Geometry over non--existent "field with one element" F1ā conceived by
Jacques Tits [Ti] half a century ago recently found an incarnation, in at least
two related but different guises. In this paper I analyze the crucial role of
roots of unity in this geometry and propose a version of the notion of
"analytic functions" over F1ā. The paper combines a focused survey with some
new constructions.
In new version, several local additions and changes are made, references
added.Comment: 30 page