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

2023/05/09 by Jackson, Faye, Otgonbayar, Misheel
#11P81 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11P82. Secondary: 05A17

paper · doi:10.48550/arxiv.2305.05753

Abstract

Let k, t be coprime integers, and let 1 ≤ r ≤ t. We let D_k^×(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^×(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.

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