  The Riemann-Siegel theta function $\vartheta(t)$ is examined for
$t\to+\infty$. Use of the refined asymptotic expansion for $\log\,\g(z)$ shows
that the expansion of $\vartheta(t)$ contains an infinite sequence of
increasingly subdominant exponential terms, each multiplied by an asymptotic
series involving inverse powers of $\pi t$. Numerical examples are given to
detect and confirm the presence of the first three of these exponentials.
