Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Schemes over $\F_1$ and zeta functions   We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for
the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose
corresponding zeta function is the complete Riemann zeta function. Then, we
develop a theory of functorial $\F_1$-schemes which reconciles the previous
attempts by C. Soul\'e and A. Deitmar. Our construction fits with the geometry
of monoids of K. Kato, is no longer limited to toric varieties and it covers
the case of schemes associated to Chevalley groups. Finally we show, using the
monoid of ad\`ele classes over an arbitrary global field, how to apply our
functorial theory of $\Mo$-schemes to interpret conceptually the spectral
realization of zeros of $L$-functions.

12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
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The categories of this paper are: cs
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio