In our companion paper [28], we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields, and we carried out these computations in a variety of cases. Here we develop a method, based on Fourier analysis and algebraic geometry, which exploits these Fourier transform formulas to yield level of distribution results, in the sense of analytic number theory. Such results are of the shape typically required for a variety of sieve methods. As an example of such an application we prove that there are $$\gg \frac{X}{\log X}$$ ≫ X log X quartic fields whose discriminant is squarefree, bounded above by X , and has at most eight prime factors.

中文翻译：

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前齐次向量空间中筛分问题的分布水平

在我们的配套论文 [28] 中，我们开发了一种有效的代数方法来计算在有限域上的前齐次向量空间上定义的某些函数的傅里叶变换，并且我们在各种情况下进行了这些计算。在这里，我们开发了一种基于傅里叶分析和代数几何的方法，它利用这些傅里叶变换公式来产生解析数论意义上的分布结果水平。此类结果具有各种筛分方法通常所需的形状。作为此类应用的一个示例，我们证明存在 $$\gg \frac{X}{\log X}$$ ≫ X log X 四次域，其判别式是无平方的，以 X 为边界，并且最多有 8 个素数因素。