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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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