Computation of the regular continued fraction for Euler's constant

Author:
Richard P. Brent

Journal:
Math. Comp. **31** (1977), 771-777

MSC:
Primary 65D20; Secondary 10-04

DOI:
https://doi.org/10.1090/S0025-5718-1977-0436547-7

MathSciNet review:
0436547

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Abstract | References | Similar Articles | Additional Information

Abstract: We describe a computation of the first 20,000 partial quotients in the regular continued fractions for Euler's constant and A preliminary step was the calculation of and to 20,700D. It follows from the continued fractions that, if or is of the form for integers *P* and *Q*, then .

**[1]**J. C. ADAMS, "On the value of Euler's constant,"*Proc. Roy. Soc. London*, v. 27, 1878, pp. 88-94.**[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 and M. S. Waterman,*Error analysis of a computation of Euler’s constant*, Math. Comp.**28**(1974), 599–604. MR**0341809**, https://doi.org/10.1090/S0025-5718-1974-0341809-5**[4]**W. A. Beyer and M. S. Waterman,*Error analysis of a computation of Euler’s constant*, Math. Comp.**28**(1974), 599–604. MR**0341809**, https://doi.org/10.1090/S0025-5718-1974-0341809-5**[5]**W. A. BEYER & M. S. WATERMAN, "Decimals and partial quotients of Euler's constant and ," Submitted to UMT file, 1976.**[6]**R. P. BRENT, "The complexity of multiple-precision arithmetic,"*Complexity of Computational Problem Solving*(R. S. Anderssen and R. P. Brent, Editors), Univ. of Queensland Press, Brisbane, 1976, pp. 126-165.**[7]**Richard P. Brent,*Multiple-precision zero-finding methods and the complexity of elementary function evaluation*, Analytic computational complexity (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1975) Academic Press, New York, 1976, pp. 151–176. MR**0423869****[8]**Richard P. Brent,*Fast multiple-precision evaluation of elementary functions*, J. Assoc. Comput. Mach.**23**(1976), no. 2, 242–251. MR**0395314**, https://doi.org/10.1145/321941.321944**[9]**R. P. BRENT, "A Fortran multiple-precision arithmetic package,"*ACM Trans. Math. Software.*(To appear.) (Also available as Tech. Report, Dept. of Comp. Sei., Carnegie-Mellon Univ., Pittsburgh, Pa.)**[10]**K. Y. Choong, D. E. Daykin, and C. R. Rathbone,*Rational approximations to 𝜋*, Math. Comp.**25**(1971), 387–392. MR**0300981**, https://doi.org/10.1090/S0025-5718-1971-0300981-0**[11]**D. Shanks,*Table errata: “Regular continued fractions for 𝜋 and 𝛾”, (Math. Comp. 25 (1971), 403); “Rational approximations to 𝜋” (ibid. 25 (1971), 387–392) by K. Y. Choong, D. E. Daykin and C. R. Rathbone*, Math. Comp.**30**(1976), no. 134, 381. MR**0386215**, https://doi.org/10.1090/S0025-5718-1976-0386215-4**[12]**L. EULER, "De numero memorabili in summatione progressionis harmonicae naturalis occurrente,"*Acta Petrop*, v. 5, part 2, 1781, p. 45.**[13]**J.W.L. GLAISHER, "History of Euler's constant,"*Messenger of Math.*, v. 1, 1872, pp. 25-30.**[14]**A. Khintchine,*Zur metrischen Kettenbruchtheorie*, Compositio Math.**3**(1936), 276–285 (German). MR**1556944****[15]**A. Ya. Khintchine,*Continued fractions*, Translated by Peter Wynn, P. Noordhoff, Ltd., Groningen, 1963. MR**0161834****[16]**Donald E. Knuth,*Euler’s constant to 1271 places*, Math. Comp.**16**(1962), 275–281. MR**0148255**, https://doi.org/10.1090/S0025-5718-1962-0148255-X**[17]**D. H. Lehmer,*Euclid’s Algorithm for Large Numbers*, Amer. Math. Monthly**45**(1938), no. 4, 227–233. MR**1524250**, https://doi.org/10.2307/2302607**[18]**P. LÉVY,*Théorie d l'Addition des Variables Aléatoires*, Gauthier-Villars, Paris, 1937, p. 320.**[19]**Dura W. Sweeney,*On the computation of Euler’s constant*, Math. Comp.**17**(1963), 170–178. MR**0160308**, https://doi.org/10.1090/S0025-5718-1963-0160308-X**[20]**J. W. WRENCH, JR., "A new calculation of Euler's constant,"*MTAC*, v. 6, 1952, p. 255.**[21]**J. W. WRENCH, JR. & D. SHANKS, "Questions concerning Khintchine's constant and the efficient computation of regular continued fractions,"*Math. Comp.*, v. 20, 1966, pp. 444-448.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0436547-7

Keywords:
Euler's constant,
Mascheroni's constant,
gamma,
rational approximation,
regular continued fractions,
multiple-precision arithmetic,
arithmetic-geometric mean,
Khintchine's law,
Lévy's law,
Gauss-Kusmin law

Article copyright:
© Copyright 1977
American Mathematical Society