In this
PRISE (Program for Research in
Science and Engeneering) project of Summer 2008,
John Lesieutre and I studied Dirichlet series
with coefficients generated by irrational
rotations. We also studied Taylor series
Our research has started by investigating, whether newer "harder"
implicit function theorems of Neuberger can be used to prove KAM results, which usually
require hard implicit function theorems.
John Neuberger's version of the implicit function theorem appeared particularly powerful.
Even after hardening that implicit function theorem, we had to realize that even in
Siegels case (a much simpler situation)the theorem does not bite. While battling this problem,
we were led to problems of Dirichlet series generated by dynamical systems. This is an area of mathematics
at the border of analytic number theory and dynamical systems theory. Using DenjoyKoksma and Fourier theory
we are able to estimate for the abscissa of convergence of such
series in the case, when the rotation number is Diophantine. Our paper estimates the abscissa of
convergence in the case of real analytic functions and functions with bounded variations.
We also find a large class of Dirichlet series, which are entire functions:
if g is an odd real analytic function and alpha is Diophantine, then
has an analytic continuation to the entire complex plane.
