Some bounds and limits in the theory of Riemann's zeta function
  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. 
J. Arias de Reyna (University of Seville), J. van de Lune (formerly at   CWI)
math.NT math.CA