  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.
