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Diophantine properties for \(q\)-analogues of Dirichlet’s beta function at positive integers. (English) Zbl 1217.11069

Let the \(q\)-analogue of Dirichlet’s beta function be defined by \[ \beta_q(s)= \sum_{k\geq 1}\,\sum_{d|k}\chi(k/d)\,d^{s-1}\,q^k, \] where \(s\) is a positive integer, \(q\) a complex number with \(|q|< 1\), and \(\chi\) is the nontrivial Dirichlet character modulo 4. The authors of this very interesting paper give a lower bound for the dimension of the vector space over \(\mathbb{Q}\) spanned by \(1,\beta_q(2),\beta_q(4),\dots, \beta_q(2m)\), where \(1/q\in\mathbb{Z}\setminus\{-1,1\}\) and \(m\) is a positive integer. One of the consequences of this theorem shows that at least one of the numbers \(\beta_q(2),\beta_q(3),\dots, \beta_q(20)\) is irrational, and that the sequence \((\beta_q(2m))\) contains infinitely many irrational numbers.

MSC:

11J72 Irrationality; linear independence over a field
11M41 Other Dirichlet series and zeta functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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