Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 

 

Euler's constant to $ 1271$ places


Author: Donald E. Knuth
Journal: Math. Comp. 16 (1962), 275-281
MSC: Primary 10.41
MathSciNet review: 0148255
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The value of Euler's or Mascheroni's constant

$\displaystyle \gamma \,=\,\underset{n\to \infty }{\mathop{\lim }}\,\,(1\,+\,\tfrac{1}{2}\,+\,\cdots \,+\,({1}/{n})\,-\,\ln \,n)$

has now been determined to 1271 decimal places, thus extending the previously known value of 328 places. A calculation of partial quotients and best rational approximations to $ \gamma $ was also made.

References [Enhancements On Off] (What's this?)

  • [1] L. Euler, ``De progressionibus harmonicis observationes,'' Euleri Opera Omnia Ser. 1, v. 14, Teubner, Leipzig and Berlin, 1925, p. 93-100.
  • [2] L. Euler, ``De summis serierum numeros Bernoullianos involventium,'' Euleri Opera Omnia Ser. 1, v. 15, Teubner, Leipzig and Berlin, 1927, p. 91-130. See especially p. 115. The calculation is given in detail on p. 569-583.
  • [3] J. W. L. Glaisher, ``On the calculation of Euler's constant,'' Proc. Roy. Soc. London, v. 19, 1870, p. 514-524.
  • [4] W. Shanks, ``On the numerical value of Euler's constant,'' Proc. Roy. Soc. London, v. 15, 1867, p. 429-432; v. 20, 1871, p. 29-34.
  • [5] J. W. L. Glaisher, ``History of Euler's constant,'' Messenger of Mathematics, v. 1, 1872, p. 25-30.
  • [6] J. C. Adams, ``On the value of Euler's constant,'' Proc. Roy. Soc. London, v. 27, 1878, p. 88-94. See also v. 42, 1887, p. 22-25.
  • [7] K. Knopp, Theory and Application of Infinite Series, Blackie and Son, London, 1951, p. 257.
  • [8] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708
  • [9] J. W. Wrench, Jr., ``A new calculation of Euler's constant,'' MTAC, v. 6, 1952, p. 255.
  • [10] David A. Pope and Marvin L. Stein, Multiple precision arithmetic, Comm. ACM 3 (1960), 652–654. MR 0116490
  • [11] Daniel Shanks and John W. Wrench Jr., Calculation of 𝜋 to 100,000 decimals, Math. Comp. 16 (1962), 76–99. MR 0136051, 10.1090/S0025-5718-1962-0136051-9
  • [12] D. J. Wheeler, The Calculation of 60,000 Digits of e by the Illiac, Digital Computer Laboratory Internal Report No. 43, University of Illinois, Urbana, 1953.
  • [13] John W. Wrench Jr., Further evaluation of Khintchine’s constant, Math. Comp. 14 (1960), 370–371. MR 0170455, 10.1090/S0025-5718-1960-0170455-1
  • [14] R. S. Lehman, A Study of Regular Continued Fractions, Ballistic Research Lab. Report 1066, Aberdeen Proving Ground, Maryland, February 1959.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10.41

Retrieve articles in all journals with MSC: 10.41


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1962-0148255-X
Article copyright: © Copyright 1962 American Mathematical Society