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Research in Number Theory

, 4:32 | Cite as

Records on the vanishing of Fourier coefficients of powers of the Dedekind eta function

  • Bernhard Heim
  • Markus Neuhauser
  • Alexander Weisse
Research

Abstract

In this paper we significantly extend Serre’s table on the vanishing properties of Fourier coefficients of odd powers of the Dedekind eta function. We address several conjectures of Cohen and Strömberg and give a partial answer to a question of Ono. In the even-power case, we extend Lehmer’s conjecture on the coefficients of the discriminant function \(\Delta \) to all non-CM-forms. All our results are supported with numerical data. For example all Fourier coefficients \(a_9(n)\) of the 9-th power of the Dedekind eta function are non-vanishing for \(n \le 10^{10}\). We also relate the non-vanishing of the Fourier coefficients of \(\Delta ^2\) to Maeda’s conjecture.

Keywords

Fourier coefficients Euler products Dedekind eta function Lehmer conjecture Maeda conjecture 

Mathematics Subject Classification

Primary 05A17 11F20 Secondary 11F30 11F37 

Notes

Acknowledgements

The authors thank the two referee’s for their valuable reports, and Neil Dummigan and Susanne Schrumpf for their useful support and comments. Several of the scientific computations had been performed at the Max-Planck-Institute of Mathematics in Bonn. We thank the Institute for providing us with their software and high performance workstations.

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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.German University of Technology in OmanMuscatSultanate of Oman
  2. 2.Max-Planck-Institute for MathematicsBonnGermany

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