On the counting function of semiprimes

TL;DR

This paper investigates the asymptotic behavior of the semiprime counting function, providing explicit series, an algorithm for constants, and extending the approach to products of multiple primes.

Contribution

It offers a detailed asymptotic series for semiprimes, an algorithm for constants, and generalizes the method to higher prime products.

Findings

Explicit asymptotic series for (x)

Algorithm for computing constants to 20 digits

Extension of approach to products of k primes

Abstract

A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function $\pi_2(x)$, the number of semiprimes less than or equal to $x$, is studied. Using a combinatorial argument, asymptotic series of $\pi_2(x)$ is determined, with all the terms explicitly given. An algorithm for the calculation of the constants involved in the asymptotic series is presented and the constants are computed to 20 significant digits. The errors of the partial sums of the asymptotic series are investigated. A generalization of this approach to products of $k$ primes, for $k\geq 3$, is also proposed.