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            <title xml:lang="en">Values of E-functions are not Liouville numbers</title>
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              <p>Shidlovskii has given a linear independence measure of values of E-functions with rational Taylor coefficients at a rational point not a singularity of the underlying differential system satisfied by these E-functions. His measure holds as well for E-functions with coefficients in an imaginary quadratic field, but not for other number fields. Recently, Beukers has proved a remarkable qualitative linear independence theorem for the values at an algebraic point of E-functions with arbitrary algebraic Taylor coefficients. But no quantitative analogue of Shidlovskii's measure has been given in Beukers' setting. The goal of this paper it to obtain such a measure, in an even more general setting where the point can be a singularity. This enables us to solve a long standing problem: the value of an E-function at an algebraic point is never a Liouville number, a result which had been obtained before only under additional assumptions. We deduce various explicit irrationality measures, in particular for values of the exponential and Bessel's J0 function at non-zero algebraic points. We also prove that the values at rational points of E-functions with rational Taylor coefficients are linearly independent over the field of algebraic numbers if and only if they are linearly independent over Q. Our methods rest upon improvements of results recently obtained by André and Beukers by means of the theory of E-operators.</p>
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