The mean values of totally real algebraic integers

Abstract

Let M p ( α ) {M_p}(\alpha ) be the pth root of the mean absolute values of the pth powers of a totally real algebraic integer α \alpha . For each fixed p > 0 p > 0 we study the set M p {\mathfrak {M}_p} of such M p ( α ) {M_p}(\alpha ) . We show that its structure is as follows: on the nonnegative real line it consists of some isolated points, followed by a small interval in which its structure is as yet undetermined. Beyond this small interval, it is everywhere dense.