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# Simultaneous Polynomial Approximations of the Lerch Function

Published:2009-12-01
Printed: Dec 2009
• Tanguy Rivoal
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## Abstract

We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite--Pad\'e approximants either of the polylogarithms or of Hurwitz zeta functions. In the former case, we recover known results, while in the latter the results are new and generalise some recent works of Beukers and Pr\'evost. Finally, we make a detailed comparison of our work with Beukers'. Such constructions are useful in the arithmetical study of the values of the Riemann zeta function at integer points and of the Kubota--Leopold $p$-adic zeta function.
 MSC Classifications: 41A10 - Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 41A21 - Pade approximation 11J72 - Irrationality; linear independence over a field