Estimates for the number of zeros of some functions with algebraic Taylor coefficients
Let \(K\) be an algebraic number field of degree \(\kappa\), \(\Lambda_{1}\), \(\Lambda_{2}\), \(\Lambda_{3}\), \(C_{1}\), \(C_{2}\), \(C_{3}\) positive real numbers, \((a_{\nu})_{\nu\geq 0}\) a sequence of elements of \(K\) and \((q_{\nu})_{\nu\geq 0}\) a sequence of non zero elements in the ring of integers \(\mathbb Z_{K}\) of \(K\). For each \(\nu\geq 0\), denote by \(a_{\nu}^{[1]},\ldots,a_{\nu}^{[\kappa]}\) the conjugates of \(a_{\nu}\), with \(a_{\nu}^{[1]}=a_{\nu}\), and assume \[ |a_{\nu}|<\Lambda_{1}C_{1}^{\nu},\qquad \prod_{k=2}^{\kappa} \bigl|a_{\nu}^{[k]}\bigr|< \Lambda_{2}C_{2}^{\nu},\qquad q_{\nu}a_{\nu}\in\mathbb Z_{K}\quad\text{and }\quad |\text{N}_{K/\mathbb Q}( q_{\nu})|< \Lambda_{3}C_{3}^{\nu}, \] where \(\text{N}_{K/\mathbb Q}:K\rightarrow\mathbb Q\) denotes the norm map. For \(R\geq 0\) and for \(F\) an analytic function in the disk \(|z|\leq R\) of \(\mathbb C\), denote by \(N(F,R)\) the number (counting multiplicities) of zeros of \(F\) in the disk \(|z|\leq R\), so that \(N(F,0)\) is the multiplicity of zero of \(F\) at the origin. The first result deals with the entire function \(f(z)=\sum_{n\geq 0} a_{\nu}z^{\nu}/\nu!\) (compare with Siegel's \(E\)-functions, where the condition \(q_{\nu}a_{\nu}\in\mathbb Z_{K}\) is replaced by the stronger requirement \(q_{n}a_{\nu}\in\mathbb Z_{K}\) for \(0\leq \nu\leq n\)): for any \(R>0\) the estimate \[ \bigl(N(f,R)-N(f,0)\bigr) \log\left(2+{N(f,0)\over C_{1}R}\right) \leq N(f,0)\bigl(1+\log(C_{1}C_{2}C_{3})\bigr)+3C_{1}R+ \log(\Lambda_{1}\Lambda_{2}\Lambda_{3}) \] holds. The second result deals with the function \(g(z)=\sum_{n\geq 0} a_{\nu}z^{\nu}\) (compare with Siegel's \(G\)-functions): for any real number \(R\) in the range \(0<R<1/(3C_{1})\), \[ \bigl(N(g,R)-N(g,0)\bigr) \log {1\over 3C_{1}R} \leq N(g,0) \log(1.5\cdot C_{1}C_{2}C_{3})+ \log(3\Lambda_{1}\Lambda_{2}\Lambda_{3}). \] The author also gives two applications of his first estimate: He gives a variant of Gel'fond's solution to Hilbert's seventh problem (Gel'fond-Schneider's theorem on the transcendence of \(\alpha^{\beta}\)), and he improves a lemma on \(E\)-functions, namely Lemma 3 of Chap.~13 of [\textit{A. B. Shidlovskij}, Transcendental numbers. De Gruyter Studies in Mathematics, 12. Berlin: Walter de Gruyter (1989; Zbl 0689.10043)]. A related work is [\textit{V. V. Zudilin}, Sb. Math. 187, 1791-1818 (1996); translation from Mat. Sb. 187, No. 12, 57-86 (1996; Zbl 0878.11030)].
scientific article; zbMATH DE number 1238024