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Hypergeometry inspired by irrationality questions. (English) Zbl 1450.11072

Let \(\zeta (k)=\sum_{n=1}^\infty \frac 1{n^k}\) be a value of Riemann’s zeta function. Then the authors prove that for any \(\lambda\in\mathbb R\), each of the sets
\[ \Bigl\{\zeta(2m+1)-\lambda \frac{2^{2m}(2^{2m+2}-1)\mid B_{2m+2}\mid}{(2^{2m+1}-1)(m+1)(2m)!} \pi^{2m+1};\quad m=1, \ldots, 19 \Bigr\}\]
and
\[ \Bigl\{\zeta(2m+1)-\lambda \frac{2^{2m}(2^{2m}-1)\mid B_{2m}\mid}{(2^{2m+1}-1)m(2m)!} \pi^{2m+1};\quad m=1,\ldots,21 \Bigr\}\]
contains at least one irrational number. Here \(B_{2m}\) denotes the \(2m\)-th Bernoulli number. The paper also includes some interesting identities concerning \(\log 2\), Catalan’s constant and \(\pi^2\).

MSC:

11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)

Software:

HYP
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References:

[1] W. N. Bailey. Generalized Hypergeometric Series (Cambridge Tracts in Mathematics, 32). Cambridge University Press, Cambridge, 1935.
[2] S. Fischler and W. Zudilin. A refinement of Nesterenko’s linear independence criterion with applications to zeta values. Math. Ann. 347(4) (2010), 739-763. · Zbl 1206.11088
[3] S. Fischler, J. Sprang and W. Zudilin. Many values of the Riemann zeta function at odd integers are irrational. C.R. Math. Acad. Sci. Paris 356(7) (2018), 707-711. · Zbl 1398.11109
[4] G. Gasper and M. Rahman. Basic Hypergeometric Series, 2nd edn (Encyclopedia of Mathematics and its Applications, 96). Cambridge University Press, Cambridge, 2004.
[5] M. Hata. Legendre type polynomials and irrationality measures. J. Reine Angew. Math. 407 (1990), 99-125. · Zbl 0692.10034
[6] Kh. Hessami Pilehrood and T. Hessami Pilehrood. On the irrationality of the sums of zeta values. Math. Notes 79(3-4) (2006), 561-571. · Zbl 1112.11035
[7] C. Krattenthaler. HYP and HYPQ—Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series. J. Symbol. Comput. 20 (1995), 737-744; the packages are freely available at http://www.mat.univie.ac.at/ kratt/.
[8] R. Marcovecchio. The Rhin-Viola method for log 2. Acta Arith. 139(2) (2009), 147-184. · Zbl 1197.11083
[9] M. Rahman and A. Verma. Quadratic transformation formulas for basic hypergeometric series. Trans. Amer. Math. Soc. 335(1) (1993), 277-302. · Zbl 0767.33011
[10] G. Rhin and C. Viola. On a permutation group related to ζ(2). Acta Arith. 77(3) (1996), 23-56. · Zbl 0864.11037
[11] T. Rivoal and W. Zudilin. Diophantine properties of numbers related to Catalan’s constant. Math. Ann. 326(4) (2003), 705-721. · Zbl 1028.11046
[12] .
[13] E. A. Rukhadze. A lower bound for the approximation of ln 2 by rational numbers. Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1987), no. 6, 25-29 (in Russian). · Zbl 0635.10025
[14] L. J. Slater. Generalized Hypergeometric Functions. Cambridge University Press, Cambridge, 1966. · Zbl 0135.28101
[15] C. Viola. Hypergeometric functions and irrationality measures. Analytic Number Theory (Kyoto, 1996) (London Mathematical Society Lecture Note Series, 247). Cambridge University Press, Cambridge, 1997, pp.353-360. · Zbl 0904.11020
[16] W. Zudilin. Irrationality of values of the Riemann zeta function. Izv. Math. 66(3) (2002), 489-542. · Zbl 1114.11305
[17] W. Zudilin. A few remarks on linear forms involving Catalan’s constant. Chebyshevskiĭ Sb. 3(2)(4) (2002), 60-70; Engl. transl., 2002, arXiv:math.NT/0210423. · Zbl 1099.11036
[18] W. Zudilin. An Apéry-like difference equation for Catalan’s constant. Electron. J. Combin. 10(1) (2003), R14, 10pp. · Zbl 1093.11075
[19] W. Zudilin. Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux 16(1) (2004), 251-291. · Zbl 1156.11327
[20] W. Zudilin. An essay on irrationality measures of π and other logarithms. Chebyshevskiĭ Sb. 5(2) (2004), 49-65; Engl. transl., 2004, arXiv:math.NT/0404523. · Zbl 1140.11036
[21] W. Zudilin. One of the odd zeta values from ζ(5) to ζ(25) is irrational. By elementary means. SIGMA 14 (2018), 028, 8pp. · Zbl 1445.11063
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