On evaluation of constants entering into estimates of linear forms (Q1589160)

The author gives explicit values of the constants in the estimation of linear forms by \textit{A. N. Korobov} [Vestn. Mosk. Univ., Ser. I 1981, No. 6, 36-40 (1981; Zbl 0477.10028)] and \textit{A. I. Galochkin} [Mat. Sb., Nov. Ser. 124, No. 3, 416-430 (1984; Zbl 0559.10028)] in certain special cases. Let \[ A(z)= A_{\beta,\alpha} (z)=\sum^\infty_{\nu=0} z^\nu \prod^\nu_{x=1} {x+\alpha \over x(x+\beta)}, \] where \(\alpha>0\), \(\alpha\in \mathbb{Q}\) with \(\alpha\), \(2\alpha\notin\mathbb{Z}\); \(\beta=2\alpha+1\). Also let \(a\in \mathbb{N}\), \(a\alpha\in \mathbb{N}\), and \(b\neq 0\), \(b\in\mathbb{Z}_\mathbb{I}\) (here \(\mathbb{I}\) is an imaginary quadratic field). Define \(\theta=1/(a^3b)\), \(\xi_1=A(\theta)\), \(\xi_2= \theta A'(\theta)\), and \[ C=\Gamma (\beta+1)2^{-1}\bigl|\exp(\theta/2) \bigr |^2\max_{j=1,2} \biggl (\bigl((\alpha+1) |\theta |\bigr)^{2-j}\bigl |A_{\beta+2-j,\alpha}(-\theta)\bigr|/\Gamma(\beta+3-j) \biggr). \] Then the inequality \[ |h_1\xi_1+h_2\xi_2|<C(H\log H)^{-1}\log \log H,\quad H=\max \bigl(|h_1|,|h_2|\bigr) \] has infinitely many solutions \((h_1,h_2)\in \mathbb{Z}^2_\mathbb{I}\); while the inequality \[ |h_1 \xi_1+h_2\xi_2|<(C-\varepsilon) (H\log H)^{-1}\log \log H,\quad H=\max \bigl( |h_1|,|h_2|\bigr) \] has only finitely many such solutions \((h_1,h_2)\), where \(\varepsilon >0\) is arbitrary.