## 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).

## Abstract

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*| > 10^{15000}.
The computations were performed using the first author's
MP package.

## Errata

On page 310, in the formula for *V*_{p}(z) after
equation (17),
(*z/k*!)^{p} should be
(*z*^{k}/k!)^{p}.
On page 312, reference 14, replace "54-61" by "55-62".

## Comments

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
*I*_{0}(2n)K_{0}(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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