Győry, Kálmán (ed.) et al., Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. 37-44 (1998).
The -conjeture asserts that if , , are integers with
then for any ,
where is the conductor of .
The present paper contains sharpening of this conjecture:
Conjecture 1. If , , are integers satisfying (1) then, for any ,
where denotes the number of distinct prime fators of and is an absolute constant.
Conjecture 2. There are absolute constants and such that, if (1) holds then, for any ,
The author discusses in detail the links between these conjectures and natural conjectures on linear forms in logarithms, in the archimedean and nonarchimedean cases. He also presents several interesting observations about these conjectures.