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Heuristic quadratic approximation for the universality theorem

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Abstract

Voronin’s Universality Theorem states grosso modo, that any non-vanishing analytic function can be uniformly approximated by certain shifts of the Riemann zeta-function ζ(s). However, the problem of obtaining a concrete approximants for a given function is computationally highly challenging. The present note deals with this problem, using a finite number n of factors taken from the Euler product definition of ζ(s). The main result of the present work is the design and implementation of a sequential and a parallel heuristic method for the computation of those approximants. The main properties of this method are: (i) the computation time grows quadratically as a function of the quotient n/m, where m is the number of coefficients calculated in one iteration of the heuristic; (ii) the error does not vary significantly as m changes and is similar to the error of the exact algorithm.

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Author notes

  1. Rafael Plaza

    Present address: Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

Authors and Affiliations

  1. School of Industrial Engineering, University of Valparaíso, Valparaíso, Chile

    Gonzalo Hernández

  2. Departamento de Informática, Universidad Técnica Federico Santa María, Valparaíso, Chile

    Rafael Plaza & Luis Salinas

Authors
  1. Gonzalo Hernández
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  2. Rafael Plaza
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  3. Luis Salinas
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Corresponding author

Correspondence to Gonzalo Hernández.

Additional information

Partially supported by FONDECYT Grant 1100805, CONICYT Anillo Project ACT 119, CCTVal–Centro Científico Tecnológico de Valparaíso and NLHPC—National Laboratory for HPC.

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Hernández, G., Plaza, R. & Salinas, L. Heuristic quadratic approximation for the universality theorem. Cluster Comput 17, 281–289 (2014). https://doi.org/10.1007/s10586-013-0312-5

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  • DOI: https://doi.org/10.1007/s10586-013-0312-5

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