Year: 2024
Type: preprint
Abstract: The Fourier coefficients $c_1(n)$ of the elliptic modular $j$-function are always even for $n \not\equiv 7 \pmod{8}$. In contrast, for $n \equiv 7 \pmod{8}$, it is conjectured that ``half" of the coefficients take odd values. In this article, we first observe in detail when $c_1(8n-1)$ is odd and show that the coefficients share the same parity as the coefficients $c_{μ_2}(n)$ of the 2nd order mock theta function $μ_2(q)$. Furthermore, we prove that this phenomenon also holds among several hauptmoduln and between hauptmoduln and even-order mock theta functions.
Source: arXiv (Cornell University)
Authors: Kang, Soon-Yi, Kim, Seonkyung, Matsusaka, Toshiki, Yoo, Jaeyeong
Institutions: Kangwon National University, Kyushu University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403345423