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

J. Arias de Reyna; J. van de Lune

doi:10.1016/j.jmaa.2012.06.017

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arXiv · 1107.5134

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

J. Arias de Reyna·J. van de Lune

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Abstract

For any real a>0 we determine the supremum of the real \sigma\ such that \zeta(\sigma+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 \sigma+it where a curve Im \zeta(\sigma+it) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real \sigma\ such that \zeta'(\sigma+it) = 0 for some real t. We find a surprising connection between the three indicated problems: \zeta(s) = 1, \zeta'(s) = 0 and turning points of \zeta(s). The almost extremal values for these three problems appear to be located at approximately the same height.

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J. Arias de Reyna, J. van de Lune. 2011-07-26. Some bounds and limits in the theory of Riemann's zeta function. https://doi.org/10.1016/j.jmaa.2012.06.017

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