The sharpening of the bounds on certain linear forms

A. I. Galochkin

Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 35-45.

Résumé

Let $g_1,\dots,g_{m-1}$, $b$, $h_1,\dots,h_m$ be the integers from some imaginary quadratic field, $b\ne0$, $\max|g_i|=g$, $\max|h_j|=H\ne0$, $P_m(x)=x^m+g_{m-1}x^{m-1}+\dots+g_1x$, $P_m(x)\ne0$ for x =$x=1,2,\dots$, $$ \psi(z)=1+\sum_{\nu=1}^\infty\Bigl[\prod_{x=1}^\nu P_m(x)\Bigr]^{-1}z^\nu. $$ Then $$ \Bigl|h_1\psi\Bigl(\frac1b\Bigr)+h_2\psi'\Bigl(\frac1b\Bigr)+\dots+h_m\psi^{(m-1)}\Bigl(\frac1b\Bigr)\Bigr|>CH^{1-m}\Bigl\{\frac{\ln\ln(H+2)}{\ln(H+2)}\Bigr\}^\gamma, $$ where $\gamma=(m-1)^2g-(m-1)\operatorname{Re}g_{m-1}+m(m^2+m-4)/2$, and $C=C(b,m,g)>0$.