  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.
