Constructive Approximation

, Volume 45, Issue 2, pp 301–310 | Cite as

A Determinantal Approach to Irrationality

  • Wadim Zudilin


It is a classical fact that the irrationality of a number \(\xi \in \mathbb R\) follows from the existence of a sequence \(p_n/q_n\) with integral \(p_n\) and \(q_n\) such that \(q_n\xi -p_n\ne 0\) for all n and \(q_n\xi -p_n\rightarrow 0\) as \(n\rightarrow \infty \). In this paper, we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement \(q_n\xi -p_n\rightarrow 0\) is weakened. Some applications are given, including a new proof of the irrationality of \(\pi \). Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.


Irrationality Rational approximation \(\pi \) Hankel determinant Fekete–Chebyshev constant Transfinite diameter 

Mathematics Subject Classification

Primary 11J72 Secondary 11C20 30C10 30E05 33C47 



There are several inspirations for this project, the most recent one being the work [4] of Francis Brown and, in particular, his remark there: “Much more optimistically still, one might hope to prove the transcendence of \(\zeta (3)\) by optimizing our polynomial forms in \(\zeta (3)\) along the lines of [14].” The other sources of inspiration include my joint work [6] with Christian Krattenthaler, Igor Rochev, and Keijo Väänänen on (related) Hankel-determinant constructions for certain q-hypergeometric series, and also the work [10] of Hartmut Monien on (unrelated) Hankel determinants on the values of Riemann’s zeta function at positive integers. I thank all these colleagues as well as Igor Pritsker for numerous helpful chats about the topic of this project. Furthermore, I am very grateful to Stéphane Fischler, whose constructive feedback was crucial at several places of the preliminary version. Special thanks go to the anonymous referee of the journal for his healthy criticism. Part of the work was done during my visit to the Max Planck Institute for Mathematics, Bonn, in March–April 2015. I am thankful to the staff and guests of the institute for creating the unique “mathemagical” atmosphere for scientific performance.


  1. 1.
    Beukers, F.: A note on the irrationality of \(\zeta (2)\) and \(\zeta (3)\). Bull. Lond. Math. Soc. 11(3), 268–272 (1979)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beukers, F.: A rational approach to \(\pi \). Nieuw archief voor wiskunde Ser. 5 1(4), 372–379 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Borwein, P.B., Pritsker, I.E.: The multivariate integer Chebyshev problem. Constr. Approx. 30(2), 299–310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brown, F.: Irrationality proofs for zeta values, moduli spaces and dinner parties. Preprint arXiv:1412.6508 [math.NT] (2014)
  5. 5.
    Heine, H.E.: Handbuch der Kugelfunktionen, 2nd edn, vol. 1. G. Reimer, Berlin (1878); vol. 2. G. Reimer, Berlin (1881)Google Scholar
  6. 6.
    Krattenthaler, C., Rochev, I., Väänänen, K., Zudilin, W.: On the non-quadraticity of values of the \(q\)-exponential function and related \(q\)-series. Acta Arith. 136(3), 243–269 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Krattenthaler, C., Rivoal, T., Zudilin, W.: Séries hypergéométriques basiques, \(q\)-analogues des valeurs de la fonction zêta et formes modulaires. J. Inst. Math. Jussieu 5(1), 53–79 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kronecker, L.: Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen. Berl. Monatsber. 1881, 535–600 (1881)zbMATHGoogle Scholar
  9. 9.
    Luque, J.-G., Thibon, J.-Y.: Hankel hyperdeterminants and Selberg integrals. J. Phys. A 36(19), 5267–5292 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Monien, H.: Hankel determinants of Dirichlet series. Preprint arXiv:0901.1883 [math.NT] (2009)
  11. 11.
    Nesterenko, Y.V.: On Catalan’s constant. Chebyshevskiĭ Sb. (Tula State Pedagogical University) 16(1(53)), 118–124 (2015). (Russian)Google Scholar
  12. 12.
    Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. II, Grundlehren Math. Wiss. 216. Springer, Berlin (1976)Google Scholar
  13. 13.
    Rivoal, T.: Nombres d’Euler, approximants de Padé et constante de Catalan. Ramanujan J. 11, 199–214 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sorokin, V.N.: A transcendence measure for \(\pi ^2\). Sb. Math. 187(12), 1819–1852 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zakharyuta, V.P.: Transfinite diameter, Chebyshev constants and capacity for a compactum in \({\mathbb{C}}^n\). Mat. Sb. (N.S.) 96(138), 374–389 (1975); English transl., Math. USSR-Sb. 25(3), 350–364 (1975)Google Scholar
  16. 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)
  17. 17.
    Zudilin, W.: On the irrationality of generalized \(q\)-logarithm. Preprint arXiv:1601.02688 [math.NT] (2016)

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

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