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On unimprovable estimates with respect to the height of some linear forms. (Russian) Zbl 0559.10028
Let I denote the field of rational numbers or an imaginary quadratic field, and denote by \(Z_ I\) the ring of integers of I. Let \(\lambda_ 1,...,\lambda_ s\in I\), each different from -1,-2,..., and let \(a\in Z_ I\) be such that \(a\neq 0\), \(a\lambda_ j\in Z_ I\), \(j=1,...,s\). Define \[ \Psi (z)=\sum^{\infty}_{\nu =0}\frac{z^{\nu}}{a^{s\nu +\nu}\nu ![\lambda_ 1+1,\nu]...[\lambda_ s+1,\nu]}, \] where \([\lambda +1,0]=1\), \([\lambda +1,\nu]=(\lambda +1)...(\lambda +\nu)\) for \(\nu =1,2,... \). The author continues his research [Mat. Zametki 8, 19-28 (1970; Zbl 0219.10035), ibid. 20, 35-45 (1976; Zbl 0336.10028)] on linear forms \(R=\sum^{s}_{k=1}h_ k \Psi^{(k)}(1/b)\), \(b,h_ k\in Z_ I\), \(b\neq 0\), giving proofs for the following sharp estimates announced in his note [Prog. Math. 31, 95-98 (1983; Zbl 0521.10027)]. There exists an effectively computable positive constant \(C_ 1=C_ 1(a,b,\lambda_ 1,...,\lambda_ s)\) such that any linear form R with \(H=\max_{0\leq k\leq s}| h_ k| >3\) satisfies \[ | R| >C_ 1 \Phi (H),\quad \Phi (H)=H^{-s} (\ln H)^{-s(1-\Delta)} (\ln \ln H)^{s(r- \Delta)}, \] and here \(\Delta\), r are explicitly given constants depending on \(\lambda_ 1,...,\lambda_ s\). Further, there exists an effectively computable positive constant \(C_ 2=C_ 2(a,b,\lambda_ 1,...,\lambda_ s)\) such that \(| R| <C_ 2 \Phi (H)\) for infinitely many forms R.
Reviewer: K.Väänänen

11J81 Transcendence (general theory)
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