Department of Mathematical and Statistical Sciences,
University of Nebraska at Omaha,
Omaha, NE, 68182,
USA

Department of Mathematics,
California State University, San Bernardino,
San Bernardino, CA, 92407,
USA
E-mail: suthakaran.ratnasingam@csusb.edu

Abstract:

In this paper, we establish new upper and lower bounds for the Dirichlet beta function β(x), with particular emphasis on its values at even integers. We derive inequalities that relate pairs of Dirichlet beta values and develop a unified approach that draws on tools from probability theory and reliability theory, together with analytic techniques, to obtain these results. Numerical comparisons illustrate the effectiveness of the proposed bounds relative to existing estimates; in several cases, our bounds are demonstrably sharper than those previously reported in the literature, including those of  [3].

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