nLab
zeta function

Idea

Many fields of mathematics have their own versions of zeta function: arithmetic and algebraic geometry, dynamical systems, graphs, operator theory. So this entry will take a lot of time to write. The most well-known example is of course the Riemann zeta function.

There are several articles trying to attach zeta functions attached to triangulated categories (and stable model categories…) and study them. Zeta functions express some motivic information, hence the categorical framework must be natural as it is for motives.

References

• $n$Lab related pages: multiple zeta values, motivic multiple zeta values, motivic integration, motive

Zeta functions in algebraic geometry

• Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228:4 (1995) 121–163, and preprint MPIM1992-50 pdf

• Nobushige Kurokawa, Zeta functions over $F_1$, Proc. Japan Acad. Ser. A Math. Sci. 81:10 (2005) 180-184 euclid

Categorical approaches

• M. Larsen?, V. A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3, 1 (2003) 85–95; Rationality criteria for motivic zeta functions, Compos. Math. 140:6 (2004) 1537–1560
• Vladimir Guletskii, Zeta functions in triangulated categories, Mathematical Notes 87, 3 (2010) 369–381, math/0605040
• M. Kontsevich, Notes on motives in finite characteristics, math.AG/0702206