The least prime ideal in a given ideal class

2018/02/17 by Naser T. Sardari, Sardari, Naser T.
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1802.06193

openalex publication_date 2018/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field with the discriminant D_K and the class number h_K, which has bounded degree over ℚ. By assuming GRH, we prove that every ideal class of K contains a prime ideal with norm less than h_K²log(D_K)² and also all but o(h_K) of them have a prime ideal with norm less than h_Klog(D_K)^2+ε. For imaginary quadratic fields K=ℚ(√(D)), by assuming Conjecture~\refpiarcor (a weak version of the pair correlation conjecure), we improve our bounds by removing a factor of log(D) from our bounds and show that these bounds are optimal.

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