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Estimates for the number of zeros of some functions with algebraic Taylor coefficients. (English. Russian original) Zbl 0968.11028
Math. Notes 61, No. 6, 687-692 (1997); translation from Mat. Zametki 61, No. 6, 817-824 (1997).
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 [A. B. Shidlovskij, Transcendental numbers. De Gruyter Studies in Mathematics, 12. Berlin: Walter de Gruyter (1989; Zbl 0689.10043)].
A related work is [V. V. Zudilin, Sb. Math. 187, 1791-1818 (1996); translation from Mat. Sb. 187, No. 12, 57-86 (1996; Zbl 0878.11030)].

##### MSC:
 11J91 Transcendence theory of other special functions
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##### References:
 [1] A. B. Shidlovskii,Transcendental Numbers [in Russian], Nauka, Moscow (1987); English translation: Walter de Gruyte Berlin-New York (1989). [2] C. L. Siegel, ”Über einige Anwendungen diophantischer Approximationen,”Abh. Preuss. Wiss. Phys.-Math. Kl., No. 1, 1–70 (1929–1930). · JFM 56.0180.05 [3] W. Hayman,Meromorphic Functions, Oxford (1964). · Zbl 0115.06203 [4] V. V. Zudilin, ”On lower bounds for polynomials of values of some entire functions,”Mat. Sb. [Russian Acad. Sci. Sb. Math.],187, No. 12, 57–86 (1996). · Zbl 0878.11030 [5] G. V. Chudnovsky, ”On some applications of Diophantine approximations,”Proc. Nat. Acad. Sci. USA,81, 1926–2930 (1984). · Zbl 0544.10034 · doi:10.1073/pnas.81.6.1926
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