A note on the Bateman-Horn conjecture

Abstract

Text

We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version of the conjecture which empirically demonstrates remarkable accuracy even for modest values of primes.

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Introduction

In 1962, Bateman and Horn [1], [2] proposed the following general conjecture concerning the distribution of primes generated from a set of polynomials.¹

Conjecture 1.1 The Bateman-Horn Conjecture

Let $f_1,f_2,\cdots ,f_M\in \mathbb{Z}[x]$ be distinct irreducible polynomials with positive leading coefficients, and let

$$\pi (f_1,f_2,\cdots ,f_M;x)=\mathrm{\#}\{n\le x:f_1(n),f_2(n),\cdots ,f_M(n)\mathit{\text{are primes}}\}$$

Suppose that $f(n)={\prod }_{i=1}^M{f}_i(n)$ does not vanish identically modulo any prime, then for large values of x we have the following asymptotic expression

$$\pi (f_1,f_2,\cdots ,f_M;x)\sim \frac{C(f_1,f_2,\cdots ,f_M)}{{\prod }_{i=1}^M\mathrm{deg}{f}_i}\underset{2}{\overset{x}{\int }}\frac{dt}{{(\log t)}^M}$$

in which

$$C(f_1,f_2,\cdots ,f_M)=\prod _p{(1-\frac{1}{p})}^{-M}(1-\frac{{\omega }_f(p)}{p})$$

where the infinite product is over all primes p, and ${\omega }_f(p)$ is the number of solutions to the congruence equation $f(n)\equiv 0(\mathit{\text{mod}}p)$.

The Bateman-Horn conjecture is very general, and many well-known conjectures, such as the Hardy-Littlewood Conjectures B, E, F, K, X, P [3], [4], are all special cases of this conjecture.

In proposing this conjecture which now bears their names, Bateman and Horn provided the following heuristic arguments [1]. Since the probability of any number x being a prime number is roughly $\frac{1}{\log x}$, the probability of $f_i(n)$ being a prime number is $\frac{1}{\log f_i(n)}$, which for large n can be approximated by $\frac{1}{(\mathrm{deg}{f}_i)\log n}$, hence the expression in equation (1.2).

Two questions naturally arise. (1) How good is the approximation of replacing $\frac{1}{\log f_i(n)}$ with $\frac{1}{(\mathrm{deg}{f}_i)\log n}$, and (2) how much better would the ‘Bateman-Horn Conjecture’ be without this approximation? To the best of this author's knowledge, there seems to be little discussion of these two questions in the literature [2].

Let us perform a naive estimation to answer the first question. Let $f_i(n)=a_i{n}^{m_i}+b_i{n}^{m_i-1}+\cdots$, then $\log f_i(n)=\log (a_i{n}^{m_i})+\log (1+O(\frac{1}{n}))=m_i\log n+\log a_i+O(\frac{1}{n})$. We therefore see that when the polynomial is not monic, the relative errors caused by ignoring the term $\log a_i$ could be potentially significant for typical counting values around $n\sim {10}^{15}$. It is interesting to note that historically most of empirical studies have been performed on monic polynomials where $\log a_i=0$, therefore the errors never showed up in those studies [5], [6]. The Hardy-Littlewood Conjecture F was expressed as a function of primes rather than the variable n [3], [4], therefore the issue with non-monic polynomials is avoided.

In section 2, we will propose a modified version of the Bateman-Horn conjecture. In section 3, we then perform empirical computations to compare the two versions of the conjecture with our numerical results.