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

2003/07/23 by Jonathan Sondow, Sondow, Jonathan
Mathematics · #11J82 (Primary) #11J86 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J82 #msc:11J86

paper · pdf · doi:10.48550/arxiv.math/0307308

12 pages, 1 figure, details of part of a talk at Journeés Arithmetiques XXIII in Graz

arxiv created 2003/07/23 · arxiv updated 2009/12/01

Abstract

The irrationality exponent μ(t) of an irrational number t, defined using the irrationality measure 1/q^μ, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure 1/β^q, we define the "irrationality base" β(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, γ. If γ is irrational and the condition turns out to be false in a certain strong sense, we prove an upper bound on μ(γ).

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