论文标题
自轭6核和二次形式
Self-conjugate 6-cores and quadratic forms
论文作者
论文摘要
在这项工作中，我们通过利用二次和模块化形式的理论来分析自轭6核分区编号$ sc_ {6}（n）$的行为。特别是，我们探索$ sc_ {6}（n）> 0 $。过去已经研究了$ sc_ {t}（n）$的阳性，当$ t> 7 $时，结果有一些肯定的结果。 Case $ t = 6 $由Hanusa和Nath分析，他们猜想$ sc_ {6}（n）> 0 $除非$ n \ in \ in \ {2、12、12、13、73 \} $。这激发了Alpoge的定理，该定理使用Duke和Schulze-Pillot的深层结果表明，使用特定三元二元格式$ Q $的代表号，$ sc_ {6}（n）> 0 $ for $ n \ gg 1 $。 近似此类表示数字涉及假想二次字段的类数字，这些数字与dirichlet $ l $ functions的值直接相关。目前，我们只能从下面限制它们。当前，这是获得三元二次形式的表示数字的更明确近似值的主要障碍，尤其是在显示$ SC_ {6}（n）$的明确阳性结果时。但是，通过假设普遍的Riemann假设，我们能够解决Hanusa和Nath的猜想。
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.
