Dirichlet Species and Arithmetic Zeta Functions

2025/02/03 by John C. Baez, Baez, John C.
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #Functional Equations Stability Results #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.01833

openalex publication_date 2025/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Though Joyal's species are known to categorify generating functions in enumerative combinatorics, they also categorify zeta functions in algebraic geometry. The reason is that any scheme X of finite type over the integers gives a "zeta species" Z_X, and any species F gives a Dirichlet series \widehatF, in such a way that \widehatZ_X is the arithmetic zeta function of X, a well-known Dirichlet series that encodes the number of points of X over each finite field. Specifically, a Z_X-structure on a finite set is a way of making that set into a semisimple commutative ring, say k, and then choosing a k-point of the scheme X. This is an elaboration of joint work with James Dolan.

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