The Curious Reluctance to Define Prime Probability Statistically
Résumé
All the known approximations of $\pi(n)$ for finite values of $n$ are derived from real-valued functions that are asymptotic to $\pi(x)$, such as $\frac{x}{log_{e}x}$, $Li(x)$ and Riemann's function $R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{(n)}li(x^{1/n})$. The degree of approximation for finite values of $n$ is determined only heuristically, by conjecturing upon an error term in the asymptotic relation that can be seen to yield a closer approximation than others to the actual values of $\pi(n)$ for computable values of $n$. None of these can, however, claim to estimate $\pi(n)$ uniquely for all values of $n$. We show that statistically the probability of $n$ being a prime is $\prod_{i = 1}^{\pi(\sqrt{j})}(1 - \frac{1}{p_{_{i}}})$, and that statistically the expected value of the number $\pi(n)$ of primes less than or equal to $n$ is given uniquely by $\sum_{j = 1}^{n}\prod_{i = 1}^{\pi(\sqrt{j})}(1 - \frac{1}{p_{_{i}}})$ for all values of $n$. We then demonstrate how this yields elementary probability-based proofs of the Prime Number Theorem, Dirichlect's Theorem, and the Twin-Prime Conjecture.
Mots clés
twin primes.
probabilistic number theory
probability model
Prime Number Theorem
primes in an arithmetical progression
prime density
Law of Large Numbers
integer factorising algorithm
Hardy-Littlewood conjecture
factorising is polynomial time
expected value
Euler's constant $\gamma$
prime counting function $\pi(n)$
polynomial time algorithm
Mertens' theorem
mutually independent prime divisors
prime counting function
prime probability function
Brocard's conjecture
Chebyshev's Theorem
complete system of incongruent residues
computational complexity
Dirichlect primes
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)
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