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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).

11J72 Irrationality; linear independence over a field
11G55 Polylogarithms and relations with \(K\)-theory
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