Notes on Mitsui's Prime Number Theorem with Siegel zeros

TL;DR

This paper refines Mitsui's Prime Number Theorem for number fields by incorporating Siegel zeros, enabling more precise estimates of prime element distribution in convex sets, which is crucial for studying linear patterns of primes.

Contribution

It introduces a refined version of Mitsui's theorem that accounts for Siegel zeros, improving the growth rate estimates of prime elements in number fields.

Findings

Enhanced prime counting estimates with Siegel zeros

Growth rate of prime norms is pseudopolynomial rather than logarithmic

Facilitates future research on linear prime patterns

Abstract

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.