Functiones et Approximatio Commentarii Mathematici

Multiple integrals and linear forms in zeta-values

Georges Rhin and Carlo Viola

Full-text: Open access


We define $n$-dimensional Beukers-type integrals over the unit hypercube. Using an $n$-dimensional birational transformation we show that such integrals are equal to suitable $n$-dimensional Sorokin-type integrals with linear constraints, and represent linear forms in $1, \zeta(2), \zeta(3), \dots, \zeta(n)$ with rational coefficients.

Article information

Funct. Approx. Comment. Math., Volume 37, Number 2 (2007), 429-444.

First available in Project Euclid: 18 December 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11J72: Irrationality; linear independence over a field
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

multiple integrals of rational functions values of the Riemann zeta-function birational transformations


Rhin, Georges; Viola, Carlo. Multiple integrals and linear forms in zeta-values. Funct. Approx. Comment. Math. 37 (2007), no. 2, 429--444. doi:10.7169/facm/1229619663.

Export citation


  • K.\thinspace Ball and T.\thinspace Rivoal, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 146 (2001), 193--207.
  • F.\thinspace Beukers, A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 11 (1979), 268--272.
  • J.\thinspace Cresson, S.\thinspace Fischler and T.\thinspace Rivoal, Séries hypergéométriques multiples et polyzêtas, Bull. Soc. Math. France, to appear.
  • S.\thinspace Fischler, Groupes de Rhin-Viola et intégrales multiples, J. Théor. Nombres Bordeaux 15 (2003), 479--534.
  • C.\thinspace Krattenthaler and T.\thinspace Rivoal, Hypergéométrie et fonction zêta de Riemann, Mem. Amer. Math. Soc., vol. 186 no. 875, March 2007.
  • Yu.\thinspace V.\thinspace Nesterenko, Integral identities and constructions of approximations to zeta-values, J. Théor. Nombres Bordeaux 15 (2003), 535--550.
  • G.\thinspace Rhin and C.\thinspace Viola, On a permutation group related to $\zeta(2)$, Acta Arith. 77 (1996), 23--56.
  • G.\thinspace Rhin and C.\thinspace Viola, The group structure for $\zeta(3)$, Acta Arith. 97 (2001), 269--293.
  • T.\thinspace Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris, Sér. I Math., 331 (2000), 267--270.
  • V.\thinspace Kh.\thinspace Salikhov and A.\thinspace I.\thinspace Frolovichev, On multiple integrals represented as a linear form in $1$, $\zeta(3)$, $\zeta(5)$, $\dots$, $\zeta(2k-1)$, Fundam. Prikl. Mat. 11 no. 6 (2005), 143--178 (russian).
  • V.\thinspace N.\thinspace Sorokin, On the measure of transcendency of the number $\pi^2$, Sb. Math. 187 no. 12 (1996), 1819--1852.
  • V.\thinspace N.\thinspace Sorokin, Apéry's theorem, Moscow Univ. Math. Bull. 53 no. 3 (1998), 48--52.
  • D.\thinspace V.\thinspace Vasilyev, Approximations of zero by linear forms in values of the Riemann zeta-function, Dokl. Belarus Acad. Sci. 45 no. 5 (2001), 36--40 (russian).
  • C.\thinspace Viola, Birational transformations and values of the Riemann zeta-function, J. Théor. Nombres Bordeaux 15 (2003), 561--592.
  • C.\thinspace Viola, The arithmetic of Euler's integrals, Riv. Mat. Univ. Parma (7) 3* (2004), 119--149.
  • S.\thinspace A.\thinspace Zlobin, Properties of the coefficients of some linear forms of generalized polylogarithms, Fundam. Prikl. Mat. 11 no. 6 (2005), 41--58 (russian).
  • W.\thinspace Zudilin, Well-poised hypergeometric service for diophantine problems of zeta values, J. Théor. Nombres Bordeaux 15 (2003), 593--626.
  • W.\thinspace Zudilin, Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux 16 (2004), 251--291. \endbibliography