Parts in k-indivisible partitions always display biases between residue classes

Abstract

Let $k,t$ be coprime integers, and let $1\le r\le t$. 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 [3], 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 in [3] 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 [1] 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.