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The Ramanujan Journal

, Volume 48, Issue 2, pp 445–458 | Cite as

On cyclotomic factors of polynomials related to modular forms

  • Bernhard HeimEmail author
  • Florian Luca
  • Markus Neuhauser
Article

Abstract

The Fourier coefficients of powers of the Dedekind eta function can be studied simultaneously. The vanishing of the coefficients varies from super lacunary (Euler, Jacobi identities) and lacunary (CM forms) to non-vanishing (Lehmer conjecture for the Ramanujan numbers). We study polynomials of degree n, whose roots control the vanishing of the nth Fourier coefficients of such powers. We prove that every root of unity appearing as any root of these polynomials has to be of order 2.

Keywords

Dedekind eta function Fourier coefficients Integer-valued polynomials Cyclotomic polynomials 

Mathematics Subject Classification

Primary 11D10 11F20 Secondary 11F30 11B83 11P84 11R18 

Notes

Acknowledgements

We thank the anonymous referee for a careful reading of this manuscript and useful comments. This work was done while all three authors were visiting the Max Planck Institute for Mathematics in Bonn in July 2017. They thank this Institution for the invitation and excellent working conditions. In addition, B.H. and M.N. thank the RWTH Aachen, the Graduate school: Experimental and constructive algebra, chaired by G. Nebe, and the German University of Technology in Oman for their support.

References

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.German University of Technology in OmanMuscatSultanate of Oman
  2. 2.Max-Planck-Institute for MathematicsBonnGermany
  3. 3.School of MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  4. 4.Department of Mathematics, Faculty of SciencesUniversity of OstravaOstrava 1Czech Republic
  5. 5.Faculty of Mathematics, Computer Science, and Natural SciencesRWTH Aachen UniversityAachenGermany

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