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
Published inMathematical modelling and analysis Vol. 24; no. 3; pp. 404 - 421
Main Authors Elaissaoui, Lahoucine, Guennoun, Zine El-Abidine
Format Journal Article
LanguageEnglish
Published Vilnius Vilnius Gediminas Technical University 06.06.2019
Subjects
Aircraft
Apéry’s constant
Dirichlet problem
Dirichlet series
harmonic series
Hurwitz zeta function
Integrals
log-tangent integrals
Mathematical functions
Mathematical research
Meromorphic functions
Riemann zeta function
Zeta functions
Morocco
Riemann zeta function
Dirichlet series
harmonic series
Apéry’s constant
Hurwitz zeta function
log-tangent integrals
Online AccessGet full text
ISSN1392-6292
1648-3510
DOI10.3846/mma.2019.025

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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 .  
Bibliography:ObjectType-Article-1
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ISSN:1392-6292
1648-3510
DOI:10.3846/mma.2019.025