Even partition functions and $2$-adic analysis
Volume 173 / 2023
Abstract
Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L. Nicolas, I. Z. Ruzsa and A. Sárközy proved that there exists a unique set $\mathcal A=\mathcal A(P)$ such that $\sum _{n\geq 0}p(\mathcal A,n)z^n\equiv P(z) \pmod 2$; that is, the partition function $p(\mathcal {A},n)$ is even from a certain point on. The problem of determining the elements of the set $\mathcal A(P)$ is not an easy one and several particular cases have already been studied; namely, when $P$ is irreducible and $\beta $ is a prime number $p$ such that the order of $2$ modulo $p$ is $p-1$, $(p-1)/2$, $(p-1)/3$ or $(p-1)/4$. In this paper, we consider the case where $P$ is irreducible such that the order of $2$ modulo $\beta $ is $\varphi (\beta )/2$ where $\varphi $ is Euler’s function.
