Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, Stephan Ramon Garcia
The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins.
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We evaluate the Bateman-Horn constant for the polynomial $x^3+x+1$.
This is an expository article on relating the Chebotarev Density Theorem to the Bateman-Horn constant.
We describe some studies related to the frequency of prime values of integer polynomials.
Worse than square-root cancellation in Bateman-Horn's conjecture
研究 Bateman-Horn 猜想在指数范围内的平均误差项，证明了其渐近性，并发现误差项比平方根抵消更差。
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.
The Bateman-Horn conjecture on average for generalized von Mangoldt Functions
研究广义von Mangoldt函数的Bateman-Horn猜想，证明对k=2,3，几乎所有Bouniakowsky多项式表示恰好为k个素数乘积的整数。
Sets of integers satisfying Bateman-Horn statistics
本文研究了满足Bateman-Horn统计的整数集，证明了随机整数集几乎必然满足该统计，并给出了强误差项。
We establish cancellation in short sums of certain special trace functions over $\mathbb{F}_q[u]$ below the Pólya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $\mathbb{F}_q[u]$-analog of Chowla's conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $\mathbb{F}_q[u]$.
Procounting measures and the Bateman--Horn conjecture
提出全局域上S-整数环的profinite完备化上的Bateman-Horn猜想的profinite版本，并基于新定义的procounting测度与经典版本进行比较。
In this paper, we determine the bound of the valency of the odd circulant graph which guarantees to be a Ramanujan graph for each fixed number of vertices. In almost of the cases, the bound coincides with the trivial bound, which comes from the trivial estimate of the largest non-trivial eigenvalue of the circulant graph. As exceptional cases, the bound in fact exceeds the trivial one by two. We then prove that such exceptionals occur only in the cases where the number of vertices has at most two prime factors and is represented by a quadratic polynomial in a finite family and, moreover, under…
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We present heuristic arguments and computational evidence based on the Bateman-Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\rm PSL}_n(q)$, ${\rm PSU}_n(q)$ and ${\rm PSp}_{2n…
The Multidimensional Bateman Horn Conjecture: Effective Estimates for the Tail of the Singular Series and an Algorithm for Computation with Guaranteed Accuracy
本文基于概率启发式和多元巴特曼-霍恩猜想，研究多项式取素数值的数量渐进行为，通过代数几何和解析数论方法，获得奇异乘积尾部的估计，并对对角多项式改进了衰减指数和常数。
Bateman-Horn, polynomial Chowla and the Hasse principle with probability 1
本文研究了在高度有序下，各种算术函数在次数为d的多项式f的值上的平均行为，从而建立了Bateman-Horn猜想、多项式Chowla猜想的平均版本，并探讨了积分Hass原理在规范形式方程中的基本问题。同时，本文还量化了渐进行为中的误差项和f的例外集的大小，均具有任意对数幂节省。
We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and Hölder in the 1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ has just six prime factors. The conjecture suggests that there are infinitely many such primes, by providing heuristic estimates for their distribution which agree clo…
In this paper, we make a conjecture (conjecture 1) related to the Bateman-Horn conjecture and proceed to study the roots of $x^2+1$ and $x^2+2$ to prime moduli, assuming the truth of the Bateman-Horn conjecture and conjecture 1 and using the Erdős-Turan-Koksma inequality.
Average Bateman--Horn for Kummer polynomials
本文证明了多项式 $n^r+k$ 在特定范围内的平均素数值分布，并据此推导出二次扩张下相关开簇的 Hasse 原理变体结论，其核心方法是引入了针对精确阶为 $r$ 的 Dirichlet 特征的新大筛法不等式。
Skolem Meets Bateman-Horn
本文围绕斯科伦问题，构造了密度至少1/8的通用斯科伦集，并基于马丁对贝特曼-霍恩猜想的统一表述证明该集合密度为1，推进了该开放问题的研究。
This paper demonstrates that from the Cramer's, Hardy-Littlewood's and Bateman-Horn's conjectures (suggest that the probability of a large positive integer being $x$ a prime - $\frac {1} {\ln(x)}$) it follows that the events consisting in a positive integer $x$ being not divisible by different primes are dependent with the ratio $0,5e^γ$ ($γ$ - Euler's constant). In establishing Hardy-Littlewood's and Bateman-Horn's conjectures, their authors followed the first suggestion by another one assuming the independence of the above-mentioned events, which on the basis of the first suggestion and Mert…
We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$) polynomials $F_1,\ldots,F_m\in\mathbf{F}_q[t][x]$, with $q$ odd, we show that the number of $f\in\mathbf{F}_q[t]$ of degree $n\ge\max(3,\mathrm{deg}_t F_1,\ldots,\mathrm{deg}_t F_m)$ such that all $F_i(t,f)\in\mathbf{F}_q[t],1\le i\le m$ are irreducible is $$\left(\prod_{i=1}^m\frac{μ_i}{N_i}\right) q^{n+1}\left(1+O_{m,\,\max\mathrm{deg} F_i,\,n}\left(q^{-1/2}\right…
The distribution of prime values of random polynomials
研究随机多项式素数取值分布，证明多项式满足Bateman-Horn猜想及Poisson尾猜想，揭示符号模式极限分布。
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $ϕ(p-1) \geq ϕ(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{…
A recent construction by Amarra, Devillers and Praeger of block designs with specific parameters depends on certain quadratic polynomials, with integer coefficients, taking prime power values. The Bunyakovsky Conjecture, if true, would imply that each of them takes infinitely many prime values, giving an infinite family of block designs with the required parameters. We have found large numbers of prime values of these polynomials, and the numbers found agree very closely with the estimates for them provided by Li's recent modification of the Bateman-Horn Conjecture. While this does not prove t…
Although we expect to find many smooth numbers (i.e., numbers with no large prime factors) among the values taken by a polynomial with integer coefficients, it is unclear what the asymptotic number of such smooth values should be; this is in contrast to the related problem of counting the number of prime values of a polynomial, for which Bateman and Horn published a conjectured asymptotic formula that is widely believed to be true. We discuss how to employ the Bateman-Horn conjecture to derive an asymptotic formula for the number of smooth values of a polynomial, with the smoothness parameter …
We heuristically analyze the Cocks-Pinch method by using the Bateman-Horn conjecture. Especially, we present the first known heuristic which suggests that any efficient construction of pairing-friendly elliptic curves can efficiently generate such curves over pairing-friendly fields, naturally including the Cocks-Pinch method. Finally, some numerical evidence is given.
We study the difference between the number of primitive roots modulo $p$ and modulo $p+k$ for prime pairs $p,p+k$. Assuming the Bateman-Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs $p,p+k$, the dominant inequality is reversed.
Numerical evidence suggests that for only about $2\%$ of pairs $p,p+2$ of twin primes, $p+2$ has more primitive roots than does $p$. If this occurs, we say that $p$ is exceptional (there are only two exceptional pairs with $5 \leq p \leq 10{,}000$). Assuming the Bateman-Horn conjecture, we prove that at least $0.47\%$ of twin prime pairs are exceptional and at least $65.13\%$ are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.
For k greater than 1 and r different from 0, let pi^k_{2r}(x) denote the number of prime pairs (p,p^k+2r) with p not exceeding (large) x. By the Bateman-Horn conjecture, the function pi^k_{2r}(x) should be asymptotic to (2/k)C^k_{2r}li_2(x), with certain specific constants C^k_{2r}. Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy-Littlewood constants C_{2r} for prime pairs (p,p+2r). The conjecture is supported by extensive numerical work.
Prime Number Theorems for Polynomials from Homogeneous Dynamics
本文借助齐次动力学、Linnik 等分布、锥逼近与上界筛法，证明矩阵行列式及 Pfaffian 等多项式的多元 Bateman–Horn 素数渐近。
Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture
本文计算了 $6.25\ imes 10^{28}$ 以内所有形如 $m^2+1$ 的素数，并据此验证了哥德巴赫的一个较少人知的猜想，同时引入“哥德巴赫冠军”概念并证明其条件性结果。
An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $π(kn)$ is prime, where $π(x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive integer $n$ the set $\{π(kn):\ k=1,2,3,\ldots\}$ contains infinitely many $P_2$-numbers which are products of at most two primes. We also prove that under the Bateman--Horn conjecture the set $\{π(4k):\ k=1,2,3,\ldots\}$ contains infinitely many primes.
Square-free values of polynomials on average
本文针对多项式无平方因子值计数的平均猜想，改进了取值长度x与多项式高度的依赖关系，方法是调整已有研究中提出的相关猜想研究方法。
We reinterpret ideas in Klein's paper on transformations of degree $11$ from the modern point of view of dessins d'enfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman-Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for som…
Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation
本文提出一类基于Heegner数的新二次多项式族，通过代数调优和Bateman-Horn猜想优化，可生成高密度素数，为解析数论及密码学、信号处理提供新方向。
Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$
该研究推导了${\ m PSL}_2(p)$（p为素数）非单位真子群的同构类、共轭类及两类自正规相关共轭类的计数公式，证明了p>37时的四类下界，结合数值搜索与数论结果给出了下界可达性与上界结论。
A number-theoretic problem concerning pseudo-real Riemann surfaces
本文证明伪实 Riemann 曲面相关整数的渐近计数定理，并据此提出素数计数猜想，且以 Bateman–Horn 猜想和计算证据加以支持。
On the Divisibility Relation $σ(n)\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture
该论文研究固定正整数h下σ(n)｜σ(n+h)的整除关系及σ(n+h)=λσ(n)的解数，结合Schinzel假设H、Bateman–Horn猜想推导相关结果并提出σ(n+h)=kσ(n)有无限多解的猜想。
Let $p$ be a prime number. A chain $\{p,2p+1,4p+3,\cdots,(p+1)2^{l(p)-1}-1\}$ is called the Cunningham chain generated by $p$ if all elements are prime number and $(p+1)2^{l(p)}-1$ is composite. Then $l(p)$ is called the length of the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the number of prime $p\leq N$ such that $l(p)\geq k$ is asymptotically equal to $B_k N/(\log N)^k$ with a real $B_k>0$ for all natural number $k$. This suggests that $l(p)=Ω(\log p/\log\log p)$. However, so far no good estimation is known. It has not even been proven whether $\limsup_{p\to\infty…
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