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Counting invertible matrices and uniform distribution

Christian Roettger (2005)

Journal de Théorie des Nombres de Bordeaux

Consider the group SL 2 ( O K ) over the ring of algebraic integers of a number field K . Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let SL 2 ( O K , t ) be the number of matrices in SL 2 ( O K ) with height bounded by t . We determine the asymptotic behaviour of SL 2 ( O K , t ) as t goes to infinity including an error term, SL 2 ( O K , t ) = C t 2 n + O ( t 2 n - η ) with n being the degree of K . The constant C involves the discriminant of K , an integral depending only on the signature of K , and the value of the Dedekind zeta function...

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