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Irrationality of \(\gamma, \zeta(m)\) and \(\beta(m)\). (English) Zbl 1499.11242

Summary: We use the uniqueness of representation of every rational number as a finite sum of reciprocal factorials to any given power \(m\in\mathbb N^*\) to prove the irrationality of Euler’s constant \(\gamma\), of all values of Riemann’s zeta function \(\zeta(m)\), \(m\ge 2\), and of all values of Dirichlet’s beta function \(\beta(m)\) with \(\beta(2)\) corresponding to Catalan’s constant.

MSC:

11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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