On the Number of Parts in Congruence Classes for Partitions into\n Distinct Parts

2021/10/29 by William Craig, Craig, William
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2110.15835

openalex publication_date 2021/10/29 · openalex created_date 2023/02/13 · openalex updated_date 2026/08/01

Abstract

For integers 0 < r \≤ t, let the function D_r,t(n) denote the number\nof parts among all partitions of n into distinct parts that are congruent to\nr modulo t. We prove the asymptotic formula D_r,t(n)
sim\n
dfrac3
^{frac 14} e^
pi
sqrt
fracn32
pi t n
^{frac 14}
left(\n
log(2) +
left(
dfrac
sqrt3
log(2)8
pi -
dfrac
pi4
sqrt3
left(\nr -
dfract2
right)
right) n^{-
frac 12}
right) as n \→ \∞. A\ncorollary of this result is that for 0 < r < s \≤ t, the inequality\nD_r,t(n) \≥ D_s,t(n) holds for all sufficiently large n. We make this\neffective, showing that for 2 \≤ t \≤ 10 the inequality D_r,t(n) \≥\nD_s,t(n) holds for all n > 8.\n

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