Abstract
The signs and vanishing of Fourier coefficients of modular forms are important properties of modular forms and are closely related. The focus of this paper is on the coefficients of the powers of the Dedekind \(\eta \)-function, in particular the discriminant function \(\Delta \) and the Ramanujan \(\tau \)-function. To all nth coefficients we attach polynomials \(P_n(x)\). The root distribution of the \(P_n(x)\) dictates the sign of all coefficients. In particular it determines the first non-sign change of \(\tau (n)\). We further show the influence of this property on the neighbours \(\eta ^{23}\) and \(\eta ^{25}\) of \(\Delta = \eta ^{24}\), which leads to a number of conjectures.
Dedicated to Murugesam Manickam on the occasion of his 60th birthday
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Acknowledgements
The authors would like to thank Robert Tröger for many useful discussions and providing us with Fig. 1. The authors thank the anonymous referee for useful comments.
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Heim, B., Neuhauser, M. (2020). Sign Changes of the Ramanujan \(\tau \)-Function. In: Ramakrishnan, B., Heim, B., Sahu, B. (eds) Modular Forms and Related Topics in Number Theory. ICNT 2018. Springer Proceedings in Mathematics & Statistics, vol 340. Springer, Singapore. https://doi.org/10.1007/978-981-15-8719-1_7
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