The least prime ideal in a given ideal class

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 2 K log(D K ) 2 and also all but o(h K ) of them have a prime ideal with norm less than h K log(D K ) 2+ǫ . For imaginary quadratic fields K = Q( √ D), by assuming Conjecture 1.2 (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.