Chapter of Book
On the sign of the real part of the Riemann zeta-function
Author/s | Arias de Reyna Martínez, Juan
Brent, Richard P. Lune, Jan van de |
Editor | Borwein, Jonathan M.
Shparlinski, Igor Zudilin, Wadim |
Department | Universidad de Sevilla. Departamento de Análisis Matemático |
Publication Date | 2013 |
Deposit Date | 2016-10-19 |
Published in |
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ISBN/ISSN | 9781461466413 9781461466420 2194-1009 2194-1017 |
Abstract | 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 ∈ ... 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). |
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