We may demonstrate that the Prime Number Theorem emerges as a natural consequence of a thought experiment in Quantum Mechanics.
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:
\[\begin{equation} \sum_{k=1}^N 1 \cdot \frac{1}{k} = \sum_{k=1}^N \frac{1}{k} \sim \ln N \tag{1} \end{equation}\]
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 \(g\):
\[\begin{equation} \exists x_k \in A_k, x_k = f \circ A_k \tag{2} \end{equation}\]
\[\begin{equation} X_N = \{g \circ x_k\}_{k=1}^N \tag{3} \end{equation}\]
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:
\[\begin{equation} \frac{N}{{N \choose 2}} \sim \frac{2}{N} \tag{4} \end{equation}\]
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:
\[\begin{equation} \mathbb{E}[K_U(X_N)] \sim \log_2(2^N) \sim N \tag{5} \end{equation}\]
so the average information gained from identifying \(\pi(N)\) photons using the prior \(f\) is given by:
\[\begin{equation} \frac{\log_2(2^N)}{\pi(N)} = \frac{N}{\pi(N)} \sim \ln N \tag{6} \end{equation}\]
Thus, the Shannon Source Coding theorem implies that the Expected Kolmogorov Complexity must satisfy:
\[\begin{equation} \mathbb{E}[K_U(X_N)] \sim \pi(N) \cdot \ln N \sim N \tag{7} \end{equation}\]
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:
\[\begin{equation} \pi(N) \sim \frac{N}{\ln N} \tag{8} \end{equation}\]
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:
\[\begin{equation} K_U(\alpha) \sim \log_2 \alpha \tag{9} \end{equation}\]
\[\begin{equation} K_U(\beta) \sim \log_2 \beta \tag{10} \end{equation}\]
which implies that:
\[\begin{equation} \forall X \sim U([1,N]), m(X \bmod \alpha = 0 \land X \bmod \beta = 0) = m(\alpha) \cdot m(\beta) \sim \frac{1}{\alpha} \cdot \frac{1}{\beta} \tag{11} \end{equation}\]
Hence \(\gcd(\alpha,\beta)=1\) which is necessary and sufficient to define the set of primes \(\mathbb{P} \subset \mathbb{N}\).
Rocke (2022, Jan. 15). Kepler Lounge: Three master keys for Probabilistic Number Theory. Retrieved from keplerlounge.com
Rocke (2022, March 4). Kepler Lounge: Elements of Quantum Probability theory. Retrieved from keplerlounge.com
Rocke (2022, March 4). Kepler Lounge: Super-determinism via Solomonoff Induction. Retrieved from keplerlounge.com
Rocke (2022, Jan. 3). Kepler Lounge: The Law of Conservation of Information. Retrieved from keplerlounge.com
Rocke (2022, Jan. 5). Kepler Lounge: Revisiting the unreasonable effectiveness of mathematics. Retrieved from keplerlounge.com
Rocke (2022, March 8). Kepler Lounge: The Von Neumann Entropy and the Riemann Hypothesis. Retrieved from keplerlounge.com
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}
}