# The Emergent Complexity of the Riemann Zeta function

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

Aidan Rocke https://github.com/AidanRocke
06-02-2023

## 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:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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($$\mathcal{\chi}$$):

$$$\sum_n p_n = 1 \tag{7}$$$

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

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

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

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

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

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

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

with explicit solution:

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

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

## Further analysis:

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

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

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

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

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

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

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:

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

where the Stieltjes constants satisfy:

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

## 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

### Citation

Rocke (2023, June 2). Kepler Lounge: The Emergent Complexity of the Riemann Zeta function. Retrieved from keplerlounge.com
@misc{rocke2023the,
}