ON THE ACCURACY OF ASYMPTOTIC APPROXIMATIONS TO THE LOG-GAMMA AND RIEMANN–SIEGEL THETA FUNCTIONS

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln \unicode[STIX]{x1D6E4}(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer [Theorie der analytischen Funktionen einer komplexen Veränderlichen, 2nd edn (Springer, Berlin, 1962)], Spira [‘Calculation of the Gamma function by Stirling’s formula’, Math. Comp. 25 (1971), 317–322], and Hare [‘Computing the principal branch of log-Gamma’, J. Algorithms 25 (1997), 221–236]. We show that $|R_{k+1}(z)/T_{k}(z)|<\sqrt{\unicode[STIX]{x1D70B}k}$ 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. We deduce similar bounds for asymptotic approximation of the Riemann–Siegel theta function $\unicode[STIX]{x1D717}(t)$ . We show that the accuracy of a well-known approximation to $\unicode[STIX]{x1D717}(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real $t>0$ from $O(\exp (-\unicode[STIX]{x1D70B}t))$ to $O(\exp (-2\unicode[STIX]{x1D70B}t))$ . We discuss a similar example due to Olver [‘Error bounds for asymptotic expansions, with an application to cylinder functions of large argument’, in: Asymptotic Solutions of Differential Equations and Their Applications (ed. C. H. Wilcox) (Wiley, New York, 1964), 16–18], and a connection with the Stokes phenomenon.

[1]  Wolfgang Gabcke,et al.  Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel , 2015 .

[2]  G. Nemes,et al.  Generalization of Binet's Gamma function formulas , 2013 .

[3]  C. Gauss,et al.  DISQUISITIONES GENERALES CIRCA SERIEM INFINITAM , 2011 .

[4]  C. Hermite.  Sur la fonction log Г(a) , 2009 .

[5]  Jonathan M. Borwein,et al.  Computational strategies for the Riemann zeta function , 2000 .

[6]  D. E. G. Hare,et al.  Computing the Principal Branch of log-Gamma , 1997, J. Algorithms.

[7]  John R. King,et al.  Stokes Phenomenon and Matched Asymptotic Expansions , 1995, SIAM J. Appl. Math..

[8]  M. Berry,et al.  The Riemann-Siegel expansion for the zeta function: high orders and remainders , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[9]  L. Gordon,et al.  The Gamma Function , 1994, Series and Products in the Development of Mathematics.

[10]  R. E. Meyer.  A Simple Explanation of the Stokes Phenomenon , 1989, SIAM Rev..

[11]  Karl Dilcher,et al.  Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials , 1987 .

[12]  Ian Tweddle,et al.  Approximating n! Historical origins and error analysis , 1984 .

[13]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[14]  Richard P. Brent,et al.  On the zeros of the Riemann zeta function in the critical strip III , 1985 .

[15]  Richard P. Brent,et al.  On the zeros of the Riemann zeta function in the critical strip , 1979 .

[16]  Richard P. Brent,et al.  High Precision Coefficients Related to the Zeta Function. , 1977 .

[17]  F. Olver.  Asymptotics and Special Functions , 1974 .

[18]  Harold M. Edwards,et al.  Riemann's Zeta Function , 1974 .

[19]  R. Spira.  Calculation of the Gamma Function by Stirling's Formula , 1971 .

[20]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[21]  H. Behnke,et al.  Theorie der analytischen Funktionen einer komplexen Veränderlichen , 1965 .

[22]  D. H. Lehmer.  Extended computation of the Riemann zeta-function , 1956 .

[23]  G. Watson.  Bessel Functions. (Scientific Books: A Treatise on the Theory of Bessel Functions) , 1923 .

[24]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[25]  J.-P. Gram,et al.  Note sur les zéros de la fonction ζ(s) de Riemann , 1903 .