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2 Computational Methods in Science and Technology

2 Kawalec A.

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On the complex magnitude of Dirichlet beta function

Kawalec A.

Computational Methods in Science and Technology

2020

Vol. 26, No. 1

21--28

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.

Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line

Kawalec A.

Computational Methods in Science and Technology

2019

Vol. 25, No. 4

161--166

In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with log(k). We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.