Approximation of numbers associated with Weierstrass elliptic functions
Let \(\wp (z)\) be the Weierstrass elliptic function with invariants \(g_ 2,g_ 3\) and periods \(2\omega_ 1,2\omega_ 2\) and let \(\zeta\) (z) be a \(\zeta\)-function with \(\eta_ i=\zeta (\omega_ i)\) \((i=1,2)\). Let \(\xi_ i\) \((i=1,...,6)\) be algebraic numbers with length \(L_ i\) and degree \(n_ i\). The author proves the following theorem by Gelfond's method. Theorem: Suppose that \(\alpha\in {\mathbb{C}}\), \(\alpha \neq 2m\omega_ 1+2h\omega_ 2\), m,h\(\in {\mathbb{Z}}\), \(n=\deg {\mathbb{Q}}(\xi_ 1,...,\xi_ 6)\), \(n_{i,k}={\mathbb{Q}}(\xi_ i,\xi_ k).\) \[ M=n(\frac{\ln L_ 1}{n_ 1}+\frac{\ln L_ 2}{n_ 2}+\frac{\ln L_ 3}{n_ 3}+\min (n_{4,5},n_{4,6},n_{5,6})\cdot (\frac{\ln L_ 4}{n_ 4}+\frac{\ln L_ 5}{n_ 5}+\frac{\ln L_ 6}{n_ 6}+1)+\ln n). \] If there exists a constant \(c>0\) such that x,y,t\(\in {\mathbb{Z}}\), \(x^ 2+y^ 2\neq 0\), \(| x|,| y| <t\), and \(\min | x\eta_ 1+y\eta_ 2| >\exp (-ct^ 3),\) then there exists an effective constant \(\Lambda >0\), such that \[ | \eta_ 1-\xi_ 1| +| \eta_ 2-\xi_ 2| +| \zeta (\alpha)-\xi_ 3| +| g_ 2-\xi_ 4| +| g_ 3-\xi_ 5| +| \wp (\alpha)-\xi_ 6| >\exp (-\Lambda M^ 2). \]
scientific article; zbMATH DE number 4210251