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

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 $d_{-}(\sigma)$.

💡 Research Summary

The paper investigates the distribution of the argument and the sign of the real part of the Riemann zeta‑function on vertical lines to the right of the critical line, i.e. for σ > ½. Two densities are introduced:

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