Log-tangent integrals and the Riemann zeta function

Log-tangent integrals and the Riemann zeta function

We show that integrals involving the log-tangent function, with respect to any square-integrable function on  , can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show a...

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Bibliographic Details
Main Authors: Lahoucine Elaissaoui, Zine El-Abidine Guennoun
Format: Article
Language:English
Published: Vilnius Gediminas Technical University 2019-06-01
Series:Mathematical Modelling and Analysis
Subjects:
Riemann zeta function
Hurwitz zeta function
Apéry’s constant
Dirichlet series
log-tangent integrals
harmonic series
Online Access:https://journals.vgtu.lt/index.php/MMA/article/view/7445
Description
Summary:We show that integrals involving the log-tangent function, with respect to any square-integrable function on  , can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series , where .
ISSN:1392-6292
1648-3510

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