Title: The G.-C. Rota approach and the Lehmer conjecture
Speaker: Bernhard Heim (German University of Technology, Oman)
Date: 14 November 2019
Time: 2:30 pm
Venue: LH-1, Mathematics Department
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.