Weixiong Li
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
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。
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.
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。
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…
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.
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…
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…
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。
Given $β\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^β x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $π(x) = \mathrm{Li}(x) + Ω\bigl(x\exp(-c'(\log x)^{\fracβ{β+1}})\bigr)$ for some $c'>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder.
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。
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 note on the Cramér-Granville model
该研究展示了一个满足Bateman-Horn猜想、哥德巴赫猜想及特定素数分布估计的集合存在性。
Liouville function, von Mangoldt function and norm forms at random binary forms
该研究分析高度排序的d次二元型取值处算术函数的平均行为，得到Chowla与Bateman-Horn猜想的平均版本，还证明当e整除d时几乎所有Châtelet簇满足有理哈塞原理，证实Colliot-Thélène猜想的平均版本。
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。