## 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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