On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions

Abstract

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\lnΓ(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer (1962), Spira (1971), and Hare (1997). We show that $|R_{k+1}(z)/T_k(z)| < \sqrt{πk}$ for nonzero $z$ in the right half-plane, where $T_k(z)$ is the $k$-th term in the asymptotic series, and $R_{k+1}(z)$ is the error incurred in truncating the series after $k$ terms. If $k \le |z|$, then the stronger bound $|R_{k+1}(z)/T_k(z)| < (k/|z|)^2/(π^2-1) < 0.113$ holds. Similarly for the asymptotic approximation of $\lnΓ(z+\frac{1}{2})$, except that a factor $η_k = 1/(1-2^{1-2k})$ multiplies some of the bounds. We deduce similar bounds for asymptotic approximation of the Riemann-Siegel theta function $\vartheta(t)$. We show that the accuracy of a well-known approximation to $\vartheta(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real $t>0$ from $O(\exp(-πt))$ to $O(\exp(-2πt))$. We discuss a similar example due to Olver (1964), and a connection with the Stokes phenomenon.