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

TL;DR

This paper investigates bounds related to the Riemann zeta function, determining supremums of certain values and revealing surprising connections between zeros, turning points, and specific function values.

Contribution

It introduces new bounds and relationships among the values, zeros, and turning points of the Riemann zeta function, highlighting their near coincidence at certain heights.

Findings

Supremum of \sigma where \zeta(\sigma+it) = a for various a values.

Supremum of real parts of turning points of \zeta(s).

Connection between zeros, turning points, and specific function values.

Abstract

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.