An algorithmic implementation of the Pi function based on a new sieve

TL;DR

This paper introduces a new algorithm based on a sieve method that efficiently generates and counts primes greater than 3 within an interval, avoiding large list generation and enabling parallel computation of the prime counting function.

Contribution

The paper presents a novel algorithmic implementation of the Pi function utilizing a new sieve, improving efficiency in prime generation and counting.

Findings

Efficient prime generation avoiding large list storage

Algorithm correctly discards non-primes based on Leopoldo's Theorem

Can be parallelized for faster computation of the prime counting function

Abstract

In this paper we propose an algorithm that correctly discards a set of numbers (from a previously defined sieve) with an interval of integers. Leopoldo's Theorem states that the remaining integer numbers will generate and count the complete list of primes of absolute value greater than 3 in the interval of interest. This algorithm avoids the problem of generating large lists of numbers, and can be used to compute (even in parallel) the prime counting function $\pi(h)$.