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Evaluating $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}L$ -functions with few known coefficients

Part of: Computational number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 June 2014

David W. Farmer  and
Nathan C. Ryan
David W. Farmer
Affiliation:
American Institute of Mathematics, 360 Portage Ave, Palo Alto, CA 94306, USA email farmer@aimath.org
Nathan C. Ryan
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA email nathan.ryan@bucknell.edu

Abstract

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We address the problem of evaluating an $L$ -function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the $L$ -function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

MSC classification

Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11Y35: Analytic computations

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 245 - 258
Copyright
© The Author(s) 2014 

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