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

For any real a > 0 we determine the supremum of the real σ such that ζ ( σ + i t ) = a for some real t . For 0 < a < 1 , a = 1 , and a > 1 the results turn out to be quite different. o determine the supremum E of the real parts of the ‘turning points’, that is points σ + i t where a curve Im ζ ( σ + i t ) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real σ such that ζ ′ ( σ + i t ) = 0 for some real t . d 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.