The Average of the Smallest Prime in a Conjugacy Class

Abstract

Abstract Let C be a conjugacy class of $S_{n}$ and K an $S_{n}$-field. Let $n_{K,C}$ be the smallest prime, which is ramified or whose Frobenius automorphism Frob$_{p}$ does not belong to C. Under some technical conjectures, we show that the average of $n_{K,C}$ is a constant. We explicitly compute the constant. For $S_{3}$- and $S_{4}$-fields, our result is unconditional. Let $N_{K,C}$ be the smallest prime for which Frob$_{p}$ belongs to C. We obtain the average of $N_{K,C}$ under some technical conjectures. For n = 3 and C = [(12)], we have the average value of $N_{K,C}$ unconditionally.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF