Kepler Lounge
The Spectral Geometry of the Prime Numbers
By exploring the correspondence between the Buckingham-Pi theorem and Unique Factorization Domains using Koopman Operators, we find that the Prime Numbers have an emergent Spectral Geometry.
Archimedes' Constant is absolutely normal
Using the theory of Algorithmic Probability, we demonstrate that Archimedes' Constant is absolutely normal.
Bell's theorem via counterfactuals
A proof of Bell's theorem via counterfactuals, which clarifies the nature of Quantum Randomness and its relation to Quantum Probability.
Random Walks and the Riemann Hypothesis
Starting from the observation that the statistical behaviour of the Mertens function is reminiscent of a one-dimensional random walk on non-zero values of the Möbius function, we demonstrate that the Riemann Hypothesis implies an entropy-bound on the Mertens function.
Gödel numbers and arithmetic functions
Gödel numbers may be used to define a programming language, where sets of propositions may be studied using arithmetic functions.
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.
The Von Neumann Entropy and the Riemann Hypothesis
Assuming that all of physics may be derived from Peano Arithmetic and that all the Quantum Information in the Universe is conserved, we may conjecture that the integers must have been specified as part of the initial state of the Universe and that they have a natural encoding using the Von Neumann Entropy.
Elements of Quantum Probability theory
An introduction to Lev Landau's approach to Quantum Probability theory.
Super-determinism via Solomonoff Induction
A super-deterministic theory of Quantum Mechanics is derived from Solomonoff's theory of Universal Induction, or what is generally known as Occam's razor.
The differential entropy of the Erdős-Kac distribution
Building upon the work of Billingsley, an entropy formula for the distribution of prime factors is carefully developed. This allows us to compare the entropy of the normal order of prime factors, which is constant, relative to extreme values where the entropy is unbounded.
Cramér's random model as a Poisson Process
In this article we demonstrate that Cramér's model is well-approximated by a Poisson Process, and that Cramér's conjecture may be deduced from an entropy bound on prime gaps.
Three master keys for Probabilistic Number Theory
Information-theoretic foundations for Probabilistic Number Theory.
An information-theoretic derivation of the Prime Number Theorem
An information-theoretic derivation of the Prime Number Theorem, using the Law of Conservation of Information.
An information-theoretic proof of the Erdős-Kac theorem
An information-theoretic adaptation of the Erdős-Kac theorem theorem, which informally states that the number of prime divisors of very large integers converges to a normal distribution.
Chebyshev's theorem via Occam's razor
An information-theoretic derivation of Chebyshev's theorem(1852), an important precursor of the Prime Number Theorem.
The Algorithmic Probability of a Prime Number
The Algorithmic Probability of a Prime Number, defined using Levin's Coding Theorem.
Revisiting the unreasonable effectiveness of mathematics
More than 60 years since Eugene Wigner's highly influential essay on the unreasonable effectiveness of mathematics in the natural sciences, it may be time for a re-appraisal.
Occam's razor
An information-theoretic formulation of Occam's razor as the Universal A Priori Probability.
The Law of Conservation of Information
Given that all physical laws are time-reversible and computable, information must be conserved as we run a simulation of the Universe forward in time.
Erdős' proof of Euclid's theorem
An information-theoretic adaptation of Erdős’ proof of Euclid’s theorem, which shows that the information content of finitely many primes is insufficient to generate all the integers.