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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]

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted $\mathscr{N}$). When K is an imaginary quadratic extension, $\mathscr{N}$ is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in $\mathscr{N}$ can be written as $\alpha/\overline{\alpha}$ for some $\alpha \in \mathscr{O}_K$, which yields another ordering of $\mathscr{N}$ given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that $\mathscr{N}$ is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that $\mathscr{N}$ is equidistributed with respect to norm, under the map β ↦ log|β|(mod log|ϵ2|) where ϵ is a fundamental unit of $\mathscr{O}_K$. [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]

The triple Euler sum defined by \[ \begin{array}{rcl} \zeta(p, q, r) & = & \sum\limits_{\ell=3}^{\infty} \frac{1}{\ell^{p}} \sum\limits_{k=2}^{\ell-1} \frac{1}{k^{q}} \sum\limits_{j=1}^{k-1} \frac{1}{j^{r}} \\ [10pt] & = & \sum\limits_{l=1}^{\infty} \sum\limits_{k=1}^{\infty} \sum\limits_{j=1}^{\infty} \frac{1}{(j+k+\ell)^{p}(j+k)^{q}j^{r}} \end{array} \] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or two when q = r =. Performing differentiations with respect to the specified variable gives the explicit evaluations of ζ(p, q, r) for general p, q, r. In particular, the values of ζ(n, 1, 1), ζ(2n + 1, 1, 2), ζ(2n + 1, 2, 1) and ζ(2n, 2, 2) are given explicitly in terms of classical double Euler sums. Also ζ(2n, 1, 2m + 1) and ζ(2n + 1, 1, 2m) are obtained for all positive integers m and n. [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]

In this paper, we give necessary and sufficient conditions for 8-ranks of narrow class groups of $F = \mathbb{Q}(\sqrt{2pq})$, distinct primes p ≡ q ≡ 1 mod 4. The results are useful for numerical computations. [ABSTRACT FROM AUTHOR]