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One computational approach in support of the Riemann hypothesis. (English) Zbl 0937.11060

The authors apply criteria for the existence of an analytic continuation into a domain of a function given on a part of the boundary to the Riemann zeta-function. From such criteria, it is shown that the Riemann Hypothesis is valid if certain explicitly defined sequences have the limiting value 1. This forms a basis for numerical computations to test the hypothesis, and the results of such experiments are reported.

MSC:

11Y35 Analytic computations
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses

Software:

Mathematica
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Full Text: DOI

References:

[1] van de Lune, J.; te Riele, H. J.J.; Winter, D. T., On the zeroes of the Riemann zeta function in the critical strip IV, Math. Comp., 46, 667-681 (1986) · Zbl 0585.10023
[2] Aizenberg, L. A.; Kutmanov, A. M., On the possibility of holomorphic extension into a domain of functions defined on a connected piece of its boundary, Math. USSR Sbornik, 72, 2, 467-483 (1992) · Zbl 0782.30003
[3] Aizenberg, L., Carleman’s Formulas in Complex Analysis (1993), Kluwer Acad: Kluwer Acad Dozdrecht
[4] Aizenberg, L., Carleman’s Formulas and Conditions of Analytic Extendability (1995), Banach Center · Zbl 0827.32009
[5] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function (1951), Clarendon Press: Clarendon Press Oxford · Zbl 0042.07901
[6] Varda, R., Theoretical and computational aspects of the Riemann hypothesis, (Scientific Computations on Mathematical Problems and Conjectures (1990), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA), Chapter 3
[7] Wolfram, S., Mathematica: A System for Doing Mathematics by Computer (1994), Addison-Wesley
[8] Odlyzko, A. M., The \(10^{20}\) zero of the Riemann zeta function and its neighbors (1989), (preprint)
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