An algorithm for the prime-counting function of primes larger than three

Abstract

The prime-counting function $\pi (x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi (x)$ with time complexity $\mathcal{O}(x^{1/2})$ without the need to introduce the non-trivial zeros of the Riemann zeta function.

Issue with double counting of composites

References in corpus (1)

• On Riemann's Paper, "On the Number of Primes Less Than a Given Magnitude"