Notes on Mitsui's Prime Number Theorem with Siegel zeros

Abstract

In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts the number of prime elements in a bounded convex set in $K{\otimes }_{\mathbf{Q}}\mathbf{R}$ and in a specified congruence class modulo an ideal $\mathfrak{q}$, 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 $e^{\sqrt{\log X}/O(1)}$ with respect to the size $X$ of the convex set as opposed to powers of $\log X$. The extra flexibility and precision are essential in our sequel work on linear patterns of prime elements. We also hope that our updated exposition will make Mitsui's work accessible to a wider mathematical audience.

arXiv commenttypos corrected. 30 pages, 3 figures