Levels of distribution for sieve problems in prehomogeneous vector spaces

TL;DR

This paper introduces a new method combining Fourier analysis and algebraic geometry to determine the distribution levels of certain functions on prehomogeneous vector spaces, aiding sieve methods in number theory.

Contribution

It develops a novel approach leveraging Fourier transforms and algebraic geometry to obtain distribution results applicable to sieve techniques in number theory.

Findings

Proves the existence of many quartic fields with squarefree discriminants and limited prime factors.

Provides a framework for level of distribution results in prehomogeneous vector spaces.

Demonstrates the application of these results to number field discriminants.

Abstract

In a companion paper, we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields, and we carried out these computations in a variety of cases. Here we develop a method, based on Fourier analysis and algebraic geometry, which exploits these Fourier transform formulas to yield level of distribution results, in the sense of analytic number theory. Such results are of the shape typically required for a variety of sieve methods. As an example of such an application we prove that there are $\gg$ X/log(X) quartic fields whose discriminant is squarefree, bounded above by X, and has at most eight prime factors.