On the complex magnitude of Dirichlet beta function
214 W Jennifer Ln, #6
Palatine, IL 60067, USA
E-mail: art.kawalec@gmail.com
Received:
Received: 10 February 2020; revised: 23 March 2020; accepted: 23 March 2020; published online: 29 March 2020
DOI: 10.12921/cmst.2020.0000005
Abstract:
In this article, we derive an expression for the complex magnitude of the Dirichlet beta function β(s) represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for β(s) valid for an even and odd kth positive integer argument and present a set of generated formulas for β(k) up to 11th order, including Catalan’s constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of β(s) valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.
Key words:
Dirichlet beta, Euler prime product, Euler-Mascheroni constant, Riemann zeta
References:
[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 9th printing, New York (1964).
[2] H.M. Edwards, Riemann’s Zeta Function, Dover Publications, Mineola, New York (1974).
[3] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford Science Publications (2008).
[4] A. Kawalec, Prime product formulas for the Riemann zeta function and related identities, math.GM/1901.09519v4 (2019).
[5] A. Kawalec, Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line, Computational Methods in Science and Technology 25(4), 161–166 (2019).
[6] LMFDB- The L-functions and Modular Forms Database, http://www.lmfdb.org/ (2019).
[7] A. Patkowski, M. Wolf, Some Remarks on Glaisher-Ramanujan Type Integrals, Computational Methods in Science and Technology 22(2), 103–108 (2016).
[8] M. Wolf, 6+ infinity new expressions for the Euler-Mascheroni constant, math.NT/1904.09855 (2019).
In this article, we derive an expression for the complex magnitude of the Dirichlet beta function β(s) represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for β(s) valid for an even and odd kth positive integer argument and present a set of generated formulas for β(k) up to 11th order, including Catalan’s constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of β(s) valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.
Key words:
Dirichlet beta, Euler prime product, Euler-Mascheroni constant, Riemann zeta
References:
[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 9th printing, New York (1964).
[2] H.M. Edwards, Riemann’s Zeta Function, Dover Publications, Mineola, New York (1974).
[3] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford Science Publications (2008).
[4] A. Kawalec, Prime product formulas for the Riemann zeta function and related identities, math.GM/1901.09519v4 (2019).
[5] A. Kawalec, Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line, Computational Methods in Science and Technology 25(4), 161–166 (2019).
[6] LMFDB- The L-functions and Modular Forms Database, http://www.lmfdb.org/ (2019).
[7] A. Patkowski, M. Wolf, Some Remarks on Glaisher-Ramanujan Type Integrals, Computational Methods in Science and Technology 22(2), 103–108 (2016).
[8] M. Wolf, 6+ infinity new expressions for the Euler-Mascheroni constant, math.NT/1904.09855 (2019).
