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Linear independence of values of polylogarithms. (Indépendance linéaire des valeurs des polylogarithmes.) (French) Zbl 1079.11038
The polylogarithms are \(\text{Li}_s(z)=\sum_{k=1}^\infty z^k/k^s\). Let \(a\geq2\) be an integer and \(\alpha=p/q\) be a rational with \(0<| \alpha| <1)\). Let \(\delta_\alpha(a)=\dim_{\mathbb Q} ({\mathbb Q}+{\mathbb Q} \text{Li}_1(\alpha)+\cdots+{\mathbb Q}\text{Li}_a(\alpha))\). For every \(\varepsilon>0\), there is a constant \(A=A(\varepsilon, p,q)\) such that if \(a\geq A\geq1\) then \(\delta_\alpha(a)\geq {1-\varepsilon\over 1+\log2}\log a\). So the \(\text{Li}_s(\alpha)\) with \(s=1,2,\ldots\) contain infinitely many \({\mathbb Q}\)-linearly independent numbers (and infinitely many irrationals). The proof rests on properties of the nearly-poised hypergeometric functions \[ N_{n,a,r}(z) = n!^{a-r} \sum_{k=1}^\infty {(k-1)(k-2)\cdots(k-rn)\over k^a(k+1)^a\cdots(k+n)^a} z^{-k} \] and Nesterenko’s criterion for linear independence. Since \(\text{Li}_s(1)=\zeta(s)\), this is an interesting complement to Rivoal’s remarkable theorem that infinitely many \(\zeta(2n+1)\) are irrational.

MSC:
11J72 Irrationality; linear independence over a field
11M41 Other Dirichlet series and zeta functions
33B15 Gamma, beta and polygamma functions
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References:
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