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On linear independence of values of generalized polylogarithms. (English. Russian original) Zbl 1023.11035
Sb. Math. 192, No. 8, 1225-1239 (2001); translation from Mat. Sb. 192, No. 8, 139-154 (2001).
This paper investigates the arithmetical nature of the values of the multipolylogarithmic functions, defined by the series $\text{Li}_{\underline{s}}(z) =\sum_{ n_1>n_2>\cdots>n_t\geq 1 } \frac{z^{n_1}}{n_1^{s_1}n_2^{s_2}\cdots n_t^{s_t}},$ where $$\underline{s}=(s_1, s_2, \ldots, s_t)$$ is any $$t$$-tuple of integers $$\geq 1$$, and $$z$$ a complex number such that $$|z|<1$$. Given an integer $$r\geq 1$$, the number of such series with $$1\leq t\leq r$$ and $$s_1+s_2+\cdots+s_t\leq r$$ is $$N_r=2^{r}-1$$: we shall denote them by $$f_1, f_2,\ldots, f_{N_r}$$, and we add to this list the function $$f_0\equiv 1$$. The author then proves the following result (which he gives for another set of functions, related to the $$\text{Li}_{\underline{s}}(z)$$ by a triangular linear system).
Theorem: If $$p/q$$ is a rational number such that $$|p/q |>(2qe^r)^{N_r}$$, then the real numbers $$f_{j}(p/q)$$ (for $$j=0, 1, \ldots, N_r$$) are linearly independent over the rationals. Furthermore, for all rational $$a_j$$ ($$j=0, 1, \ldots, N_r$$) such that $$a=\max(|a_i|: j=0, 1, \cdots,N_r) >0,$$ we have $\left|\sum_{j=0}^{N_r}a_jf_j(p/q)\right |\geq \frac{c}{a^{\mu}},$ where $$c>0$$ is an effectively computable constant and $\mu=\frac{\log|p|+ r+4} {(1/N_r)\log|p/q|-\log |q|-r -1}.$ The method is based on the explicit construction of diagonal Hermite-Padé approximants of type II of a certain Angelesco-Nikishin system: as the author points out, the exact asymptotical behavior of these approximations is completely described by the divisors of certain meromorphic functions on a related Riemann surface. Here, he uses a more direct approach, with estimations similar to those of Nikishin, Hata and others. Although the theorem can not be applied to $$z=1$$, the author’s techniques might help to prove a similar result for the polyzeta numbers $$\zeta(s_1, s_2, \ldots, s_t)=\text{Li}_{\underline{s}}(1)$$, or at least to give a nontrivial lower bound for the dimension of the linear space that these numbers span over the rationals (in the spirit of the Kontsevich-Zagier conjecture).

##### MSC:
 11J72 Irrationality; linear independence over a field 11G55 Polylogarithms and relations with $$K$$-theory
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