On the Sign of the Real Part of the Riemann Zeta Function

Abstract

We consider the distribution of the argument of the Riemann zeta function on vertical lines with real part greater than 1/2, and in particular two densities related to the argument and to the real part of the zeta function on such lines. Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function associated with the argument. We give explicit expressions for the densities in terms of this characteristic function. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of the densities. © Springer Science+Business Media New York 2013.