Dirichlet Species and Arithmetic Zeta Functions

TL;DR

This paper explores how Joyal's species categorify both generating functions and zeta functions, linking combinatorics, algebraic geometry, and number theory through the concept of zeta species and Dirichlet series.

Contribution

It introduces the notion of zeta species associated with schemes over integers, connecting species structures to arithmetic zeta functions and Dirichlet series.

Findings

Zeta species categorify arithmetic zeta functions.

Species structures correspond to points on schemes over finite fields.

The framework unifies combinatorics and algebraic geometry concepts.

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 $\widehat{F}$, in such a way that $\widehat{Z}_X$ 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.