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.

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.

Tables

Table 1. Table 1: Approximate values of the constants for small d 𝑑 d

$d$	${\alpha }_d={\hat{\mu }}_d$	${\beta }_d$	${\gamma }_d$	${\mu }_d$	${\sigma }_d^2$	${\hat{\sigma }}_d^2$
$2$	$0.17971$	$0.00731$	$0.16765$	$1.07192$	$0.07187$	$0.17239$
$3$	$0.05653$	$0.00127$	$0.05557$	$1.01714$	$0.01705$	$0.05525$
$4$	$0.02255$	$0.00027$	$0.02243$	$1.00519$	$0.00517$	$0.02227$
$5$	$0.00989$	$0.00006$	$0.00988$	$1.00169$	$0.00169$	$0.00983$
$6$	$0.00456$	$0.00001$	$0.00456$	$1.00056$	$0.00056$	$0.00454$

Equations

$$\#{\mathcal{F}}_d(H)={\gamma }_d(2H{)}^{d+1}+\{\begin{array}{ll}O(H^d)& \text{if d>2}\\ O(H^2(\log H{)}^2)& \text{if d=2}.\end{array}$$

$${\alpha }_d:=\sum _{p\text{ prime}}\frac{(p-1{)}^2}{p^{d+2}},{\beta }_d:=\sum _{p\text{ prime}}{\left(\frac{(p-1{)}^2}{p^{d+2}}\right)}^2,{\gamma }_d:=1-\prod _{p\text{ prime}}\left(1-\frac{(p-1{)}^2}{p^{d+2}}\right),$$

$${\mu }_d:=\underset{H\to \infty }{\lim }\frac{{\sum }_{f\in {\mathcal{F}}_d(H)}\psi (f)}{{\sum }_{f\in {\mathcal{F}}_d(H)}1}=\frac{{\alpha }_d}{{\gamma }_d},$$

$${\sigma }_d^2:=\underset{H\to \infty }{\lim }\frac{{\sum }_{f\in {\mathcal{F}}_d(H)}(\psi (f)-{\mu }_d{)}^2}{{\sum }_{f\in {\mathcal{F}}_d(H)}1}=\frac{{\alpha }_d+{\alpha }_d^2-{\beta }_d-{\mu }_d{\alpha }_d}{{\gamma }_d}.$$

$$\#{\mathcal{H}}_d(s,H)=\frac{(2H{)}^{d+1}{\varphi }^2(s)}{s^{d+2}}+O\left(\frac{2^{\omega (s)}{H}^d}{s^{d-1}}\right).$$

$$\sum _{f\in {\mathcal{F}}_d(H)}\psi (f)=(2H{)}^{d+1}{\alpha }_d+\{\begin{array}{ll}O(H^2)& \text{if d>2}\\ O(H^2\log \log H)& \text{if d=2}.\end{array}$$

$$\sum _{f\in {\mathcal{F}}_d(H)}\psi (f)$$

$$=\sum _{p\le H}\left[\frac{(2H{)}^{d+1}{\varphi }^2(p)}{p^{d+2}}+O\left(\frac{2^{\omega (p)}{H}^d}{p^{d-1}}\right)\right]$$

$$=(2H{)}^{d+1}\sum _{p\le H}\frac{(p-1{)}^2}{p^{d+2}}+\sum _{p\le H}O\left(\frac{H^d}{p^{d-1}}\right)$$

$$=(2H{)}^{d+1}\sum _p\frac{(p-1{)}^2}{p^{d+2}}-(2H{)}^{d+1}\sum _{p>H}\frac{(p-1{)}^2}{p^{d+2}}+\sum _{p\le H}O\left(\frac{H^d}{p^{d-1}}\right).$$

$$(2H{)}^{d+1}\sum _{p>H}\frac{(p-1{)}^2}{p^{d+2}}=O\left(H^{d+1}{\int }_H^{\infty }\frac{(x-1{)}^2}{x^{d+2}}dx\right)=O\left(H^{d+1}{H}^{-d+1}\right)=O\left(H^2\right).$$

$$H^d\sum _{p\le H}\frac{1}{p^{d-1}}=\{\begin{array}{ll}O\left(H^2\right)& \text{if d>2}\\ O\left(H^2\log \log H\right)& \text{if d=2},\end{array}$$

$$\sum _{f\in {\mathcal{F}}_d(H)}\psi (f{)}^2=({\alpha }_d+{\alpha }_d^2-{\beta }_d)(2H{)}^{d+1}+\{\begin{array}{ll}O(H^2)& \text{if d>2}\\ O(H^2(\log \log H{)}^2)& \text{if d=2}.\end{array}$$

$$\tau (f,p)=\{\begin{array}{ll}1& \text{if f is p-Eisenstein}\\ 0& \text{otherwise},\end{array}$$

$$\sum _{f\in {\mathcal{F}}_d(H)}\psi (f{)}^2$$

$$=\sum _{f\in {\mathcal{F}}_d(H)}\left(\sum _{p\text{ prime}}\tau (f,p)\sum _{q\text{ prime}}\tau (f,q)\right)$$

$$=\sum _{f\in {\mathcal{F}}_d(H)}\left(\sum _{p,q\text{ prime}}\tau (f,p)\tau (f,q)\right)$$

$$=\sum _{p,q\text{ prime}}\left(\sum _{f\in {\mathcal{F}}_d(H)}\tau (f,p)\tau (f,q)\right).$$

$$\sum _{f\in {\mathcal{F}}_d(H)}\psi (f{)}^2$$

$$\sum _{\begin{array}{c}pq\le H\\ p\ne q\end{array}}\#\mathcal{H}(pq,H)$$

$$=(2H{)}^{d+1}\sum _{\begin{array}{c}pq\le H\\ p\ne q\end{array}}\frac{(p-1{)}^2(q-1{)}^2}{p^{d+2}{q}^{d+2}}+O\left(\sum _{\begin{array}{c}p,q\text{ prime}\\ pq\le H\end{array}}\frac{H^d}{(pq{)}^{d-1}}{2}^{\omega (pq)}\right)$$

$$=(2H{)}^{d+1}\sum _{\begin{array}{c}pq\le H\end{array}}\frac{(p-1{)}^2(q-1{)}^2}{p^{d+2}{q}^{d+2}}-(2H{)}^{d+1}\sum _{\begin{array}{c}{p}^2\le H\end{array}}{\left(\frac{(p-1{)}^2}{p^{d+2}}\right)}^2+O\left(H^d\sum _{\begin{array}{c}p,q\text{ prime}\\ pq\le H\end{array}}\frac{1}{(pq{)}^{d-1}}\right).$$

$$(2H{)}^{d+1}\sum _{\begin{array}{c}pq\le H\end{array}}\frac{(p-1{)}^2(q-1{)}^2}{p^{d+2}{q}^{d+2}}$$

$$=(2H{)}^{d+1}{\left(\sum _p\frac{(p-1{)}^2}{p^{d+2}}\right)}^2+O\left(H^{d+1}\sum _{p>H}\frac{1}{p^d}\right)$$

$$=(2H{)}^{d+1}{\alpha }_d^2+O(H^2).$$

$$(2H{)}^{d+1}\sum _{\begin{array}{c}{p}^2\le H\end{array}}{\left(\frac{(p-1{)}^2}{p^{d+2}}\right)}^2$$

$$=(2H{)}^{d+1}{\beta }_d+O(H^{3/2}).$$

$$H^d\sum _{\begin{array}{c}p,q\text{ prime}\\ pq\le H\end{array}}\frac{1}{(pq{)}^{d-1}}=\{\begin{array}{ll}O(H^2),& \text{if d>2}\\ O(H^2{\left(\log \log H\right)}^2),& \text{if d=2}.\end{array}$$

$${\sigma }_d^2$$

$$=\underset{H\to \infty }{\lim }\frac{1}{\#{\mathcal{F}}_d(H)}\sum _{f\in {\mathcal{F}}_d(H)}\left(\psi (f{)}^2-2\psi (f){\mu }_d+{\mu }_d^2\right)$$

$$=\underset{H\to \infty }{\lim }\frac{1}{\#{\mathcal{F}}_d(H)}\left[\sum _{f\in {\mathcal{F}}_d(H)}\psi (f{)}^2-2{\mu }_d\sum _{f\in {\mathcal{F}}_d(H)}\psi (f)+{\mu }_d^2\sum _{f\in {\mathcal{F}}_d(H)}1\right].$$

$$\frac{1}{{\gamma }_d}\left[({\alpha }_d+{\alpha }_d^2-{\beta }_d)-2{\mu }_d{\alpha }_d+{\mu }_d^2{\gamma }_d\right],$$

$${\alpha }_d=\frac{1}{2^{d+2}}+O\left(\frac{1}{3^d}\right),{\beta }_d=\frac{1}{2^{2(d+2)}}+O\left(\frac{1}{3^{2d}}\right),{\gamma }_d=\frac{1}{2^{d+2}}+O\left(\frac{1}{3^d}\right).$$

Full text

On the number of primes for which a polynomial is Eisenstein

Shilin Ma (Carleton College)

Kevin J. McGown (California State University, Chico)

Devon Rhodes (California State University, Chico)

Mathias Wanner (Villanova University)

1 Introduction

For an integer $d\geq 2$, let $f(x)=a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots+a_{1}x+a_{0}$ be a polynomial with integer coefficients. We say that $f$ is Eisenstein if there exists a prime $p$ such that $p\mid a_{i}$ for $i=0,1,\ldots,d-1$, $p^{2}\nmid a_{0}$, and $p\nmid a_{d}$. The well-known fact that Eisenstein polynomials are irreducible is often encountered in an undergraduate algebra course. See [1] for a fascinating history of this result, which was proved independently by Schönemann and Eisenstein.

Dobbs and Johnson (see [2]) posed some probabilistic questions concerning Eisenstein polynomials. In particular, one could ask: What is the probability that a randomly chosen polynomial is Eisenstein? Dubickas answers this question in [4] by providing an asymptotic expression for the number of monic Eisenstein polynomials of fixed degree and bounded height. Later Heyman and Shparlinski (see [6]) gave an asymptotic expression for the number of Eisenstein polynomials (monic or not) of fixed degree and bounded height but with a stronger error term. We mention in passing that there are generalizations and variations one may consider; some results in this area include [5, 7, 8, 3].

Our paper builds naturally on [6] so we begin by stating their result. Define the height of a polynomial $f$ to be $\max\{|a_{0}|,|a_{1}|,\ldots,|a_{d}|\}$. Let $\mathcal{F}_{d}(H)$ be the set of Eisenstein polynomials of degree $d$ and height at most $H$.

Theorem 1** (Heyman–Shparlinski).**

We have

[TABLE]

Let $\psi(f)$ denote the number of primes for which $f$ is Eisenstein. Our aim is to study the statistics of this function. We establish the following result, which gives an expression for the mean and variance of the function $\psi(f)$ as $f$ ranges over all Eisenstein polynomials of a fixed degree.

Theorem 2**.**

If

[TABLE]

then we have

[TABLE]

We note in passing that $\alpha_{d}$ and $\beta_{d}$ can be expressed as finite linear combinations of values of the prime zeta function $P(s)=\sum_{p}p^{-s}$. Throughout this paper, the variables $p$ and $q$ will always denote primes. See Section 3 for additional comments on $\alpha_{d}$, $\beta_{d}$, $\gamma_{d}$, $\mu_{d}$, $\sigma_{d}^{2}$, including a table of numerical values for various values of $d$.

2 Proofs

As usual we let $\omega(n)$ denote the number of distinct prime factors of $n$ and let $\phi(n)$ denote the Euler phi-function. Following [6], we let $\mathcal{H}_{d}(s,H)$ be the number of polynomials of degree $d$ and height at most $H$ satisfying $s\mid a_{i}$ for $i=0,1,\ldots,d-1$, $\gcd(a_{0}/s,s)=1$, and $\gcd(a_{d},s)=1$.

Lemma 3**.**

We have

[TABLE]

Proof.

See Lemma 5 of [6]. ∎

Lemma 4**.**

We have

[TABLE]

Proof.

We rewrite the sum in question as a sum over primes and apply Lemma 3; this yields

[TABLE]

The splitting of $\sum_{p\leq H}$ into $\sum_{p}$ and $\sum_{p>H}$ is justified since $\sum_{p}$ converges absolutely. It remains to bound the second and third terms in the last line above. We bound the second term using the integral test to obtain

[TABLE]

For the third term, we find

[TABLE]

where we have used Mertens’ Theorem (see, for example, [9]) in the case of $d=2$. ∎

Lemma 5**.**

We have

[TABLE]

Proof.

If we define

[TABLE]

then the first sum can be rewritten as

[TABLE]

The inner sum above represents the number of polynomials of height at most $H$ that are Eisenstein for both $p$ and $q$, but the fact that $p$ may equal $q$ complicates matters. Consequently, we have

[TABLE]

The first sum on the right-hand side above is exactly what appears in Lemma 4, and therefore is it equal to the right-hand side of (2). It remains to deal with the second sum, which equals

[TABLE]

For the first term, as in the proof of Lemma 4, we have

[TABLE]

For the second term,

[TABLE]

Finally, for the third term, we have

[TABLE]

Putting this all together proves the lemma. ∎

Proof of Theorem 2.

The part of the theorem concerning the mean $\mu_{d}$ follows immediately from Lemma 4 and Theorem 1. Now we consider the variance:

[TABLE]

By Lemma 4, Lemma 5, and Theorem 1, the limit above equals

[TABLE]

which simplifies to the desired expression. ∎

3 Remarks on the constants

It is not hard to show that

[TABLE]

It then follows that $\lim_{d\to\infty}\mu_{d}=1$ and $\lim_{d\to\infty}\sigma_{d}^{2}=0$, as one would expect. If one was interested in the mean $\hat{\mu_{d}}$ and variance $\hat{\sigma}_{d}^{2}$ of $\psi(f)$ as $f$ ranges over all polynomials, instead of just Eisenstein polynomials, one would obtain the simpler expressions $\hat{\mu}_{d}=\alpha_{d}$ and $\hat{\sigma}_{d}^{2}=\alpha_{d}-\beta_{d}$. We will not prove this explicitly but it essentially follows from the proof of Theorem 2. In this case, one observes that $\lim_{d\to\infty}\hat{\mu_{d}}=0$ and $\lim_{d\to\infty}\hat{\sigma}_{d}^{2}=0$, as expected.

Acknowledgement

This research was completed as part of the Research Experience for Undergraduates and Teachers program at California State University, Chico funded by the National Science Foundation (DMS-1559788).