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Some bounds and limits in the theory of Riemann's zeta function

 

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Opened Access Some bounds and limits in the theory of Riemann's zeta function
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Author: Arias de Reyna Martínez, Juan
Van de Lune, Jan
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2012-12-01
Published in: Journal of Mathematical Analysis and Applications, 396 (1), 199-214.
Document type: Article
Abstract: For any real a > 0 we determine the supremum of the real σ such that ζ(σ+it) = a for some real t. For 0 < a < 1, a = 1, and a > 1 the results turn out to be quite different. We also determine the supremum E of the real parts of the ‘turning points’, that is points σ + it where a curve Im ζ(σ + it) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real σ such that ζ 0 (σ + it) = 0 for some real t. We find a surprising connection between the three indicated problems: ζ(s) = 1, ζ 0 (s) = 0 and turning points of ζ(s). The almost extremal values for these three problems appear to be located at approximately the same height.
Cite: Arias de Reyna Martínez, J. y Van de Lune, J. (2012). Some bounds and limits in the theory of Riemann's zeta function. Journal of Mathematical Analysis and Applications, 396 (1), 199-214.
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URI: http://hdl.handle.net/11441/42131

DOI: http://dx.doi.org/10.1016/j.jmaa.2012.06.017

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