Structural properties of Dirichlet series with harmonic coefficients
An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sum...
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| Published in | The Ramanujan journal Vol. 45; no. 2; pp. 569 - 600 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.02.2018
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1382-4090 1572-9303 |
| DOI | 10.1007/s11139-017-9906-5 |
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| Abstract | An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of
L
-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers. |
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| AbstractList | An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of
L
-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers. An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers. |
| Author | Alkan, Emre Göral, Haydar |
| Author_xml | – sequence: 1 givenname: Emre surname: Alkan fullname: Alkan, Emre email: ealkan@ku.edu.tr organization: Department of Mathematics, Koç University – sequence: 2 givenname: Haydar surname: Göral fullname: Göral, Haydar organization: Department of Mathematics, Koç University |
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| Keywords | Polylogarithm 11M41 Harmonic number 30B50 Functional equation 33B10 30D05 Dirichlet series Gauss sum Catalan’s constant |
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| References | Berndt (CR7) 1985 Ramanujan (CR20) 1957 Borwein, Zucker, Boersma (CR10) 2008; 15 Conway, Guy (CR12) 1996 Neukirch (CR19) 1999 Wang, Jia (CR22) 2014; 35 CR18 CR9 Davidson, Donsig (CR13) 2010 Jackson (CR16) 1994 Alkan (CR4) 2014; 410 Alkan (CR3) 2013; 7 Chen, Cheng (CR11) 2015; 38 Jordan (CR17) 1973; 79 Sitaramachandra (CR21) 1987; 25 Apostol, Vu (CR6) 1984; 19 De Doelder (CR14) 1991; 37 Wolstenholme (CR24) 1862; 5 Alkan (CR2) 2011; 26 Hardy, Wright (CR15) 1979 Wei, Gong (CR23) 2014; 34 Alkan (CR5) 2015; 143 Berndt (CR8) 1998 Alkan (CR1) 1994; 101 9906_CR9 E Alkan (9906_CR2) 2011; 26 KR Davidson (9906_CR13) 2010 E Alkan (9906_CR5) 2015; 143 9906_CR18 CP Chen (9906_CR11) 2015; 38 PJ Doelder De (9906_CR14) 1991; 37 GH Hardy (9906_CR15) 1979 J Neukirch (9906_CR19) 1999 C Wei (9906_CR23) 2014; 34 TM Apostol (9906_CR6) 1984; 19 JM Borwein (9906_CR10) 2008; 15 J Wolstenholme (9906_CR24) 1862; 5 BC Berndt (9906_CR7) 1985 BC Berndt (9906_CR8) 1998 R Sitaramachandra (9906_CR21) 1987; 25 W Wang (9906_CR22) 2014; 35 S Ramanujan (9906_CR20) 1957 PF Jordan (9906_CR17) 1973; 79 JH Conway (9906_CR12) 1996 E Alkan (9906_CR4) 2014; 410 E Alkan (9906_CR1) 1994; 101 E Alkan (9906_CR3) 2013; 7 D Jackson (9906_CR16) 1994 |
| References_xml | – year: 2010 ident: CR13 publication-title: Real Analysis and Applications. Theory in Practice. Undergraduate Texts in Mathematics doi: 10.1007/978-0-387-98098-0 – volume: 79 start-page: 681 year: 1973 end-page: 683 ident: CR17 article-title: Infinite sums of psi functions publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1973-13259-8 – ident: CR18 – year: 1999 ident: CR19 publication-title: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften – year: 1979 ident: CR15 publication-title: An Introduction to the Theory of Numbers – volume: 143 start-page: 3743 year: 2015 end-page: 3752 ident: CR5 article-title: Special values of the Riemann zeta function capture all real numbers publication-title: Proc. Am. Math. 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| SubjectTerms | Coefficients Combinatorics Dirichlet problem Field Theory and Polynomials Fourier Analysis Functional equations Functions (mathematics) Functions of a Complex Variable Mathematical analysis Mathematics Mathematics and Statistics Number Theory Real numbers Representations |
| Title | Structural properties of Dirichlet series with harmonic coefficients |
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