Numerical information processing under the global rule expressed by the Euler-Riemann ζ function defined in the complex plane.
When nonzero, the ζ function is intimately connected with numerical information processing. Two other functions play a key role, namely, η(s)=∑n≥1(-1)n+1/ns and λ(s)=∑(n≥0)1/(2n+1)s. The paper opens on a survey of some of the seminal work of Euler [Mémoires Acad. Sci., Berlin 1768, 83 (1749)] and of the amazing theorem by Voronin [Math. USSR, Izv. 9, 443 (1975)] Then, as a follow-up of Chatelin [Qualitative Computing. A Computational Journey into Nonlinearity (World Scientific, Singapore, in press)], we present a fresh look at the triple (η,ζ,λ) which suggests an elementary analysis based on the distances of the three complex numbers z, z/2, and 2/z to 0 and 1. This metric approach is used to contextualize any nonlinear computation when it is observed at a point describing a complex plane. The results applied to ζ, η, and λ shed a new epistemological light about the critical line. The suggested interpretation related to ζ carries computational significance.
Chatelin
Françoise
F
University of Toulouse and CERFACS, 42 Avenue Gaspard Coriolis, Toulouse, Cedex 31057, France.