Structural properties of Dirichlet series with harmonic coefficients

• Published in: The Ramanujan journal Vol. 45; no. 2; pp. 569 - 600
• Main Authors: Alkan, Emre, Göral, Haydar
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2018; Springer Nature B.V
• Subjects: 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; Polylogarithm; 11M41; Harmonic number; 30B50; Functional equation; 33B10; 30D05; Dirichlet series; Gauss sum; Catalan’s constant
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-017-9906-5

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.