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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, \({}_r\phi_s\)
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References:
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