Distribution of the full rank in residue classes for odd moduli

TL;DR

This paper investigates how the full ranks of marked Durfee symbols are distributed across residue classes for various moduli, revealing equidistribution properties and deriving related congruences.

Contribution

It establishes that k-marked Durfee symbols evenly distribute across certain residue classes, and demonstrates how to derive congruences from these distributions.

Findings

k-marked Durfee symbols of n are equally distributed among residue classes with gcd conditions

Constructs congruences based on distribution properties

Provides a specific theorem for 4-marked symbols modulo multiples of 3

Abstract

The distribution of values of the full ranks of marked Durfee symbols is examined in prime and nonprime arithmetic progressions. The relative populations of different residues for the same modulus are determined: the primary result is that k-marked Durfee symbols of n equally populate the residue classes a and b mod 2k+1 if gcd(a,2k+1)=gcd(b,2k+1). These are used to construct a few congruences. The general procedure is illustrated with a particular theorem on 4-marked symbols for multiples of 3.