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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
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