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On linear independence of theta values. (English) Zbl 1066.11031

In this paper, the linear independence (over \(\mathbb{Q}\)) of the Tschakaloff function \[ T_q(z):=\sum_{n=0}^\infty z^n q^{-n(n-1)/2} \] is not only studied at distinct (arithmetically characterized) points \(z\), but also for different (a.c.) values of the parameter \(q\). More precisely, the main result can be described as follows.
Suppose \(\alpha_{i,j}\in \mathbb{Q}^\times\) for \(i=1,\dots ,\ell;j=1,\dots ,m_i\) satisfying \(\alpha_{i,j_1} /\alpha_{i,j_2} \not \in q^\mathbb{Z}\) for \(j_1 \neq j_2\), where \(q\in \mathbb{Z}, |q|>1\). Let \(L\in \mathbb{Z}, L\geq \ell-1\), and put \(m:=\sum_{i=1}^\ell m_i\). Then there is an (effectively computable) \(\gamma(L,m)\in \mathbb{R}_+\) such that, for any \(s_1,\dots ,s_\ell \in \mathbb{Z}\) with \(\gamma(L,m) <s_1<s_2<\dots<s_\ell \leq s_1 +L\) and for any \(\beta \in \mathbb{Q}^\times\), the \(m+1\) numbers \(1,f_{i,j}(\beta) \enspace (i,j\) as before) are linearly independent over \(\mathbb{Q}\). Here the entire transcendental functions \(f_{i,j}\) are defined by \(f_{i,j}(z):= T_{q^{s_i}}(\alpha_{i,j}(z/q)^{s_i})\). Several corollaries are given.
The proof of the above main result uses some ideas of K. Väänänen [Math. Ann. 325, 123–136 (2003; Zbl 1025.11023)] consisting in a tricky application of certain Padé type approximations of the second kind for the functions \(f_{i,j}\).

MSC:

11J72 Irrationality; linear independence over a field

Citations:

Zbl 1025.11023
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