Zeta functions over \(\mathbb F_1\). (English) Zbl 1141.11316

Kurokawa, Nobushige

Proc. Japan Acad., Ser. A 81, No. 10, 180-184 (2005).

Summary: We show basic properties of zeta functions over the one element field starting from an algebraic set over the integer ring. We calculate several examples and we investigate special values via the associated \(K\)-group identified as the stable homotopy group of spheres.

MSC:

11M41	Other Dirichlet series and zeta functions
11G25	Varieties over finite and local fields
11S40	Zeta functions and \(L\)-functions
11S70	\(K\)-theory of local fields
14G15	Finite ground fields in algebraic geometry

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