Self-conjugate 6-cores and quadratic forms

TL;DR

This paper investigates the positivity of self-conjugate 6-core partition numbers using quadratic and modular forms, proving the conjecture under the assumption of the Generalized Riemann Hypothesis.

Contribution

It proves Hanusa and Nath's conjecture on the positivity of $sc_6(n)$ assuming GRH, connecting partition theory with deep results in quadratic forms and L-functions.

Findings

Proves $sc_6(n) > 0$ for large $n$ under GRH

Identifies main obstacles in explicit bounds for representation numbers

Connects positivity of partitions to class numbers and L-functions

Abstract

In this work, we analyze the behavior of the self-conjugate 6-core partition numbers $sc_{6}(n)$ by utilizing the theory of quadratic and modular forms. In particular, we explore when $sc_{6}(n) > 0$. Positivity of $sc_{t}(n)$ has been studied in the past, with some affirmative results when $t > 7$. The case $t = 6$ was analyzed by Hanusa and Nath, who conjectured that $sc_{6}(n) > 0$ except when $n \in \{2, 12, 13, 73\}$. This inspires a theorem of Alpoge, which uses deep results from Duke and Schulze-Pillot to show that $sc_{6}(n) > 0$ for $n \gg 1$ using representation numbers of a particular ternary quadratic form $Q$. Approximating such representation numbers involves class numbers of imaginary quadratic fields, which are directly related to values of Dirichlet $L$-functions. At present, we can only ineffectively bound these from below. This is currently the main hurdle in obtaining more explicit approximations for representation numbers of ternary quadratic forms, and in particular in showing explicit positivity results for $sc_{6}(n)$. However, by assuming the Generalized Riemann Hypothesis we are able to settle Hanusa and Nath's conjecture.