Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Some new algorithms for high-precision computation of Euler's constant


Authors: Richard P. Brent and Edwin M. McMillan
Journal: Math. Comp. 34 (1980), 305-312
MSC: Primary 10-04; Secondary 10A40, 68C05
MathSciNet review: 551307
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Abstract: We describe several new algorithms for the high-precision computation of Euler's constant $ \gamma = 0.577 \ldots $ Using one of the algorithms, which is based on an identity involving Bessel functions, $ \gamma $ has been computed to 30,100 decimal places. By computing their regular continued fractions we show that, if $ \gamma $ or $ \exp (\gamma )$ is of the form $ P/Q$ for integers P and Q, then $ \vert Q\vert > {10^{15000}}$.


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  • [1] M. ABRAMOWITZ & I. A. STEGUN, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D. C., 1964. MR 0167642 (29:4914)
  • [2] M. BEELER, R. W. GOSPER & R. SCHROEPPEL, "Hakmem," Memo No. 239, M.I.T. Artificial Intelligence Lab., Cambridge, Mass., 1972, pp. 70-71.
  • [3] W. A. BEYER & M. S. WATERMAN, "Error analysis of a computation of Euler's constant," Math. Comp., v. 28, 1974, pp. 599-604. MR 49 #6555. MR 0341809 (49:6555)
  • [4] W. A. BEYER & M. S. WATERMAN, "Decimals and partial quotients of Euler's constant and In 2," UMT 19, Math. Comp., v. 28, 1974, p. 667. Errata: Math. Comp., MTE 549, v. 32, 1978, pp. 317-318. MR 0341809 (49:6555)
  • [5] R. P. BRENT, "The complexity of multiple-precision arithmetic," Complexity of Computational Problem Solving (R. S. Anderssen and R. P. Brent, Eds.), Univ. of Queensland Press, Brisbane, 1976, pp. 126-165.
  • [6] R. P. BRENT, "Multiple-precision zero-finding methods and the complexity of elementary function evaluation," Analytic Computational Complexity (J. F. Traub, Ed.), Academic Press, New York, 1976, pp. 151-176. MR 52 #15938, 54 #11843. MR 0423869 (54:11843)
  • [7] R. P. BRENT, "Computation of the regular continued fraction for Euler's constant," Math. Comp., v. 31, 1977, pp. 771-777. MR 55 #9490. MR 0436547 (55:9490)
  • [8] R. P. BRENT, "$ \gamma $ and $ \exp (\gamma )$ to 20700D and their regular continued fractions to 20000 partial quotients," UMT 1, Math. Comp., v. 32, 1978, p. 311.
  • [9] R. P. BRENT, "A Fortran multiple-precision arithmetic package," ACM Trans. Math. Software, v. 4, 1978, pp. 57-70.
  • [10] R. P. BRENT, "Euler's constant and its exponential to 30,100 decimals," Computing Research Group, Australian National University, Sept. 1978. Submitted to Math. Comp. UMT file.
  • [11] R. P. BRENT & E. M. McMILLAN, "The first 29,000 partial quotients in the regular continued fractions for Euler's constant and its exponential," submitted to Math. Comp. UMT file.
  • [12] J. W. L. GLAISHER, "History of Euler's constant," Messenger of Math., v. 1, 1872, pp. 25-30.
  • [13] A. YA. KHINTCHINE (A. JA. HINČIN), Continued Fractions, 3rd ed., (English transl. by P. Wynn), Noordhoff, Groningen, 1963. MR 28 #5038. MR 0161834 (28:5038)
  • [14] G. F. B. RIEMANN, "Zur Theorie der Nobili'schen Farbenringe," Poggendorff's Annalen der Physik und Chemie, Bd. 95, 1855, pp. 130-139. (Reprinted in Bernhard Riemann's Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, Teubner, Leipzig, 1876, pp. 54-61.)
  • [15] A. SCHÖNHAGE & V. STRASSEN, "Schnelle Multiplikation grosser Zahlen," Computing, v. 7, 1971, pp. 281-292. MR 0292344 (45:1431)
  • [16] D. W. SWEENEY, "On the computation of Euler's constant," Math. Comp., v. 17, 1963, pp. 170-178. MR 28 #3522. MR 0160308 (28:3522)
  • [17] A. VAN DER POORTEN, "A proof that Euler missed-Apéry's proof of the irrationality of $ \zeta (3)$," Mathematical Intelligence, v. 1, 1979, pp. 196-203.
  • [18] H. WALL, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948. MR 0025596 (10:32d)
  • [19] G. N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, London, 1944. MR 0010746 (6:64a)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1980-0551307-4
Keywords: Euler's constant, Mascheroni's constant, gamma, Bessel functions, rational approximation, regular continued fractions, multiple-precision arithmetic, Gauss-Kusmin law
Article copyright: © Copyright 1980 American Mathematical Society