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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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