# On the number of primes for which a polynomial is Eisenstein

## TL;DR

This paper investigates the distribution of primes for which a polynomial is Eisenstein, providing formulas for the mean and variance of this count across polynomials of fixed degree ordered by height.

## Contribution

It introduces explicit expressions for the mean and variance of the number of primes making a polynomial Eisenstein, extending previous asymptotic results.

## Key findings

- Derived formulas for mean and variance of primes for Eisenstein polynomials
- Extended asymptotic analysis to include statistical measures
- Analyzed distribution properties of Eisenstein primes across polynomial families

## Abstract

Previously Heyman and Shparlinski gave an asymptotic formula with error term for the number of Eisenstein polynomials of fixed degree and bounded height. Let $\psi(f)$ denote the number of primes for which a polynomial $f$ is Eisenstein. We give expressions for the mean and variance of the function $\psi$ for each fixed degree, where the polynomials are ordered according to their height.