×

Integral identities and constructions of approximations to zeta-values. (English) Zbl 1090.11047

Apéry’s proof of the irrationality of \(\zeta(3)\) uses rational approximations \(u_n/v_n\) derived from a recurrence relation. Simpler approaches were later found using integral representations by Beukers \[ \int_0^1\int_0^1\int_0^1{x^n(1-x)^ny^n(1-y)^nz^n(1-z)^n\over (1-(1-xy)z)^{n+1}}\,dx\,dy\,dz = 2(v_n\zeta(3)-u_n) \] and Nesterenko \[ {1\over 2\pi i}\int_C{\Gamma^4(-s)\Gamma^2(n+1+s)\over \Gamma^2(n+1-s)}\,ds = 2(v_n\zeta(3)-u_n) \] (where \(C\) is the vertical straight line from \(-{1\over2}-i\infty\) to \(-{1\over2}+i\infty\)). This paper establishes an identity between multidimensional generalisations of such pairs of integrals. The integrals are then connected to the construction of linear forms in the polylogarithms \(L_k(z)=\sum_{\nu=1}^\infty z^\nu/\nu^k\) and since \(L_k(1)=\zeta(k)\) this leads to small linear forms in the values of the \(\zeta\)-function at integer points. The constructions therefore give explicit integral representations of the linear forms.

MSC:

11J72 Irrationality; linear independence over a field
11G55 Polylogarithms and relations with \(K\)-theory
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Beukers, F., A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc.11 (1979), 268-272. · Zbl 0421.10023
[2] Gutnik, L.A., The irrationality of certain quantities involving ζ(3). Acta Arith.42 (1983), 255-264. · Zbl 0474.10026
[3] Luke, Yu L., Mathematical functions and their approximations. Academic Press, New York, 1975. · Zbl 0318.33001
[4] Nesterenko, YU., A few remarks on ζ(3). 59 (1996), 625-636. · Zbl 0888.11028
[5] Hessami, T.- Pilerhood, Linear independence of vectors with polylogarithmic coordinates. Vestnik Moscow University Ser.1 (1999), no6, 54-56. · Zbl 0983.11044
[6] Rivoal, T., La fonction Zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris331 (2000), 267-270. · Zbl 0973.11072
[7] Slater, L.J., Generalized hypergeometric functions. Cambridge Univ. Press, 1966. · Zbl 0135.28101
[8] Whittaker, E.T., Watson, G.N., A course of modern analysis. CambdidgeUniversity Press, 1927. · JFM 53.0180.04
[9] Zudilin, V.V., On irrationality of values of Riemann zeta function. Izvestia of Russian Acad. Sci.66, 2002, 1-55. · Zbl 1114.11305
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.