## 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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### References:

 [1] Apéry (R.), Irrationalité de $$\zeta(2)$$ et $$\zeta(3)$$, Astérisque 61 (1979), 11-13. · Zbl 0401.10049 [2] Peter Bundschuh and Keijo Väänänen, Arithmetical investigations of a certain infinite product, Compositio Math. 91 (1994), no. 2, 175 – 199. · Zbl 0802.11027 [3] Peter Bundschuh and Wadim Zudilin, Rational approximations to a \?-analogue of \? and some other \?-series, Diophantine approximation, Dev. Math., vol. 16, SpringerWienNewYork, Vienna, 2008, pp. 123 – 139. · Zbl 1213.11146 [4] Peter Bundschuh and Wadim Zudilin, Irrationality measures for certain \?-mathematical constants, Math. Scand. 101 (2007), no. 1, 104 – 122. · Zbl 1153.11034 [5] George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. · Zbl 1129.33005 [6] Jouhet (F.) and Mosaki (E.), Irrationalité aux entiers impairs positifs d’un $$q$$-analogue de la fonction zêta de Riemann, Int. J. Number Theory, to appear. · Zbl 1204.11110 [7] Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. · Zbl 0553.10019 [8] C. Krattenthaler and T. Rivoal, On a linear form for Catalan’s constant, South East Asian J. Math. Math. Sci. 6 (2008), no. 2, 3 – 15. · Zbl 1217.11070 [9] C. Krattenthaler, T. Rivoal, and W. Zudilin, Séries hypergéométriques basiques, \?-analogues des valeurs de la fonction zêta et séries d’Eisenstein, J. Inst. Math. Jussieu 5 (2006), no. 1, 53 – 79 (French, with English and French summaries). · Zbl 1089.11038 [10] Nesterenko (Yu. V.), On the linear independance of numbers (in Russian) Vest. Mosk. Univ., Ser. I, no. 1 (1985), 46-54; English transl. in Mosc. Univ. Math. Bull. 40.1 (1985), 69-74. [11] Yu. V. Nesterenko, Modular functions and transcendence questions, Mat. Sb. 187 (1996), no. 9, 65 – 96 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 9, 1319 – 1348. · Zbl 0898.11031 [12] Tanguy Rivoal, Nombres d’Euler, approximants de Padé et constante de Catalan, Ramanujan J. 11 (2006), no. 2, 199 – 214 (French, with French summary). · Zbl 1152.11337 [13] T. Rivoal and W. Zudilin, Diophantine properties of numbers related to Catalan’s constant, Math. Ann. 326 (2003), no. 4, 705 – 721. · Zbl 1028.11046 [14] Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. · Zbl 0889.05001 [15] Walter Van Assche, Little \?-Legendre polynomials and irrationality of certain Lambert series, Ramanujan J. 5 (2001), no. 3, 295 – 310. · Zbl 1035.11032
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