Cyclotomy and analytic geometry over F_1

Understand

Geometry over non--existent "field with one element" $F_1$ 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 $F_1$.
• The paper combines a focused survey with some new constructions.
• In new version, several local additions and changes are made, references added.