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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.

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The exponential function is generally easier to understand than than the logarithm, since it is analytic on all of $\mathbb{C}$ and satisfies a simple first-order ODE. Typically, the logarithm is then defined as the inverse of the exponential function. So it seems natural to first try to understand the "exponential function over $\mathbb{F}_1$" (whatever that might mean!) and then base your understanding of the "logarithm over $\mathbb{F}_1$ off of that. –  Michael Joyce Mar 28 at 16:15
    
@M. Joyce: thanks for your comment. But the idea (or my idea) is to try to construct a $\mathbb F_1$-logarithm starting from algebraic object as in the complex world: the complex log is the integral of $dz/z$, 'the' rational $\mathbb C^*$-invariant form on $\mathbb C^*$. –  Lucien from IHP Mar 28 at 16:34
    
A perfectly reasonable approach. I don't know enough about $\mathbb{F}_1$ to have any idea what approach is more tractable. It is certainly possible that an analogue of integrating against a differential form is easier to generalize. Still the non-simply connectedness of $\mathbb{C}^*$ and the impossibility of defining an entire logarithm in the case of $k = \mathbb{C}$ makes me think that the "topology of $\mathbb{F}_1^*$" might be quite complex. –  Michael Joyce Mar 28 at 16:43

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