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Schemes over $$\mathbb F_{1}$$ and zeta functions. (English) Zbl 1201.14001
The authors determine the real counting function for the hypothetical “curve” $$C$$ over $${\mathbb{F}}_1$$, whose corresponding zeta function is the complete Riemann zeta function. Such a counting function exists as a distribution, is positive on $$(1, \infty)$$ and takes the value $$- \infty$$ at $$q=1$$ as expected from the infinite genus of $$C$$.
As an application, the authors apply their functorial theory in order to interpret conceptually the spectral realization of zeros of L-functions.

##### MSC:
 14A15 Schemes and morphisms 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
##### Keywords:
absolute point; counting functions; zeta functions
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##### References:
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