Refined asymptotics of the Riemann-Siegel theta function

The Riemann-Siegel theta function $\vartheta (t)$ is examined for $t\to +\mathrm{\infty }$. Use of the refined asymptotic expansion for $\log \text{g}(z)$ 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.