Published online by Cambridge University Press: 20 November 2018
The behavior of the dynamical zeta function
${{Z}_{D}}(s)$
related to several strictly convex disjoint obstacles is similar to that of the inverse
$Q(s)\,=\,\frac{1}{\zeta (s)}$
of the Riemann zeta function
$\zeta \left( s \right)$. Let
$\prod \left( s \right)$ be the series obtained from
${{Z}_{D}}(s)$
summing only over primitive periodic rays. In this paper we examine the analytic singularities of
${{Z}_{D}}(s)$
and
$\prod \left( s \right)$ close to the line
$\Re s={{s}_{2}},$ where
${{s}_{2}}$
is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions
${{Z}_{D}}(s),$
$\prod \left( s \right)$ has a singularity at
$s\,=\,{{s}_{2}}$.