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The G.-C. Rota approach and the Lehmer conjecture

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Bernhard Heim
German University of Technology, Oman (GUtech) and RWTH Aachen
Die, 2019-08-06 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp,
R. Troeger, and A. Weisse.
The Lehmer conjecture and Serre's lacunary theorem describe the vanishing properties of the Fourier
coefficients of even powers of the Dedekind eta function.
G.-C. Rota proposed to translate and study problems in number theory and combinatorics to and via
properties of polynomials. We follow G.-C. Rota's advice. This leads to several new results and
improvement of known results. This includes Kostant's non-vanishing results attached to simple complex
Lie algebras, a new non-vanishing zone of the Nekrasov-Okounkov formula (improving a result of G. Han),
a new link between generalized Laguerre and Chebyshev polynomials, strictly sign-changes results of
reciprocals of the cubic root of Klein's absolute $j$-invariant, and hence the $j$-invariant itself.
Finally we give an interpretation of the first non-sign change of the Ramanujan $\tau(n)$ function by
the root distribution of a certain family of polynomials in the spirit of G.-C. Rota.

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