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Mathematics of Computation

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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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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