[Paper Review] Some information-theoretic computations related to the distribution of prime numbers

Ioannis Kontoyiannis|ArXiv.org|Oct 22, 2007

Computability, Logic, AI Algorithms15 references3 citations

TL;DR

This paper presents an information-theoretic proof of the asymptotic result $\sum_{p\leq n}\frac{\log p}{p} \sim \log n$ as $n \to \infty$, using entropy arguments and the approximate independence of prime exponents in the factorization of random integers. It establishes rigorous finite-$n$ bounds by combining entropy-based lower and upper bounds, offering a novel, elementary derivation of a classical number-theoretic result related to the prime number theorem.

ABSTRACT

We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result: The sum of (log p)/p, taken over all primes p not exceeding n, is asymptotic to log n as n tends to infinity. We also give finite-n bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.

Motivation & Objective

• To provide a new, elementary proof of the asymptotic relation $\sum_{p\leq n}\frac{\log p}{p} \sim \log n$ using information-theoretic principles.
• To establish finite-$n$ bounds refining this asymptotic result, improving upon classical estimates.
• To demonstrate the utility of entropy and probabilistic modeling in number theory, particularly in analyzing the distribution of prime numbers.
• To bridge information theory and number theory by showing how entropy of integer factorizations yields insights into prime sum behavior.

Proposed method

• Model a uniformly random integer $N \in \{1, \dots, n\}$ via its prime factorization $N = \prod_{p \leq n} p^{X_p}$, inducing a joint distribution on the exponents $X_p$.
• Use the fact that the entropy of $N$ equals the entropy of the exponents $\{X_p\}$, so $H(N) = \log n = H(X_p : p \leq n)$.
• Approximate the distribution of $X_p$ as geometric and the joint distribution as approximately independent, leading to the entropy bound $\log n \approx \sum_{p \leq n} H(X_p)$.
• Derive a lower bound on $C(n) = \sum_{p \leq n} \frac{\log p}{p}$ using the entropy approximation and summation-by-parts identities.
• Establish an upper bound via a refined entropy argument using a squarefree decomposition $N = M^2 \prod p^{Y_p}$, where $Y_p \in \{0,1\}$, to bound $H(M)$ and $H(Y_p)$.
• Combine the lower and upper bounds to rigorously prove $C(n) \sim \log n$ and derive finite-$n$ inequalities.

Experimental results

Research questions

• RQ1 Can information-theoretic entropy arguments be used to prove classical results in number theory, such as the asymptotic behavior of $\sum_{p \leq n} \frac{\log p}{p}$?
• RQ2 How can the approximate independence of prime exponents in random integer factorizations be leveraged to derive bounds on prime sum functions?
• RQ3 What finite-$n$ bounds can be established for $\sum_{p \leq n} \frac{\log p}{p}$ using entropy-based reasoning?
• RQ4 Can entropy-based methods yield a new, elementary proof of Chebyshev's result without relying on complex analysis?

Key findings

• The paper proves $\sum_{p \leq n} \frac{\log p}{p} \sim \log n$ as $n \to \infty$ using only elementary information-theoretic arguments and entropy identities.
• A finite-$n$ lower bound is established: $C(n) \geq \left(1 - \frac{1}{16}\right)\left(1 - \frac{1}{1 + \log 16}\right)\log n + C(16) - T(16)$, valid for $n \geq 16$, with $C(16) \approx 3.05$.
• The asymptotic equivalence is confirmed via a liminf bound: $\liminf_{n \to \infty} \frac{C(n)}{\log n} \geq \frac{(N_0 - 1)\log N_0}{N_0(1 + \log N_0)}$ for any $N_0 \geq 2$, and taking $N_0 \to \infty$ yields the full asymptotic.
• The proof relies on the entropy of the random variable $N$ being equal to $\log n$, and the entropy of the exponents $X_p$ being approximated by the entropy of geometric distributions.
• The method avoids complex analysis and provides a self-contained, information-theoretic derivation of a result historically proven by Chebyshev.
• The approach also yields a new bound on the prime counting function: $\pi(n) \geq \frac{\log n}{2\log 2}$ for $n \geq 2$, derived from a squarefree decomposition of $N$.