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


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

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Author: Arias de Reyna Martínez, Juan
Brent, Richard P.
Van de Lune, Jan
Coordinator/Director: Borwein, Jonathan M.
Shparlinski, Igor
Zudilin, Wadim
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2013
Published in: Number Theory and Related Fields
ISBN/ISSN: 9781461466413
Document type: Chapter of Book
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 ∈ [−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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DOI: 10.1007/978-1-4614-6642-0 3

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