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

49. R. P. Brent and E. M. McMillan, Some new algorithms for high-precision computation of Euler's constant, Mathematics of Computation 34 (1980), 305-312. MR 82g:10002.

Abstract: dvi (3K), pdf (80K), ps (28K),

Paper: pdf (617K).


We describe several new algorithms, more efficient than those of [40], for the high-precision computation of Euler's constant gamma = 0.577...

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 |Q| > 1015000. The computations were performed using the first author's MP package.


On page 310, in the formula for Vp(z) after equation (17), (z/k!)p should be (zk/k!)p.

On page 312, reference 14, replace "54-61" by "55-62".


It is not known whether gamma or exp(gamma) is rational; hence the interest in lower bounds on Q.

A nice introductory paper on Euler's constant by Gourdon and Sebah is available here. In 1999, Demichel and Gourdon used formula (13) of [49] to compute Euler's constant to 108,000,000 decimal digits. In December 2006, Yee found 116,580,041 decimal digits using the same formula (evaluated with binary splitting). This and other record computations are summarised here.

More recently, Richard Kreckel found 900,000,000 decimal digits and Shigeru Kondo found 10,000,000,000 decimal digits.

As of June 2010 the record seems to be 29,844,489,545 decimal digits by Alexander J. Yee.

Formula (13) of [49] is implemented in Zimmermann's MPFR package.

A rigorous error bound for the asymptotic expansion (14) of I0(2n)K0(2n), following the suggestion given on slides 24-26 of my July 2010 talk Ramanujan and Euler's constant (in memory of Edwin M. McMillan), is at [256] (joint with Fredrik Johansson).

An interesting connection with the work of Ramanujan is described in Brent [139].

McMillan is better known for the discovery of neptunium and plutonium: see Jackson and Panofsky, "Edwin Mattison McMillan 1907-1991", Biographical Memoirs of the National Academy of Sciences (USA), 69 (1996).

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