On the Sign of the Real Part of the Riemann Zeta Function

We consider the distribution of argζ (σ + it) on fixed lines σ > 1 2 , and in particular the density d(σ) = lim T→+∞ 1 2T |{t ∈ [−T,+T ] : |argζ (σ + it)|> π/2}| , and the closely related density d−(σ) = lim T→+∞ 1 2T |{t ∈ [−T,+T ] : Rζ (σ + it) < 0}| . Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function ψσ (x) associated with argζ (σ + it). We give explicit expressions for d(σ) and d−(σ) in terms of ψσ (x). Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of d(σ) and d−(σ).