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The exponentially improved large-$a$ expansion for the Hurwitz zeta function$\zeta(s,a)$ is exploited to examine the expansion of the periodic zetafunction $F(a,s)$ in the upper half-plane of the variable $a$. It is shown thata double Stokes phenomenon takes place in the vicinity of the positiveimaginary $a$-axis as $|a|\ra\infty$. This is a consequence of the fact thatconstituent parts of $F(a,s)$ involve two Hurwitz zeta functions resulting intwo parallel Stokes lines at unit distance apart. Numerical calculationsconfirm the theoretical predictions.

The Stokes phenomenon associated with the periodic zeta function $F(a,s)$