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

TL;DR

This paper analyzes the distribution of the argument of the Riemann zeta-function on lines with real part greater than 1/2, providing explicit formulas and algorithms to compute the density of negative real parts and argument magnitudes.

Contribution

It derives explicit formulas for the distribution densities of the argument and real part of the zeta-function, and develops algorithms for their numerical evaluation.

Findings

Explicit expression for the characteristic function of arg ζ(σ+it)

Formulas for densities d(σ) and d_{-}(σ) in terms of the characteristic function

Practical algorithms for accurate numerical computation of these densities

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 $d_{-}(\sigma)$.