COMPUTATION OF THE REGULAR CONTINUED FRACTION FOR EULER’S CONSTANT

RICHARD P. BRENT

Abstract

We describe a computation of the first 20, 000 partial quotients in the regular continued fraction fo Euler’s constant γ = 0.577 … and exp(γ) = 1.781 . . .. A preliminary step was the calculation of γ and exp(γ) to 20, 700D. It follows from the continued fractions that, if γ or exp(γ) is of the form P/Q for integers P and Q, then |Q| > 10^{10000}.

Comments

Only the Abstract is given here. The full paper appeared as [2]. The result improved on an earlier (and only partially correct) result of Beyer and Waterman [1]. An interesting connection with the work of Ramanujan is described in [4]. For a sequel which introduced some more efficient algorithms and extended the computation, see [3].

References

1. W. A. Beyer and M. S. Waterman, “Error analysis of a computation of Euler’s constant”, Math. Comp. 28 (1974), 599–604. Also “Decimals and partial quotients of Euler’s constant and ln 2”, reviewed in RMT 19, Math. Comp. 28 (1974), 667.

2. 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.

3. 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. rpb049.

4. R. P. Brent, An asymptotic expansion inspired by Ramanujan, Technical Report CMA-MR02-93/SMS-10-93, Centre for Mathematics and its Applications, ANU, February 1993, 7 pp. Also “Ramanujan and Euler’s constant”, to be presented at Mathematics of Computation 1943–1993, Vancouver, August 1993. rpb139.

Computer Sciences Laboratory, Australian National University, Canberra

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, rational approximation, regular continued fraction, multiple-precision arithmetic, arithmetic-geometric mean, Khintchine’s law, Lévy’s law, Gauss-Kusmin law..

Received September 27, 1976.

Daniel Shanks made several useful suggestions and drew my attention to the disagreement with Beyer and Waterman’s computation.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1977, American Mathematical Society.

Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb040a typeset using AMS-L^{A}TEX