The Curious Reluctance to Define Prime Probability Non-Heuristically

Bhupinder Singh Anand (1)

1 Independent Researcher

The reluctance to define the probability of a number being prime non-heuristically is curious, since we can define the residues $i > r_{i}(n) \geq 0$ for all $n \geq 2$ and all $i \geq 2$ such that $r_{i}(n) = 0$ if, and only if, $i$ is a divisor of $n$, and show: (i) that $\mathbb{M}_{i} = \{(0, 1, 2, \ldots, i-1),\ r_{i}(n), \frac{1}{i}\}$ is a probability model for $r_{i}(n)$; and (ii) that the joint non-heuristic probability $\mathbb{P}(r_{p_{_{i}}}(n) = 0\ \cap\ r_{p_{_{j}}}(n) = 0)$ of two primes $p_{i} \neq p_{j}$ dividing any integer $n$ is the product $\mathbb{P}(r_{p_{_{i}}}(n) = 0).\mathbb{P}(r_{p_{_{j}}}(n) = 0)$. We conclude that the non-heuristic probability of $n$ being a prime $p$ is given by the non-heuristic prime probability function $\mathbb{P}(n \in \{p\}) = \prod_{i = 1}^{\pi(\sqrt{n})}(1 - \frac{1}{p_{i}}) \sim \frac{2e^{-\gamma}}{log_{e}n}$. By the Law of Large Numbers, the number $\pi(n)$ of primes less than or equal to $n$ is therefore non-heuristically approximated by $\pi_{_{L}}(n) = \sum_{j = 1}^{n}\prod_{i = 1}^{\pi(\sqrt{j})}(1 - \frac{1}{p_{i}})$. We show that, in the interval $(p_{_{n}}^{2},\ p_{_{n+1}}^{2})$, the non-heuristic approximation $\pi_{_{L}}(x)$ of $\pi(x)$ is a straight line with gradient $\prod_{i = 1}^{n}(1 - \frac{1}{p_{i}})$; and that the function $\pi_{_{L}}(x)/\frac{x}{log_{e}x}$ is differentiable with derivative $(\pi_{_{L}}(x)/\frac{x}{log_{e}x})' \in o(1)$. We conclude by the Law of Large Numbers that $\pi(x) \sim \pi_{_{L}}(x)$ since $p_{_{n+1}}^{2} - p_{_{n}}^{2} \rightarrow \infty$; and that both $\pi_{_{L}}(x)/\frac{x}{log_{e}x}$ and $\pi(x)/\frac{x}{log_{e}x}$ do not oscillate as $x \rightarrow \infty$. Chebyshev's Theorem, $\pi(x) \asymp \frac{x}{log_{e}x}$, then yields an elementary probability-based proof of the Prime Number Theorem $\pi(x) \sim \frac{x}{log_{e}x}$. We also give an elementary probability-based proof that the number $\pi_{_{(a, d)}}(n)$ of Dirichlect primes of the form $a+m.d$ which are less than or equal to $n$, where $a, d$ are co-prime and $1 \leq a < d = q_{_{1}}^{\alpha_{_{1}}}.q_{_{2}}^{\alpha_{_{2}}} \ldots q_{_{k}}^{\alpha_{_{k}}}$ ($q_{_{i}}$ prime), is non-heuristically approximated by the non-heuristic Dirichlect prime counting function $\pi_{_{D}}(n) = \prod_{i = 1}^{k}{\frac{1}{q_{_{i}}^{\alpha_{_{i}}}}}.\prod_{i = 1}^{k}(1 - \frac{1}{q_{_{i}}})^{-1}.\pi_{_{L}}(n) \rightarrow \infty$. We finally give an elementary probability-based proof that the number $\pi_{_{2}}(n)$ of twin primes $\leq n$ is approximated by the non-heuristic twin-prime counting function $\pi_{_{T}}(n) = \sum_{j = 1}^{n}\mathbb{P}(j \in \{p\}\ \cap\ j+2 \in \{p\})$; and conclude by the Law of Large Numbers that there are infinitely many twin primes since we show that $\pi_{_{2}}(n) \sim \pi_{_{T}}(n) \sim e^{-2\gamma}.\frac{n}{log_{e}^{^{2}}n}$.

Mots clés: non-heuristic prime counting function; non-heuristic prime probability function; Brocard's conjecture; Chebyshev's Theorem; complete system of incongruent residues; computational complexity; Dirichlect primes; Euler's constant $\gamma$; expected value; factorising is polynomial time; Hardy-Littlewood conjecture; integer factorising algorithm; Law of Large Numbers; Mertens' theorem; mutually independent prime divisors; polynomial time algorithm; prime counting function $\pi(n)$; prime density; primes in an arithmetical progression; Prime Number Theorem; probability model; probabilistic number theory; twin primes.

Domaines: Mathématiques [math]

Bhupinder Singh Anand. The Curious Reluctance to Define Prime Probability Non-Heuristically: An elementary probability-based approach to estimating prime counting functions non-heuristically. 2015. ⟨hal-01199385v1⟩