Refined asymptotics of the Riemann-Siegel theta function

Abstract

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