On the sign of the real part of the Riemann zeta-function
This work provides a theoretical and computational tool for understanding the distribution of the Riemann zeta-function's real part, relevant to number theory.
The authors derive explicit expressions for the densities of sign changes of the real part of the Riemann zeta-function on fixed lines σ > 1/2, using classical results of Bohr and Jessen, and provide a practical algorithm for numerical evaluation.
We consider the distribution of $\argζ(σ+it)$ on fixed lines $σ> \frac12$, 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 $ψ_σ(x)$ associated with $\argζ(σ+it)$. We give explicit expressions for $d(σ)$ and $d_{-}(σ)$ in terms of $ψ_σ(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(σ)$ and $d_{-}(σ)$.