On the algebraic independence of certain numbers connected with the exponential and the elliptic functions
At least two of the four numbers \(\pi\), log \(\pi\), \(\sum_{n\geq 0}\pi^{-n^ 2}\), \(\sum_{n\geq 0}(-1)^ n \pi^{-n^ 2}\) are algebraically independent, and at least two of the four numbers \(\pi\), log 2, \(\sum_{n\geq 0}2^{-n^ 2}\), \(\sum_{n\geq 0}(-1)^ n 2^{- n^ 2}\) are algebraically independent. These results as well as a few other nice ones in the same vein, follow from the main theorem of this paper, which is the following. Let \(\wp\) be a Weierstrass elliptic function with invariants \(g_ 2\), \(g_ 3\), and a pair of fundamental periods \((\omega_ 1,\omega_ 2)\); let \(\alpha\) be any non-zero complex number, and let K be a field of transcendence type \(<7/3\). Then at least one of the seven numbers \(\alpha\), \(g_ 2\), \(g_ 3\), \(\omega_ 1\), \(\omega_ 2\), \(e^{\alpha \omega_ 1}\), \(e^{\alpha \omega_ 2}\) is transcendental over K. \{Reviewer's remark. The main result (not the corollaries) had been already stated by \textit{G. V. Chudnovsky} [Contributions to the theory of transcendental numbers (Math. Surv. Monogr., No.19) (1984; Zbl 0594.10024); see Chapter 7, {\S} 4, theorem 4.1]. However some of the other results announced by Chudnovsky in the same section of his book are not clear - for instance his theorem 4.8, where he considers ''algebraic points'' of elliptic functions without assuming that the invariants are algebraic. On the other hand, in the result quoted above, \textit{R. Tubbs} has shown that it is possible to replace the assumption on the transcendence type by the condition that K has transcendence degree at most 1 over \({\mathbb{Q}}\), using the method of his paper [Algebraic groups and small transcendence degree. I, J. Number Theory 25, 279-307 (1987)].\}
scientific article; zbMATH DE number 3979168