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 $\sigma >\frac{1}{2}$, and in particular the density \[d(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\argζ(σ+it)| > π/2\}|\,,\] and the closely related density \[d_{-}(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: \Reζ(σ+it) < 0\}|\,.\] 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 $d_{-}(\sigma )$.

arXiv comment22 pages, 3 tables. To appear in Proceedings of the International Number Theory Conference in Memory of Alf van der Poorten (Newcastle, Australia, 2011)

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• On the value-distribution of the Riemann zeta-function on the critical line
• X-Ray of Riemann zeta-function
• A note on the real part of the Riemann zeta-function
• A method for locating where the real part of the Riemann zeta function becomes negative for its real argument greater than one

Cited by in corpus (1)

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