeta function might satisfy. The idea is that the branches for the inverse of zeta define in gneral complex eigenvalues of the modified Dirac operator and that the zeta function assignable to these eigenvalues or subset of them co-incides with the original zeta at certain points of complex plane. At least at origin which corresponds physically complete quantum criticality. I call this condition self-referentiality.

The minimal outcome is following. One might define quite a general zeta function from the formula

zeta(s)= sum_k (zeta^{-1}_k(0)/2)^{-s}

where zeta^(-1)_k(0) correspond to real zeros of zeta in question.

In the case of Riemann Zeta the substitution of real zeros -2m gives sum_k m^(-s) which is indeed the definition of Riemann zeta.

In the more general case the expression would give strong self consistency constraints on real zeros. It would be interesting to look whether one could construct zeta functions on real axis by assuming finite number of zeros and numerically iterating the definition in the hope that the real zeros converge to fixed values.

Matti

Hmm, interesting. Something to muse over!

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