We prove a version of Schanuel's theorem in the noncommutative case: we provide an asymptotic formula for the number of one-dimensional left subspaces of of height at most , where is a finite dimensional rational division algebra, a positive integer and a real number. The height, as considered in a previous paper, is defined with the help of a maximal order in and a positive anti-involution. We give a completely explicit main term involving class number, regulator, discriminant and zeta function of . We also compute an explicit error term.
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Gaël Rémond, Christine Zehrt-Liebendörfer, Le théorème de Schanuel pour un corps non commutatif. Rend. Sem. Mat. Univ. Padova 130 (2013), pp. 221–282