Counting Algebraic Units with Bounded Height

Abstract

Let K be an algebraic number field of degree d and let U denote its group of units. Suppose U has rank r and regulator R. Let U(q) denote the number of units in U with height less than q. We obtain an asymptotic formula for U(q) of the shape U(q) = A(log q)r + O((log q)r − 1 − δ), where δ is given explicitly in terms of d and R.