On the sign of the real part of the Riemann zeta-function

Abstract

We consider the distribution of $\arg\zeta(\sigma+it)$ on fixed lines $\sigma > \frac12$, and in particular the density $$d(\sigma) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\arg\zeta(\sigma+it)| > \pi/2\}|\,,$$ and the closely related density $$d_{-}(\sigma) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: \Re\zeta(\sigma+it) < 0\}|\,.$$ Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function $\psi_\sigma(x)$ associated with $\arg\zeta(\sigma+it)$. We give explicit expressions for $d(\sigma)$ and $d_{-}(\sigma)$ in terms of $\psi_\sigma(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(\sigma)$ and $d_{-}(\sigma)$.Comment: 22 pages, 3 tables. To appear in Proceedings of the International Number Theory Conference in Memory of Alf van der Poorten (Newcastle, Australia, 2011