MacMahonesque partition functions detect sets related to primes

TL;DR

This paper extends MacMahon's partition functions to detect prime powers and primes in arithmetic progressions, building on recent work that identified primes using these functions.

Contribution

It introduces modifications to existing MacMahonesque functions to also detect prime cubes and primes in arithmetic progressions, expanding their applicability.

Findings

Successfully adapted MacMahonesque functions to detect prime cubes.

Extended the detection capabilities to primes in arithmetic progressions.

Built on prior work to generalize prime detection methods.

Abstract

Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are infinitely many such expressions in these functions. Here, we show how to modify and adapt their construction to detect cubes of primes as well as primes in arithmetic progressions.