All the information in the integers is in the primes

TL;DR

This paper interprets Chebyshev's asymptotic result through information theory, linking the entropy of integers to prime distributions and extending it to algorithmic information theory, revealing deep connections between number theory and information content.

Contribution

It provides a novel information-theoretic interpretation of Chebyshev's asymptotic involving primes and extends the result to algorithmic information theory using recent insights about prime distributions and randomness.

Findings

Asymptotic equivalence of integer information content and prime-based sum

Prime numbers have a uniform source distribution

Upper bounds on the algorithmic information of typical integers

Abstract

Building upon the work of Chebyshev, Shannon and Kontoyiannis, it may be demonstrated that Chebyshev's asymptotic result: \begin{equation} \ln N \sim \sum_{p \leq N} \frac{1}{p} \cdot \ln p \end{equation} has a natural information-theoretic interpretation as the information-content of a typical integer $Z \sim U([1,N])$, sampled under the maximum entropy distribution, is asymptotically equivalent to the information content of all primes $p \leq N$ weighted by their typical frequency. This result is established by showing that the correct information-theoretic interpretation of the entropy $\ln p$ may be deduced from the author's recent observation that the prime numbers have a uniform source distribution. Moreover, using Chaitin's theorem that almost all integers are algorithmically random we may extend this asymptotic result to the setting of algorithmic information theory and derive an upper-bound on the algorithmic information of a typical integer.