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

For any real a>0a>0 we determine the supremum of the real σσ such that ζ(σ+it)=aζ(σ+it)=a for some real tt. For 01a>1 the results turn out to be quite different.We also determine the supremum EE of the real parts of the ‘turning points’, that is points σ+itσ+it where a curve Imζ(σ+it)=0 has a vertical tangent. This supremum EE (also considered by Titchmarsh) coincides with the supremum of the real σσ such that ζ′(σ+it)=0ζ′(σ+it)=0 for some real tt.We find a surprising connection between the three indicated problems: ζ(s)=1ζ(s)=1, ζ′(s)=0ζ′(s)=0 and turning points of ζ(s)ζ(s). The almost extremal values for these three problems appear to be located at approximately the same height.