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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
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