## Linear independence of vectors with polylogarithmic coordinates.(English. Russian original)Zbl 0983.11044

Mosc. Univ. Math. Bull. 54, No. 6, 40-42 (1999); translation from Vestn. Mosk. Univ., Ser I 1999, No. 6, 54-56 (1999).
A construction that allows one to obtain joint approximations of values of polylogarithmic functions is described in the paper. Using these approximation forms, we establish linear independence over $$\mathbb{Q}$$ of unit vectors of the space $$\mathbb{R}^p$$ and $$p$$-dimensional vectors whose coordinates are values of polylogarithms at rational points.
Consider the function $$L_i(z)= \sum_{\kappa=1}^\infty \frac{z^\kappa} {\kappa^i}$$, $$i\geq 1$$.
Theorem 1. Let $$p\in \mathbb{N}$$, $$Q_p^-= \min\{q\in \mathbb{N}: e^{2p-1}< (A_{q,p-1}+ q^{1/p}+ \sqrt{2} q^{1/2p} B_{q,p-1})^p\}$$, where $$A_{q,p-1}= (1+ q^{2/p}- 2q^{1/p} \cos \frac{\pi}{p})^{1/2}$$, $$B_{q,p-1}= (q^{1/p}- \cos \frac{\pi}{p}+ A_{q,p-1})^{1/2}$$; $$\overline{a}_i= (C_{i-1}^{i-1}\cdot L_i(-1/q), C_i^{i-1}\times L_{i+1}(-1/q),\dots, C_{i+p-2}^{i-1}\cdot L_{i+p-1}(-1/q))\in \mathbb{R}^p$$, $$i=1,\dots, p$$. Then for any natural $$q\geq Q_p^-$$ no $$\mathbb{Q}$$-linear combination of the vectors $$\overline{a}_1,\dots, \overline{a}_p$$ lies in $$\mathbb{Q}^p$$.
Previously, a similar problem for $$m$$ vectors of the space $$\mathbb{R}^n$$ $\overline{a}_i= (L_i(a/b), L_{i+1}(a/b),\dots,L_{i+n-1}(a/b)),\;i=1,\dots,m,$ was considered by Gutnik. In our particular case $$(m=n=p)$$, the construction technique and estimates of linear approximating forms differ from the ones proposed by Gutnik. As a result, a better bound for the values $$Q_p^-$$ is obtained. Corollaries for the particular values $$p=2,3$$ are as follows:
Corollary 1. For any natural $$q$$, the vectors $\overline{a}_1= (L_1(-1/q),L_2(-1/q)),\;\overline{a}_2= (L_2(-1/q),2L_3(-1/q)),\;\overline{e}_1= (1,0),\;\overline{e}_2= (0,1)$ are linearly independent over $$\mathbb{Q}$$.
Corollary 2. For any natural $$q$$, one of the numbers $$L_3(-1/q)$$, $$L_2(-1/q)$$ is irrational.
Corollary 3. For any natural $$q\geq 4$$, there is an irrational number among the numbers $$L_5(-1/q)$$, $$L_4(-1/q)$$, $$L_3(-1/q)$$.

### MSC:

 11J72 Irrationality; linear independence over a field 11J91 Transcendence theory of other special functions 33B15 Gamma, beta and polygamma functions