The evaluation of is considerably more difficult. By contrast, for , not even a quadratically convergent algorithm is known. Knuth obtained 1271 digits of Euler's constant in 1962, using Euler-Maclaurin summation. They also calculated the first 29200 partial quotients in the regular continued fraction expansion for and deduced that if is a rational number, then its integer denominator must exceed . This is compelling evidence that Euler's constant is not rational.
Progress continues in computing .
Ekatherine Karatsuba was so kind to send me a number of her papers. Fast algorithms like her FEE method and Brent's binary splitting method were used subsequently to compute 1000000 digits of Euler's constant. Plouffe gave a highly accurate approximation of in the Inverse Symbolic Calculator web pages. See also Gourdon and Sebah's essay from Numbers, Constants and Computation . Return to the MathSoft Constants page .