论文标题
平均贝特曼 - kummer多项式
Average Bateman--Horn for Kummer polynomials
论文作者
论文摘要
对于任何$ r \ in \ mathbb {n} $，几乎全部$ k \ in \ mathbb {n} $小于$ x^r $，我们表明polyenmial $ f（n）= n^r + k $以$ n $范围为$ n $ ranges prime的数量为$ n $ ranges，范围为1到$ x $。结果，我们推断出有关HASSE原理的变体的语句以及某些开放品种的积分原理，以由$ n_ {k/\ mathbb {q}}的等式定义的某些开放品种定义，\ mathbf {z}}（\ mathbf {z}）= t^r +k \ k \ neq 0 $ k/k/k/k/kmathbb iS a use a a us a a us a a qudriestion。我们证明中的一个关键要素是针对精确顺序$ r $的Dirichlet字符的新大筛子不平等。
For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$ smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes the expected number of prime values as $n$ ranges from 1 to $x$. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\mathbb{Q}}(\mathbf{z}) = t^r +k \neq 0$ where $K/\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.
