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

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

2011-07-26

arxiv.org/abs/1107.5134v1

Mathematics

Number Theory

28 pages 1 figure

Scientific paper

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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