Quantum Probability and the Prime Number Theorem

We may demonstrate that the Prime Number Theorem emerges as a natural consequence of a thought experiment in Quantum Mechanics.

Aidan Rocke https://github.com/AidanRocke
03-09-2022

In the following analysis, we shall demonstrate that the Prime Number Theorem emerges as natural consequence of a thought experiment in Quantum Mechanics.

Let’s consider the outcome of $$N \in \mathbb{N}$$ Quantum Measurements where each experiment may be viewed as a black box where a photon either appears or it does not.

In order to define the black boxes we must define a choice function on $$\{A_k\}_{k=1}^N$$ where $$A_k = [1,k]$$. Hence, given the choice $$x_k \in A_k$$ each box $$A_k$$ contributes one bit of information upon evaluation of the correctness of that choice.

It follows that if we condition on any choice function $$f$$, the expected information gained from observing a single photon is on the order of:

$$$\sum_{k=1}^N 1 \cdot \frac{1}{k} = \sum_{k=1}^N \frac{1}{k} \sim \ln N \tag{1}$$$

as a photon may appear in any of the $$N$$ boxes independently of all the other photons. In fact, it is easy to see that any choice of prior $$f$$ is informationless as the photons have a maximum entropy distribution.

Now, given our choice function $$f$$ we may define the sequence of observations $$X_N$$ in terms of the observation function $$g$$:

$$$\exists x_k \in A_k, x_k = f \circ A_k \tag{2}$$$

$$$X_N = \{g \circ x_k\}_{k=1}^N \tag{3}$$$

where $$g \circ x_k = 1$$ if a photon is observed at the slit $$x_k \in A_k$$ and $$g \circ x_k = 0$$ otherwise.

By definition, the detectors cover a fraction:

$$$\frac{N}{{N \choose 2}} \sim \frac{2}{N} \tag{4}$$$

of the total number of slits.

In this context, if we define $$X_N = \{g \circ x_k\}_{k=1}^N \in \{0,1\}^N$$ each observation contributes one bit of information which implies:

$$$\mathbb{E}[K_U(X_N)] \sim \log_2(2^N) \sim N \tag{5}$$$

so the average information gained from identifying $$\pi(N)$$ photons using the prior $$f$$ is given by:

$$$\frac{\log_2(2^N)}{\pi(N)} = \frac{N}{\pi(N)} \sim \ln N \tag{6}$$$

Thus, the Shannon Source Coding theorem implies that the Expected Kolmogorov Complexity must satisfy:

$$$\mathbb{E}[K_U(X_N)] \sim \pi(N) \cdot \ln N \sim N \tag{7}$$$

as our detectors only detect photons and not the absence of photons, and the detection of $$\pi(N)$$ photons occurs simultaneously.

Curiously, this yields the Prime Number Theorem:

$$$\pi(N) \sim \frac{N}{\ln N} \tag{8}$$$

indicating that the prime numbers were drawn from a maximum entropy distribution.

Moreover, as the locations $$\alpha, \beta \in \mathbb{N}$$ where $$g \circ \alpha = g \circ \beta = 1$$ are algorithmically random, we may deduce that:

$$$K_U(\alpha) \sim \log_2 \alpha \tag{9}$$$

$$$K_U(\beta) \sim \log_2 \beta \tag{10}$$$

which implies that:

$$$\forall X \sim U([1,N]), m(X \bmod \alpha = 0 \land X \bmod \beta = 0) = m(\alpha) \cdot m(\beta) \sim \frac{1}{\alpha} \cdot \frac{1}{\beta} \tag{11}$$$

Hence $$\gcd(\alpha,\beta)=1$$ which is necessary and sufficient to define the set of primes $$\mathbb{P} \subset \mathbb{N}$$.

References:

1. Rocke (2022, Jan. 15). Kepler Lounge: Three master keys for Probabilistic Number Theory. Retrieved from keplerlounge.com

2. Rocke (2022, March 4). Kepler Lounge: Elements of Quantum Probability theory. Retrieved from keplerlounge.com

3. Rocke (2022, March 4). Kepler Lounge: Super-determinism via Solomonoff Induction. Retrieved from keplerlounge.com

4. Rocke (2022, Jan. 3). Kepler Lounge: The Law of Conservation of Information. Retrieved from keplerlounge.com

5. Rocke (2022, Jan. 5). Kepler Lounge: Revisiting the unreasonable effectiveness of mathematics. Retrieved from keplerlounge.com

6. Rocke (2022, March 8). Kepler Lounge: The Von Neumann Entropy and the Riemann Hypothesis. Retrieved from keplerlounge.com

Citation

For attribution, please cite this work as

Rocke (2022, March 9). Kepler Lounge: Quantum Probability and the Prime Number Theorem. Retrieved from keplerlounge.com

BibTeX citation

@misc{rocke2022quantum,
author = {Rocke, Aidan},
title = {Kepler Lounge: Quantum Probability and the Prime Number Theorem},
url = {keplerlounge.com},
year = {2022}
}