On the Accuracy of 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 Γ(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)| < √ 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 ≤ |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 ln Γ(z + 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 ϑ(t). We show that the accuracy of a well-known approximation to ϑ(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.

Stirling’s Original Asymptotic Series from a Formula Like One of Binet’s and its Evaluation by Sequence Acceleration

“…The Gamma function and the factorial function, invented in the 1700's, have been very thoroughly studied. Richard Brent's article [8] points out some facts, known to Hermite and to Gauss, that were not covered in the survey [6], which looked at about 100 references. One learns therefore that it is difficult to claim a result (formula or proof) is truly new; we are worried in particular that Gauss knew of our Binet-like formula proved here.…”

Refined asymptotics of the Riemann-Siegel theta function

“…valid as t → +∞, where B 2k denote the Bernoulli numbers. Recently, Brent [2] employed an alternative representation of ϑ(t), obtained by application of the reflection and duplication formulas for the gamma function, in the form…”