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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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