SOME NEW ALGORITHMS FOR HIGH-PRECISION COMPUTATION OF EULER’S CONSTANT

RICHARD P. BRENT AND EDWIN M. MCMILLAN

Abstract

We describe several new algorithms for the high-precision computation of Euler’s constant γ = 0.577 . . .. Using one of the algorithms, which is based on an identity involving Bessel functions, γ has been computed to 30, 100 decimal places. By computing their regular continued fractions, we show that, if γ or exp(γ) is of the form P/Q for integers P and Q, then |Q| > 10^{15000}.

Comments

Only the Abstract is given here. The full paper appeared as [2]. For earlier work, see [1]. An interesting connection with the work of Ramanujan is described in [3].

References

1. R. P. Brent, “Computation of the regular continued fraction for Euler’s constant”, Math. Comp. 31 (1977), 771–777. MR 55#9490, Zbl 369.10002. See also “γ and exp(γ) to 20700D and their regular continued fractions to 20000 partial quotients”, UMT 1, ibid 32 (1978), 311. rpb040.

2. R. P. Brent and E. M. McMillan, “Some new algorithms for high-precision computation of Euler’s constant”, Math. Comp. 34 (1980), 305–312. MR 82g:10002. Also Report TR LBL-8729, Lawrence Berkeley Laboratory; and Report TR-CS-79-03, DCS, ANU (January 1979), 16 pp. See also “Euler’s constant and its exponential to 30,100 decimals”, and “The first 29,000 partial quotients in the regular continued fraction for Euler’s constant and its exponential”, Math. Comp. UMT File. rpb049.

3. R. P. Brent, An asymptotic expansion inspired by Ramanujan, Report CMA-MR02-93/SMS-10-93, Centre for Mathematics and its Applications, ANU, February 1993, 7 pp. rpb139.

(Brent) Department of Computer Science, Australian National University, Canberra

(McMillan) Lawrence Berkeley Laboratory, University of California, Berkeley

1991 Mathematics Subject Classification. Primary 11-04; Secondary 11A55, 11J72, 65-04, 65A05, 68D20.

Key words and phrases. Euler’s constant, Mascheroni’s constant, gamma, Bessel functions, rational approximation, regular continued fraction, multiple-precision arithmetic, Gauss-Kusmin law.

Received January 22, 1979; revised May 15, 1979.

The work of the first author was supported in part by National Science Foundation grand 1-442427-21164-2 at the University of California, Berkeley. This work was also supported by the U. S. Department of Energy under Contract W-7405-ENG-48.

This work was initiated while the first author was visiting the Department of Electrical Engineering and Computer Sciences at the University of California, Berkeley. The computations were performed at the Australian National University.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1980, American Mathematical Society.

Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb049a typeset using AMS-L^{A}TEX