All the information in the integers is in the primes

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.