## 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

### Keywords:

Linear independence; Theta series; Chakalov function

Zbl 1025.11023
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