English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We establish an explicit bound for the least prime occurring in theChebotarev density theorem without any restriction. Let $L/K$ be any Galoisextension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be aconjugacy class in the Galois group of $L/K$. We show that there exists anunramified prime $\mathfrak{p}$ of $K$ such that $\sigma_{\mathfrak{p}}=C$ and$N \mathfrak{p} \le d_{L}^{B}$ with $B= 310$. This improves the value$B=12\,577$ as proven by Ahn and Kwon. In comparison to previous works on thesubject, we modify the weights to detect the least prime, and we use a newversion of Tur\'an's power sum method which gives a stronger Deuring-Heilbronn(zero-repulsion) phenomenon. In addition, we refine the analysis of how thelocation of the potential exceptional zero for $\zeta_L(s)$ affects the finalresult. We also use Fiori's numerical verification for $L$ up to a certaindiscriminant height. Finally, we provide a lower bound for the number ofunramified primes $\mathfrak{p}$ of $K$ such that $\sigma_{\mathfrak{p}}=C$.

Primes in the Chebotarev density theorem for all number fields