A determinantal approach to irrationality. (English) Zbl 1367.11062

If we want to prove the irrationality of the real number \(\zeta\) then we usually consider sequences \(\{ p_n\}_{n=1}^\infty\) and \(\{ q_n\}_{n=1}^\infty\) of integers such that \(\lim_{n\to\infty}(q_n\zeta - p_n)=0\) where \(\zeta -\frac{ p_n}{q_n}\not= 0\) for all positive integers \(n\). The author of this interesting paper deals with the modification of this method and presents a new proof of the irrationality of the number \(\pi\) and other numbers. The method makes use of special integrals as well as properties of Hankel and Vandermonde determinants.


11J72 Irrationality; linear independence over a field
30C10 Polynomials and rational functions of one complex variable
Full Text: DOI arXiv


[1] Beukers, F.: A note on the irrationality of \[\zeta (2)\] ζ(2) and \[\zeta (3)\] ζ(3). Bull. Lond. Math. Soc. 11(3), 268-272 (1979) · Zbl 0421.10023
[2] Beukers, F.: A rational approach to \[\pi\] π. Nieuw archief voor wiskunde Ser. 5 1(4), 372-379 (2000) · Zbl 1173.11358
[3] Borwein, P.B., Pritsker, I.E.: The multivariate integer Chebyshev problem. Constr. Approx. 30(2), 299-310 (2009) · Zbl 1246.11075
[4] Brown, F.: Irrationality proofs for zeta values, moduli spaces and dinner parties. Preprint arXiv:1412.6508 [math.NT] (2014) · Zbl 1370.11103
[5] Heine, H.E.: Handbuch der Kugelfunktionen, 2nd edn, vol. 1. G. Reimer, Berlin (1878); vol. 2. G. Reimer, Berlin (1881) · Zbl 0103.29304
[6] Krattenthaler, C., Rochev, I., Väänänen, K., Zudilin, W.: On the non-quadraticity of values of the \[q\] q-exponential function and related \[q\] q-series. Acta Arith. 136(3), 243-269 (2009) · Zbl 1232.11080
[7] Krattenthaler, C., Rivoal, T., Zudilin, W.: Séries hypergéométriques basiques, \[q\] q-analogues des valeurs de la fonction zêta et formes modulaires. J. Inst. Math. Jussieu 5(1), 53-79 (2006) · Zbl 1089.11038
[8] Kronecker, L.: Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen. Berl. Monatsber. 1881, 535-600 (1881) · JFM 13.0114.02
[9] Luque, J.-G., Thibon, J.-Y.: Hankel hyperdeterminants and Selberg integrals. J. Phys. A 36(19), 5267-5292 (2003) · Zbl 1058.33019
[10] Monien, H.: Hankel determinants of Dirichlet series. Preprint arXiv:0901.1883 [math.NT] (2009) · Zbl 1232.11080
[11] Nesterenko, Y.V.: On Catalan’s constant. Chebyshevskiĭ Sb. (Tula State Pedagogical University) 16(1(53)), 118-124 (2015). (Russian)
[12] Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. II, Grundlehren Math. Wiss. 216. Springer, Berlin (1976) · Zbl 0338.00001
[13] Rivoal, T.: Nombres d’Euler, approximants de Padé et constante de Catalan. Ramanujan J. 11, 199-214 (2006) · Zbl 1152.11337
[14] Sorokin, V.N.: A transcendence measure for \[\pi^2\] π2. Sb. Math. 187(12), 1819-1852 (1996) · Zbl 0876.11035
[15] Zakharyuta, V.P.: Transfinite diameter, Chebyshev constants and capacity for a compactum in \[{\mathbb{C}}^nCn\]. Mat. Sb. (N.S.) 96(138), 374-389 (1975); English transl., Math. USSR-Sb. 25(3), 350-364 (1975) · Zbl 0324.32009
[16] Zudilin, W.: A few remarks on linear forms involving Catalan’s constant. Chebyshevskiĭ Sb. (Tula State Pedagogical University) 3, no. 2 (4), 60-70 (2002); English transl., arXiv:math/0210423 [math.NT] (2002) · Zbl 1099.11036
[17] Zudilin, W.: On the irrationality of generalized \[q\] q-logarithm. Preprint arXiv:1601.02688 [math.NT] (2016) · Zbl 1415.11099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.