论文标题
关于Mitsui的质数定理的注释与Siegel Zeros
Notes on Mitsui's Prime Number Theorem with Siegel zeros
论文作者
论文摘要
在这些注释中，我们从1957年开始完善Mitsui的质数定理，该定理对于一个数字字段$ k $，可以预测$ k \ otimes _ {\ mathbf q} \ mathbf r $中有多少个质量元素，通过结合潜在的siegel siegel zeros of Hecke l-functions。这允许模量$ \ mathbf n（\ mathfrak q）$的规范以伪元素速率增长，相对于凸的尺寸$ x $，而不是$ \ log x $。额外的灵活性和精度对于我们将来对素数元素线性模式的研究的应用至关重要。我们还希望我们的最新博览会能使Mitsui的工作能够被更广泛的数学受众访问。
In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts how many prime elements there are in bounded convex sets in $K \otimes_{\mathbf Q} \mathbf R$, by incorporating potential Siegel zeros of Hecke L-functions. This allows the norm of the modulus $\mathbf N (\mathfrak q) $ to grow at a pseudopolynomial rate with respect to the size $X$ of the convex set as opposed to powers of $\log X$. The extra flexibility and precision will be essential in our future application to the study of linear patterns of prime elements. We also hope that our updated exposition will make Mitsui's work accessible to a wider mathematical audience.
