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

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.