Parts in $k$-indivisible Partitions Always Display Biases between Residue Classes

Abstract

Let $k, t$ be coprime integers, and let $1 \leq r \leq t$. We let $D_k^\times(r,t;n)$ denote the total number of parts among all $k$-indivisible partitions (i.e., those partitions where no part is divisible by $k$) of $n$ which are congruent to $r$ modulo $t$. In previous work of the authors, an asymptotic estimate for $D_k^\times(r,t;n)$ was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of $L$-functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing to conclude that there is always a bias towards one congruence class or another modulo $t$ among all parts in $k$-indivisible partitions of $n$ as $n$ becomes large.