Some bounds and limits in the theory of Riemann's zeta function
arXiv:1107.5134 · doi:10.1016/j.jmaa.2012.06.017
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'(σ+it) = 0 for some real t. We find a surprising connection between the three indicated problems: ζ(s) = 1, ζ'(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.
arXiv comment28 pages 1 figure