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

Juan Arias de Reyna; Richard P. Brent; Jan van de Lune

doi:10.1007/978-1-4614-6642-0_3

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

Juan Arias de Reyna, Richard P. Brent, Jan van de Lune

Research output: Chapter in Book/Report/Conference proceeding › Conference Paper › peer-review

Abstract

We consider the distribution of the argument of the Riemann zeta function on vertical lines with real part greater than 1/2, and in particular two densities related to the argument and to the real part of the zeta function on such lines. Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function associated with the argument. We give explicit expressions for the densities in terms of this characteristic function. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of the densities.

Original language	English
Title of host publication	Number Theory and Related Fields
Subtitle of host publication	In Memory of Alf van der Poorten
Publisher	Springer New York LLC
Pages	75-97
Number of pages	23
ISBN (Print)	9781461466413
DOIs	https://doi.org/10.1007/978-1-4614-6642-0_3
Publication status	Published - 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	43
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

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10.1007/978-1-4614-6642-0_3

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title = "On the Sign of the Real Part of the Riemann Zeta Function",

abstract = "We consider the distribution of the argument of the Riemann zeta function on vertical lines with real part greater than 1/2, and in particular two densities related to the argument and to the real part of the zeta function on such lines. Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function associated with the argument. We give explicit expressions for the densities in terms of this characteristic function. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of the densities.",

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On the Sign of the Real Part of the Riemann Zeta Function. / de Reyna, Juan Arias; Brent, Richard P.; van de Lune, Jan.
Number Theory and Related Fields: In Memory of Alf van der Poorten. Springer New York LLC, 2013. p. 75-97 (Springer Proceedings in Mathematics and Statistics; Vol. 43).

Research output: Chapter in Book/Report/Conference proceeding › Conference Paper › peer-review

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AB - We consider the distribution of the argument of the Riemann zeta function on vertical lines with real part greater than 1/2, and in particular two densities related to the argument and to the real part of the zeta function on such lines. Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function associated with the argument. We give explicit expressions for the densities in terms of this characteristic function. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of the densities.

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

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