Quasi-modularity in MacMahon partition variants and prime detection

초록

Introduced in 1920 as a generalization of the sum-of-divisors function, the MacMahon partition function has recently attracted renewed interest due to its connection with quasi-modular forms. In this article, we explicitly establish the quasi-modularity of the level N MacMahon function, originally studied by MacMahon himself, by employing the polynomials introduced by Lehmer in his investigation of cyclotomic polynomials. As applications, we explore analogues and extensions of the prime-detecting expressions proposed by Craig, van Ittersum, and Ono.

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