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A lower bound of the dimension of the vector space spanned by the special values of certain functions. (English) Zbl 1430.11121
Summary: Let $$K$$ be a number field. Fix a finite set of analytic functions $$\mathbf{f}_{\infty}:=\{f_{1,\infty}(x),\dotsc,f_{s,\infty}(x) \}$$ defined on $$\{x\in \mathbb{C} \mid |x|>1\}$$ (resp. $$\mathbb{C}_p$$-valued functions $$\mathbf{f}_{p}:=\{f_{1,p}(x),\dotsc,f_{s,p}(x) \}$$ defined on $$\{x\in \mathbb{C}_p \mid |x|_p>1\}$$). For $$\beta\in K$$, we denote the $$K$$-vector space spanned by $$f_{1,\infty}(\beta),\dotsc,f_{s,\infty}(\beta)$$ by $$V_K(\mathbf{f}_{\infty},\beta)$$ (resp. $$f_{1,p}(\beta),\dotsc,f_{s,p}(\beta)$$ by $$V_K(\mathbf{f}_{p},\beta)$$). In this article, under some assumptions for $$\mathbf{f}_{\infty}$$ (resp. $$\mathbf{f}_{p}$$), we give an estimation of a lower bound of the dimension of $$V_K(\mathbf{f}_{\infty},\beta)$$ (resp. $$V_K(\mathbf{f}_{p},\beta)$$) (see Theorem 2.4 for Archimedean case and Theorem 8.6 for $$p$$-adic case). Applying our estimation, we give a lower bound of the dimension of the $$K$$-vector space spanned by the special values of the Lerch functions over a number field in $$\mathbb{C}$$ (see Theorem 1.1 and Remark 1.2) and the $$p$$-adic analog of the above result (see Theorem 1.3 and Remark 1.4). Furthermore, we also give a lower bound of the $$K$$-vector space spanned by the special values of certain $$p$$-adic functions related with $$p$$-adic Hurwitz zeta function (see Theorem 1.5).

##### MSC:
 11M35 Hurwitz and Lerch zeta functions 11J72 Irrationality; linear independence over a field
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