Equidistribution and arithmetic $\Lambda$-distributions

TL;DR

This paper introduces an abstract framework for equidistribution in $\lambda$-probability spaces, linking it to motivic Euler products and applying it to analyze asymptotic distributions of function field $L$-functions and zeta functions.

Contribution

It develops a new abstract notion of equidistribution for $\lambda$-probability spaces and connects it to motivic Euler products, enabling the computation of $\Lambda$-distributions for function field $L$-functions.

Findings

Established a framework for equidistribution in $\lambda$-probability spaces.

Derived formulas for $\sigma$-moment generating functions as motivic Euler products.

Computed asymptotic $\Lambda$-distributions for families of function field $L$-functions.

Abstract

We formulate an abstract notion of equidistribution for families of $\lambda$-probability spaces parameterized by admissible $\mathbb{Z}$-sets. Under the assumption of equidistribution, we show that the $\sigma$-moment generating functions of certain infinite sums of random variables can be computed as motivic Euler products. Combining this result with earlier generalizations of Poonen's sieve, we compute the asymptotic $\Lambda$-distributions for several natural families of function field $L$-functions and zeta functions.