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In this paper, we study the notion of norm-Euclidean ideal class, which was defined by Lenstra [Euclidean ideal classes, Astérisque 61 (1979) 121-131]. Using a slight modification of an algorithm determining among other properties the Euclidean minimum described in [P. Lezowski, Computation of the Euclidean minimum of algebraic number fields, preprint (2011); http://hal.archives-ouvertes.fr/hal-00632997/en/], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari [The Euclidean condition in pure cubic and complex quartic fields, Math. Comput. 33 (1979) 389-398], we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class. Finally, we show that ${\mathbf{Q}}(\sqrt{2}, \sqrt{35})$, which is known to admit a non-principal Euclidean ideal class, has no norm-Euclidean ideal class. [ABSTRACT FROM AUTHOR]

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted M∞(α) for an algebraic number α. We show that the computation of M∞(α) can be reduced to a certain search through a finite set. Although it is an open problem to record the points of this set in general, we provide some examples where it is reasonable to compute and our result can be used to determine M∞(α). [ABSTRACT FROM AUTHOR]

We prove that for almost all real primitive characters χd of modulus |d|, the least positive integer nχd at which χd takes a value not equal to 0 and 1 satisfies nχd ≪ log|d|, and give a quite precise estimate on the size of the exceptional set. [ABSTRACT FROM AUTHOR]