In '*Cyclotomy and analytic geometry over F1*', Manin proposes a version of the notion of `*analytic function*' over the 'field with one element $\mathbb F_1$'.

**Question 1:** can somebody explain or give an insight of what it is?

The logarithm is a basic analytic function. It seems natural to expect from any `good' theory of 'analytic functions' to have his own version of the logarithm.

**Question 1:** is there a version of the logarithm over $\mathbb F_1$?
If yes, how it looks like?

Thanks in advance.