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Geometric sieve over number fields for higher moments.

Giacomo Micheli¹, Severin Schraven², Simran Tinani³

¹Department of Mathematics, University of South Florida, 4202 E. Fowler Avenue, Tampa, FL 33620 USA.

Research in Number Theory

|August 7, 2023

Summary

This summary is machine-generated.

This study introduces an effective geometric sieve to calculate higher moments of densities for subsets within number fields. The method computes density, mean, and variance for Eisenstein polynomials, extending prior research.

Keywords:

Densities Expected Values Number Fields Variance

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Area of Science:

Number Theory
Algebraic Geometry

Background:

The geometric sieve is a tool for computing subset densities.
Existing methods are limited to specific number fields or lower moments.

Purpose of the Study:

To develop an effective criterion for computing higher moments of densities over general number fields.
To extend the geometric sieve for density computations to algebraic integer rings and higher moments.

Main Methods:

A generalized geometric sieve is developed for computing higher moments of densities.
The method is applied to finite dimensional free modules over algebraic integers in number fields.

Main Results:

An effective criterion for computing all higher moments of densities is established.
The geometric sieve is extended to general number fields and higher moments, surpassing previous limitations.

Conclusions:

The developed geometric sieve provides a unified and effective approach for density moment computations.
This work generalizes and enhances existing results on geometric sieves for number theoretic applications.