On the vanishing of the Fourier coefficients of CM eta quotients
Graduate project, Sun Yat-sen University, 2023
T. Huber, C. Liu, J. McLaughlin, D. Ye, M. Yuan and S. Zhang, On the vanishing of the coefficients of CM eta quotients, Proceedings of the Edinburgh Mathematical Society, 66 (2023), 1202-1216.
Project description
In this project, we studied and characterized the vanishing of the Fourier coefficients of CM eta quotients. For example, we proved that if \(a(n)\) is defined by
\[\sum_{n=1}^{\infty}a(n)q^{n}=q\prod_{n=1}^{\infty}(1-q^{n})^{3}(1-q^{7n})^{3},\]then \(a(n)\) vanishes, i.e., \(a(n)=0\) if and only if \(n=p^{e_{p}}n_{0}\) with \(\gcd(p,n_{0})=1\), \(p\equiv 3,5,6\pmod{7}\) and \(e_{p}\) being odd.