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Simultaneous Polynomial Approximations of the Lerch Function |
Canad. J. Math. 61(2009), 1341-1356
Published:2009-12-01
Printed: Dec 2009
Printed: Dec 2009
- Tanguy Rivoal
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 |