Mathematics – Number Theory
Scientific paper
2005-07-12
Pr\~A\c{opyright}vost, Marc Legendre modified moments for Euler's constant. J. Comput. Appl. Math. 219 (2008), no. 2, 484--492
Mathematics
Number Theory
Scientific paper
Following earlier results of Sondow, we propose another criterion of irrationality for Euler's constant $\gamma$. It involves similar linear combinations of logarithm numbers $L\_{n,m}$. To prove that $\gamma$ is irrational, it suffices to prove that, for some fixed $m$, the distance of $d\_n L\_{n,m}$ ($d\_n$ is the least common multiple of the $n$ first integers) to the set of integers $\mathbf{Z}$ does not converge to 0. A similar result is obtained by replacing logarithms numbers by rational numbers: it gives a sufficient condition involving only rational numbers. Unfortunately, the chaotic behavior of $d\_n$ is an obstacle to verify this sufficient condition. All the proofs use in a large manner the theory of Pad\'e approximation.
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