From Golomb to Schinzel

1. Introduction

1.1. The Schinzel and Bateman-Horn Conjectures

Let p denote a prime number. In a paper with Sierpiński [12], Schinzel proposed the following conjecture, which is known as Schinzel’s hypothesis (H).

Conjecture 1.1. 

Let $k\ge 1$ and let $f_1\left(x\right),\cdots ,f_k\left(x\right)\in \mathbb{Z}\left[x\right]$ be irreducible polynomials with positive leading coefficients. Assume that there does not exist any integer $n>1$ dividing all the products $f_1\left(m\right)\cdots f_k\left(m\right)$ for every integer m. Then there are infinitely many natural numbers n such that $f_1\left(n\right),\cdots ,f_k\left(n\right)$ are all primes.

The Schinzel hypothesis (H) was recently studied on the average by Skorobogatov and Sofos [13]. Let $f_1\left(x\right),\cdots ,f_k\left(x\right)\in \mathbb{Z}\left[x\right]$ be as in Conjecture 1.1, then they proved that when ordered by height $100\%$ of these polynomials take simultaneously prime values at least once. For more results see their paper and references therein.

A related and quantitative form of Schinzel’s hypothesis is the Bateman-Horn conjecture. Let $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ be irreducible polynomials of degree $h_1,\cdots ,h_k$ and with positive leading coefficients. We write $f=f_1{f}_2\cdots f_k$. Assume that there does not exist a prime number p that divides $f\left(n\right)$ for every positive integer n. Let

$${\pi }_f\left(x\right)=\#\{n\le x:f_1\left(n\right),\cdots ,f_k\left(n\right)\mathrm{are}\mathrm{all}\mathrm{prime}\}.$$

The Bateman-Horn conjecture is the following.

Conjecture 1.2 

([3]). Let $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ be as above, then as $x\to \infty$,

$${\pi }_f\left(x\right)\sim {\displaystyle \frac{c\left(f\right)}{h_1{h}_2\cdots h_k}}·{\displaystyle \frac{x}{{log}^{k}x}}$$

(1.1)

where

$$c\left(f\right)=\prod _p{\displaystyle \frac{1-N_f\left(p\right)/p}{{(1-1/p)}^k}}$$

(1.2)

and $N_f\left(p\right)$ is the number of solutions of the congruence $f\left(n\right)\equiv 0(modp)$.

Remark 1.3.

Let $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ be as in Conjecture 1.2, then Bateman and Horn proved in [3] that $c\left(f\right)$ converges and is positive.

For an excellent survey and relevant historical literature on the Bateman-Horn conjecture we refer to the recent expository article [1].

It is clear the Bateman–Horn conjecture includes many special cases. For a single linear polynomial, it is Dirichlet’s theorem on primes in arithmetic progressions, which in turn contains the prime number theorem ($f\left(x\right)=x$) as a special case. For $k\ge 2$ or for non-linear polynomials the conjecture is open. The simplest case of non-linear polynomials is the case $f\left(x\right)=x(x+2)$ of the twin prime conjecture. The Bateman–Horn conjecture also includes the Hardy-Littlewood prime tuples conjecture [7]. Indeed Hardy and Littlewood [7] also proposed many other conjectures in their Partitio Numerorum III, many of which are special cases of the Bateman-Horn conjecture.

Let $\mathsf{\Lambda }\left(n\right)$ be the von Mangoldt function and set

$${\psi }_f\left(x\right)=\sum _{n\le x}\mathsf{\Lambda }\left(f_1\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right).$$

(1.3)

By partial summation we have as in [2,4]

Proposition 1.4. 

The Bateman-Horn conjecture 1.2 is equivalent to

$${\psi }_f\left(x\right)=c\left(f\right)x+o\left(x\right).$$

(1.4)

An important property of ${\psi }_f\left(x\right)$ is the following.

Proposition 1.5 

([4], Theorem 1). We have ${\psi }_f\left(x\right)=O\left(x\right)$.

1.2. Main Result

We now state the main result in this paper. While we are unable to prove the quantitative Bateman-Horn conjecture we are going to show the weaker Schinzel’s hypothesis is true.

Theorem 1.6. 

Conjecture 1.1 is true.

We will prove this by Golomb’s method, together with a contradiction argument.

For the case of twin primes, that is for $f\left(x\right)=x(x+2)$, Conjecture 1.1 implies

$$\underset{n\to \infty }{lim\; inf}(p_{n+1}-p_n)=2$$

where $p_n$ is the nth prime. The previous results on bounded prime gaps was made by Zhang [14] by using the sieve method, who proved that

$$\underset{n\to \infty }{lim\; inf}(p_{n+1}-p_n)<7·{10}^7.$$

Later on Maynard [10], Tao and the Polymath Project [11] reduced the bound of Zhang. The current record is the following

$$\underset{n\to \infty }{lim\; inf}(p_{n+1}-p_n)\le 246.$$

Acknowledgements

This work was started during my stay at Nagoya University. I would like to thank Professor Keith Conrad for sending me a copy of the paper [5], which was (and is) not available in the internet, also for his insightful corrections and comments on an earlier version of this paper.

Special thanks to the staff of the library of Department of Science of Nagoya University, who kindly allowed me to use this library.

2. Golomb’s Method

For positive integers $a,b$ by $(a,b)$ we mean the greatest common divisor of a and b. Let $\mu \left(n\right)$ be the Möbius function and $\omega \left(n\right)$ the number of distinct prime divisors of n.

We investigate Schinzel’s hypothesis and the Bateman-Horn conjecture by Golomb’s method. There are mainly three papers on the Golomb method. The first is Golomb’s thesis [5] (see also [6]), the second is Conrad [4] and the third is Hindry and Rivoal [9]. The papers [5] and [9] are by using the power series, while Conrad [4] is by using the Dirichlet series. Since one can turn a power series to a Dirichlet series and vice versa, the two approaches are equivalent. In this paper we shall use the power series.

The key of the Golomb method is the following identity of Golomb.

Lemma 2.1 

([5,6]). Let $a_i>1$ and $(a_i,a_j)=1$ for $i\ne j$ and let $A=a_1{a}_2\cdots a_k$. Then we have

$$\mathsf{\Lambda }\left(a_1\right)\mathsf{\Lambda }\left(a_2\right)\cdots \mathsf{\Lambda }\left(a_k\right)={\displaystyle \frac{{(-1)}^k}{k!}}\sum _{d|A}\mu \left(d\right){log}^{k}d.$$

(2.1)

Golomb used this identity to study the twin prime conjecture by a way analogous to Wiener’s proof of the prime number theorem. That is, let $n>2$ be even, then by (2.1) we have

$$2\mathsf{\Lambda }(n-1)\mathsf{\Lambda }(n+1)=\sum _{d\left|\right(n^2-1)}\mu \left(d\right){log}^{2}d.$$

(2.2)

For $\left|z\right|<1$ let

$$G\left(z\right)=2\sum _{n\ge 1}\mathsf{\Lambda }(n-1)\mathsf{\Lambda }(n+1)z^n,$$

then we have [5,6]

$$(1-z)G\left(z\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{d=1}{(d,2)=1}}}^{\infty }{\displaystyle \frac{(1-z)\mu \left(d\right){log}^{2}d}{1-z^{2d}}}\sum _{i=1}^{2^{\omega \left(d\right)}}{z}^{a_i}$$

(2.3)

where the $a_i$ are the $2^{\omega \left(d\right)}$ even roots of the congruence $a^2\equiv 1(modd)$ between 0 and $2d$. If the termwise limit could be justified, then we would have

$$\underset{z\to 1^{-}}{lim}(1-z)G\left(z\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{d=1}{(d,2)=1}}}^{\infty }{\displaystyle \frac{\mu \left(d\right){log}^{2}d·2^{\omega \left(d\right)}}{2d}}=4\prod _{p>2}\left(1-{\displaystyle \frac{1}{{(p-1)}^2}}\right),$$

(2.4)

which would lead to a proof of the Bateman-Horn conjecture for the case of twin primes.

We now turn to the general Bateman-Horn conjecture. The identity (2.1) requires that $(a_i,a_j)=1$ for $i\ne j$, for this Hindry and Rivoal [9] introduced the hypothesis F: for all integers $n\ge 1$, $(f_i\left(n\right),f_j\left(n\right))=1$ for $1\le i\ne j\le k$. They proved the following result.

Lemma 2.2 

([9], Théorème 3). Let $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ be as in Conjecture 1.2. If the Bateman-Horn conjecture 1.2 is true for all $f_1,\cdots ,f_k$ that satisfies the hypothesis F, then it is true for all $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ that as in Conjecture 1.2.

Note that Conrad [4] also addressed this coprime issue, see [4]. Since what we are going to prove is Schinzel’s hypothesis but not Bateman-Horn, we need a version for Conjecture 1.1. As we shall see this can be deduced from the proof of [9] and we have the following.

Lemma 2.3. 

Let $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ be as in Conjecture 1.1. If Conjecture 1.1 is true for all $f_1,\cdots ,f_k$ that satisfies the hypothesis F, then it is true for all $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ that as in Conjecture 1.1.

Proof. 

Remember that $f=f_1\cdots f_k$ and

$${\pi }_f\left(x\right)=\#\{n\le x:f_1\left(n\right),\cdots ,f_k\left(n\right)\mathrm{are}\mathrm{all}\mathrm{prime}\}.$$

Let f be as in Conjecture 1.1, then one sees from the proof of [9] that there associates a family $f_{n_0}$ satisfying the hypothesis F such that

$${\pi }_f\left(x\right)\sim {\displaystyle \frac{\#{\mathcal{N}}_0}{N_0}}{\pi }_{f_0}\left(x\right).$$

(2.5)

For notations $n_0,{\mathcal{N}}_0,N_0$ see the proof of [9, Théorème 3]. Thus if ${\pi }_{f_0}\left(x\right)\sim Q\left(x\right)$, then ${\pi }_f\left(x\right)\sim {\displaystyle \frac{\#{\mathcal{N}}_0}{N_0}}Q\left(x\right)$. It is remarked there that $Q\left(x\right)$ does not depend on the choice of $n_0$. Also $\#{\mathcal{N}}_0\ne 0$ by Lemma 2.2 since otherwise the Bateman-Horn version (Lemma 2.2) would be false. Therefore if ${\pi }_{f_0}\left(x\right)$ is unbounded then so is ${\pi }_f\left(x\right)$. The proof is complete. □

Henceforth in the following we let $f_1,\cdots ,f_k\in \mathbb{Z}\left[x\right]$ be as in Conjecture 1.2 or Conjecture 1.1 that satisfies the hypothesis F. Remember that $f=f_1\cdots f_k$.

Now by (2.1) we have

$$\mathsf{\Lambda }\left(f_1\left(n\right)\right)\mathsf{\Lambda }\left(f_2\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)={\displaystyle \frac{{(-1)}^k}{k!}}\sum _{d\left|f\right(n)}\mu \left(d\right){log}^{k}d.$$

(2.6)

Consider the absolutely convergent series for $\left|z\right|<1$:

$$G_f\left(z\right)={(-1)}^{k}k!\sum _{n\ge 1}\mathsf{\Lambda }\left(f_1\left(n\right)\right)\mathsf{\Lambda }\left(f_2\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)z^n,$$

(2.7)

then as in Hindry and Rivoal [9] we have

$$G_f\left(z\right)=\sum _{d\ge 1}{\displaystyle \frac{\mu \left(d\right){log}^{k}d}{1-z^d}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d{z}^n.$$

(2.8)

If the termwise limit could be justified, then we would have

$$\underset{z\to 1^{-}}{lim}(1-z)G_f\left(z\right)=\sum _{d\ge 1}\mu \left(d\right){log}^{k}d\underset{z\to 1^{-}}{lim}{\displaystyle \frac{1-z}{1-z^d}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d{z}^n=\sum _{d\ge 1}\mu \left(d\right){log}^{k}d{\displaystyle \frac{N_f\left(d\right)}{d}}$$

(2.9)

where $N_f\left(d\right)$ is the number of solutions of the congruence $f\left(n\right)\equiv 0(modd)$. Furthermore Conrad proved the following result, in a slightly different but equivalent form.

Proposition 2.4 

([4], Theorem 7). The right side of (2.9) converges and

$$\sum _{d\ge 1}\mu \left(d\right){log}^{k}d{\displaystyle \frac{N_f\left(d\right)}{d}}=c\left(f\right)>0$$

where $c\left(f\right)$ is as in (1.2).

Thus if the termwise limit could be justified, the Bateman-Horn conjecture would be proved. The termwise limit can be done for the case $f\left(x\right)=x$ of the prime number theorem. Hindry and Rivoal [9] also proved Dirichlet’s theorem on primes in arithmetic progressions by using Golomb’s method and thus the termwise limit can be done for linear polynomials. However for nonlinear polynomials the termwise limit is elusive.

3. Proof of Theorem 1.6

Recall that

$${\psi }_f\left(x\right)=\sum _{n\le x}\mathsf{\Lambda }\left(f_1\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right),$$

(3.1)

and for $\left|z\right|<1$,

$$G_f\left(z\right)={(-1)}^{k}k!\sum _{n\ge 1}\mathsf{\Lambda }\left(f_1\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)z^n.$$

(3.2)

The function $\mathsf{\Lambda }\left(n\right)$ is not so good since it takes values not only for primes but also for prime powers. Thus we let

$${\theta }_f\left(x\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n\le x}{f_1\left(n\right),\cdots ,f_k\left(n\right)\mathrm{prime}}}}log{f}_1\left(n\right)\cdots log{f}_k\left(n\right).$$

(3.3)

Baier [2] proved that as $x\to \infty$, the asymptotic

$${\psi }_f\left(x\right)\sim c\left(f\right)x$$

is equivalent to

$${\theta }_f\left(x\right)\sim c\left(f\right)x$$

by showing that

$${\psi }_f\left(x\right)-{\theta }_f\left(x\right)=O\left(\sqrt{x}{(logx)}^{k+1}\right).$$

(3.4)

Now corresponding to ${\theta }_f\left(x\right)$ we let

$$\begin{array}{ccc}\hfill H_f\left(z\right)& =& {(-1)}^{k}k!\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n\ge 1}{f_1\left(n\right),\cdots ,f_k\left(n\right)\mathrm{prime}}}}log{f}_1\left(n\right)\cdots log{f}_k\left(n\right)z^n\hfill \end{array}$$

(3.5)

$$\begin{array}{ccc}& =& {(-1)}^{k}k!\sum _{n\ge 1}\mathsf{\Lambda }\left(f_1\left(n\right)\right)|\mu \left(f_1\left(n\right)\right)|\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)\left|\mu \left(f_k\left(n\right)\right)\right|z^n.\hfill \end{array}$$

(3.6)

Now to prove Conjecture 1.1 our idea is this: suppose it is not true, that is $H_f\left(1\right)<\infty$, then we shall derive ${lim}_{z\to 1^{-}}(1-z)H_f\left(z\right)=c\left(f\right)>0$ and thus leads to a contradiction.

Recall that

$$\mathsf{\Lambda }\left(f_1\left(n\right)\right)\mathsf{\Lambda }\left(f_2\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)={\displaystyle \frac{{(-1)}^k}{k!}}\sum _{d\left|f\right(n)}\mu \left(d\right){log}^{k}d.$$

(3.7)

Then proceeding as in Hindry and Rivoal [9] we have

$$\begin{array}{ccc}\hfill H_f\left(z\right)& =& \sum _{n\ge 1}\left(\sum _{d\left|f\right(n)}\mu \left(d\right){log}^{k}d\right)|\mu \left(f_1\left(n\right)\right)|\cdots |\mu \left(f_k\left(n\right)\right)|z^n\hfill \end{array}$$

(3.8)

$$\begin{array}{ccc}& =& \sum _{d\ge 1}{\displaystyle \frac{\mu \left(d\right){log}^{k}d}{1-z^d}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d|\mu \left(f_1\left(n\right)\right)|\cdots |\mu \left(f_k\left(n\right)\right)|z^n.\hfill \end{array}$$

(3.9)

Notice that

$$\sum _{n\ge 1}\mathsf{\Lambda }\left(f_1\left(n\right)\right)|\mu \left(f_1\left(n\right)\right)|\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)\left|\mu \left(f_k\left(n\right)\right)\right|z^n=\sum _{n\ge 1}\mathsf{\Lambda }\left(f_1\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)z^n$$

(3.10)

$$+\sum _{n\ge 1}\left(\mathsf{\Lambda }\left(f_1\left(n\right)\right)|\mu \left(f_1\left(n\right)\right)|\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)\left|\mu \left(f_k\left(n\right)\right)\right|-\mathsf{\Lambda }\left(f_1\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)\right)z^n$$

It follows from (3.4) that

$$\underset{z\to 1^{-}}{lim}(1-z)\sum _{n\ge 1}\left(\mathsf{\Lambda }\left(f_1\left(n\right)\right)|\mu \left(f_1\left(n\right)\right)|\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)\left|\mu \left(f_k\left(n\right)\right)\right|-\mathsf{\Lambda }\left(f_1\left(n\right)\right)\cdots \mathsf{\Lambda }\left(f_k\left(n\right)\right)\right)z^n=0.$$

(3.11)

Explicitly it follows from (3.8)-(3.10) and (3.6)-(3.7) that

$$H_f\left(z\right)=\sum _{d\ge 1}{\displaystyle \frac{\mu \left(d\right){log}^{k}d}{1-z^d}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d{z}^n+\sum _{d\ge 1}{\displaystyle \frac{\mu \left(d\right){log}^{k}d}{1-z^d}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d\left(|\mu \left(f_1\left(n\right)\right)|\cdots |\mu \left(f_k\left(n\right)\right)|-1\right)z^n$$

(3.12)

and by (3.11) (which is in turn followed from (3.4)), that

$$\underset{z\to 1^{-}}{lim}(1-z)\sum _{d\ge 1}{\displaystyle \frac{\mu \left(d\right){log}^{k}d}{1-z^d}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d\left(|\mu \left(f_1\left(n\right)\right)|\cdots |\mu \left(f_k\left(n\right)\right)|-1\right)z^n=0.$$

(3.13)

Therefore from (3.12) and (3.13) we have

$$\underset{z\to 1^{-}}{lim}(1-z)H_f\left(z\right)=\underset{z\to 1^{-}}{lim}\sum _{d=1}^{\infty }\mu \left(d\right){log}^{k}d\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d{\displaystyle \frac{(1-z)z^n}{1-z^d}}.$$

(3.14)

Now suppose on the contrary that Conjecture 1.1 is not true, that is $H_f\left(1\right)<\infty$, then by the definition (3.5), $H_f\left(z\right)$ is a polynomial and thus absolutely convergent everywhere in the complex plane. Consequently the termwise limit is applicable for (3.14) and we have

$$\underset{z\to 1^{-}}{lim}(1-z)H_f\left(z\right)=\sum _{d=1}^{\infty }\mu \left(d\right){log}^{k}d\underset{z\to 1^{-}}{lim}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n=1}{d\left|f\right(n)}}}^d{\displaystyle \frac{(1-z)z^n}{1-z^d}}=\sum _{d\ge 1}\mu \left(d\right){log}^{k}d{\displaystyle \frac{N_f\left(d\right)}{d}}>0.$$

But this implies that $H_f\left(1\right)=\infty$ and we obtain a contradiction.