The Emergent Complexity of the Riemann Zeta function

A proof that the Riemann Zeta function is the statistical signature of Open-Ended Evolution.

Introduction:

In the following analysis, we postulate three self-evident axioms for Open-Ended Evolution. From these axioms we demonstrate that the Riemann Zeta function is the statistical signature of Open-Ended Evolution.

The main line of reasoning is taken from Zipf’s law, unbounded complexity and open-ended evolution[1] though it is worth noting that some arguments which were found to be erroneous were fixed by the author.

Constraints on Open-Ended Evolution:

We shall consider three reasonable constraints on a necessary and sufficient model for Open-Ended Evolution that concern computability, algorithmic randomness and path-dependence(aka heredity):

1. Information results from the growth of genome complexity through a combination of gene duplication and interactions with the external world. This process of information growth must therefore be a path-dependent process.

2. Algorithmic Probability allows us to distinguish predictable from unpredictable sequences in a meaningful way.

3. The algorithmic definition based on the use of a program matches our intuition that evolution may be captured by a computational description.

We shall generally focus on dynamical systems whose description may be made in terms of finite binary strings $\sigma_t$ at each time step $t$ over evolutionary time. If $\sigma_t$ is the description of the system at time $t$, let the sequence:

$$\begin{equation} \Sigma(t) = \{\sigma_1,\sigma_2,...,\sigma_t \} \end{equation}$$

be the history of the system until time $t$ in arbitrary time units.

Axioms for Open-Ended Evolution:

Given the above constraints on Open-Ended Evolution we may formulate the following necessary and sufficient axioms in terms of Kolmogorov Complexity relative to a Universal Turing Machine $U$:

Axiom 1. Open-Ended

We say that the process that generates $\sigma_t$ is open-ended if:

$$\begin{equation} \frac{K_U(\Sigma(t))}{t} \leq \frac{K_U(\Sigma(t+1))}{t+1} \tag{1} \end{equation}$$

for all $t \in \mathbb{N}$. Of all open-ended processes that obey (1) we are interested in those whose complexity is unbounded.

Axiom 2. Unbounded

We say that the process generating $\sigma_t$ has an unbounded complexity if for any $N \in \mathbb{N}$ there is a time $t$ such that:

$$\begin{equation} \frac{K_U(\Sigma(t))}{t} > N \tag{2} \end{equation}$$

These two axioms imply that information is always being added by the generative process in the long-term. The knowledge of the history up to time $t$ is not enough to predict what will happen next.

Axiom 3. Heredity

Evolutionary processes attempt to minimize the action:

$$\begin{equation} S(\Sigma(t) \rightarrow \Sigma(t+1)) \equiv K_U(\Sigma(t) | \Sigma(t+1)) \tag{3} \end{equation}$$

This axiom defines an Algorithmic Least-Action principle that imposes that the information carried between successive steps is maximized as much as other constraints allow, turning the generative process into a path-dependent one. Moreover, in consideration of the previous axioms we may deduce the following fundamental inequality:

$$\begin{equation} \frac{K(\Sigma(t))}{t} \leq K(\sigma_{t+1}|\sigma_{t}) \leq K(\sigma_{t+1}) \tag{*} \end{equation}$$

Now, in addition to these axioms we need Kolmogorov’s Lemma which may be derived from these axioms as well as the hypothesis that a Universal Wave Function simulates the Observable Universe.

Kolmogorov’s Lemma: Expected Kolmogorov Complexity equals Shannon Entropy

Before demonstrating that the Riemann Zeta function is a fundamental signature of Open-Ended Evolution, we will need the key lemma:

$$\begin{equation} \mathbb{E}[K_U(X)] = H(X) + \mathcal{O}(1) \tag{4} \end{equation}$$

which is demonstrated in the article: Lesser known miracles of Levin’s Universal Distribution.

Given that the information-theoretic properties of the Shannon Entropy are invariant to the choice of base of the logarithm we may observe that:

$$\begin{equation} \forall \lambda \in \mathbb{R^*}_{+}, \mathbb{E}[K_U(X)] \propto \lambda \cdot H(X) \tag{5} \end{equation}$$

which motivates our analysis of the typical information of an observable: $\langle \ln n \rangle$.

Power laws in infinite state space:

Given a countable set of observable Combinatorial Objects(ex. words, proteins, Lego) generated by a Universal Grammar we may define their un-normalised frequency counts using the integers $\mathbb{N}$.

The maximum entropy approach to characterizing the appropriate frequency distribution estimates the probabilities $p_n$ by maximizing the Shannon Entropy:

$$\begin{equation} S = -\sum_n p_n \ln p_n \tag{6} \end{equation}$$

subject to a number of constraints that represent epistemic limits on the underlying generative process.

If we define two reasonable constraints, a Unitarity constraint on the Universal Wave Function and the Asymptotic Equipartition Property($\mathcal{\chi}$):

$$\begin{equation} \sum_n p_n = 1 \tag{7} \end{equation}$$

$$\begin{equation} \langle \ln n \rangle = \sum_{n=1}^\infty p_n \ln n = \mathcal{\chi} \tag{8} \end{equation}$$

we may now maximize the Shannon Entropy subject to these constraints using the method of Lagrange Multipliers, so we find:

$$\begin{equation} \hat{S} = -z\big(\sum_{n=1}^\infty p_n \ln n - \mathcal{\chi}\big) - \lambda \big(\sum_{n=1}^\infty p_n - 1\big) - \sum_{n=1}^\infty p_n \ln p_n \tag{9} \end{equation}$$

and if we apply the change of variables $\lambda: Z \mapsto \ln Z - 1$:

$$\begin{equation} \hat{S} = -z\big(\sum_{n=1}^\infty p_n \ln n - \mathcal{\chi}\big) - (\ln Z - 1) \big(\sum_{n=1}^\infty p_n - 1\big) - \sum_{n=1}^\infty p_n \ln p_n \tag{10} \end{equation}$$

…varying with respect to $p_n$ yields the extremality condition:

$$\begin{equation} -z \ln n - \ln Z - \ln p_n = 0 \tag{11} \end{equation}$$

with explicit solution:

$$\begin{equation} \forall z > 1, p_n = \frac{n^{-z}}{\zeta(z)} \tag{12} \end{equation}$$

where $Z = \zeta(z)$ is the renormalisation factor.

Further analysis:

The typical information of the observable $\mathcal{\chi}$ is therefore given by:

$$\begin{equation} \mathcal{\chi}(z) = \langle \ln n \rangle = \frac{\sum_{n=1}^\infty n^{-z} \ln n}{\zeta(z)} = \frac{-d \zeta(z)/dz}{\zeta(z)} = \frac{-d \ln \zeta(z)}{dz} \tag{13} \end{equation}$$

At maximum entropy, by imposing the extremality condition we find:

$$\begin{equation} \hat{S}(z) = S(z) = -\sum_{n=1}^\infty p_n \ln p_n = \ln \zeta(z) + z \mathcal{\chi}(z) \tag{14} \end{equation}$$

Finally, we may deduce the typical frequency using the typical information:

$$\begin{equation} \exp \langle \ln n \rangle = \prod_{n=1}^\infty n^{p_n} \tag{15} \end{equation}$$

which is the geometric mean of the integers with exponents weighted by the probabilities $p_n$. To characterize $\zeta(z)$ in terms of its unique singularity at $z=1$, we may observe that:

$$\begin{equation} \forall z \in \mathbb{C} \setminus \{1\}, \zeta(z) = \frac{1}{z-1} + \sum_{n=1}^\infty \frac{(-1)^n}{n!}(z-1)^n \tag{16} \end{equation}$$

where the Stieltjes constants satisfy:

$$\begin{equation} \gamma_n = \frac{(-1)^n n!}{2 \pi} \int_{0}^{2 \pi} e^{-nix} \zeta(e^{ix}+1) dx \tag{17} \end{equation}$$

Implications for non-equilibrium turbulence:

When I meet God, I’m going to ask him two questions: why relativity? And why turbulence? I really believe he’ll have an answer for the first.-Heisenberg

Does turbulence need God? That is a difficult question. What I can say is that non-equilibrium turbulence requires the Riemann Zeta function.

References:

1. Corominas-Murtra Bernat, Seoane Luís F. and Solé Ricard. Zipf’s Law, unbounded complexity and open-ended evolution. Journal of the Royal Society. 2018.

2. Matt Visser. Zipf’s law, power laws, and maximum entropy. Arxiv. 2012.

3. Rocke (2023, April 19). Kepler Lounge: Lesser known miracles of Levin’s Universal Distribution. Retrieved from keplerlounge.com