### 11 september 2006,
Jens Marklof (University of Bristol, U.K.),
Spectral Theta Series

**Abstract.**
The theta series $\vartheta(z)=\sum \exp(2\pi\i n^2 z)$ is a classical
example of a modular form. In this talk we argue that the trace
$\vartheta_P(z)=\Tr\, \exp(2\pi\i P^2 z)$, where $P$ is a first-order
self-adjoint pseudo-differential operator with periodic
bicharacteristic flow, may be viewed as a natural generalization. In
particular, we establish approximate functional relations under the
action of the modular group. This allows a detailed analysis of the
asymptotics of $\vartheta_P(z)$ near the real axis, and the proof of
logarithm laws and limit theorems for its value distribution.