A density-based approach for non-heuristic approximations of prime counting functions

TL;DR

This paper introduces a density-based, non-heuristic method for approximating prime counting functions by analyzing the densities of special integer sets related to primes.

Contribution

It proposes a novel density-based approach that non-heuristically approximates prime counts using the densities of integers not divisible by small primes.

Findings

Expected counts of primes, Dirichlet primes, and twin primes grow to infinity.

Density-based estimates align with known prime distribution patterns.

Method provides a non-heuristic alternative to classical asymptotic approximations.

Abstract

All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). 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) within a finite range of values of n. By considering the density of each of the set of (i) all integers n, (ii) Dirichlect integers n = a+md, and (iii) Twin integers (n, n+2), which are not divisible by any of the first k primes, we show that---based on their respective densities---the expected number of such integers in the initial interval (1, n) of length n non-heuristically approximates the number of (a) primes, (b) Dirichlect primes, and (c) Twin primes, respectively, which are less than or equal to n. We further show that, in each case, the estimate tends to infinity.