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# An infinite sequence of inequalities involving special values of the Riemann zeta function

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## Abstract

In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions. An interesting asymptotic formula related to the n-th complete homogeneous symmetric function is conjectured in this context.

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The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers. Keywords: Jacobi-Stirling numbers, Legendre-Stirling numbers, symmetric functions, combinatorial interpretations. 1
Elementary evaluation of ζ (2n)
• B C Berndt
B. C. Berndt, Elementary evaluation of ζ (2n), Math. Magazine, 48, 3 (1975), 148-154.
e-mail: mircea.merca@profinfo.edu.ro Corresponding Author: Mircea Merca
• Mircea Merca
Mircea Merca, Academy of Romanian Scientists, Splaiul Independentei 54, Bucharest, 050094 Romania e-mail: mircea.merca@profinfo.edu.ro Corresponding Author: Mircea Merca