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The Prime Number Theorem is a theorem that describes the distribution of the primes. It says that the number of primes less than or equal to a real number $x$ is asymptotic to $\frac{x}{\ln x}$.

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information-theoretic derivation of the prime number theorem

You may be interested in this arxiv paper [1], "Some information-theoretic computations related to the distribution of prime numbers", Ioannis Kontoyiannis, 2007. It discusses Chebyshev's 1852 result, …
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