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 $\ln \Gamma (z)$ 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 $|R_{k+1}(z)/T_k(z)|<\sqrt{\pi k}$ 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 $|R_{k+1}(z)/T_k(z)|<(k/|z|{)}^2/({\pi }^2-1)<0.113$ holds. Similarly for the asymptotic approximation of $\ln \Gamma (z+\frac{1}{2})$, except that a factor ${\eta }_k=1/(1-2^{1-2k})$ 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 $O(\exp (-\pi t))$ to $O(\exp (-2\pi t))$. We discuss a similar example due to Olver (1964), and a connection with the Stokes phenomenon.

23 pages, 2 tables, corrected typos and added new results in v2

References in corpus (1)

• Asymptotic approximation of central binomial coefficients with rigorous error bounds

Cited by in corpus (2)

• Accurate estimation of sums over zeros of the Riemann zeta-function
• Asymptotic approximation of central binomial coefficients with rigorous error bounds