An irrationality measure for Liouville numbers and conditional measures for Euler's constant

Details An irrationality measure for Liouville numbers and conditional measures for Euler's constant An irrationality measure for Liouville numbers and conditional measures for Euler's constant

2003-07-23

arxiv.org/abs/math/0307308v1

Mathematics

Number Theory

12 pages, 1 figure, details of part of a talk at Journe\'es Arithme\'tiques XXIII in Graz

Scientific paper

The irrationality exponent $\mu(t)$ of an irrational number t, defined using the irrationality measure $1/q^\mu$, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure $1/\beta^q$, we define the "irrationality base" $\beta(t)$, which distinguishes among Liouville numbers and is 1 for non-Liouville numbers. We give some properties and examples. Assuming a condition on certain linear forms in logarithms, for which we present numerical evidence supplied by P. Sebah, we prove an upper bound on the irrationality base of Euler's constant, $\gamma$. If $\gamma$ is irrational and the condition turns out to be false in a certain strong sense, we prove an upper bound on $\mu(\gamma)$.

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