Abstract. Let K be a cyclic number field of prime degree . Heilbronn showed that for a given there are only finitely many such fields that are normEuclidean. In the case of = 2 all such norm-Euclidean fields have been identified, but for = 2, little else is known. We give the first upper bounds on the discriminants of such fields when > 2. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results.
In this paper we examine Grosswald's conjecture on g(p), the least primitive root modulo p. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that g(p) < √ p − 2 for all p > 409. Our method also shows that under GRH we have ĝ(p) < √ p − 2 for all p > 2791, where ĝ(p) is the least prime primitive root modulo p.
We give an explicit version of a result due to D. Burgess. Let χ be a non-principal Dirichlet character modulo a prime p. We show that the maximum number of consecutive integers for which χ takes on a particular value is less than πe √ 6 3 + o(1) p 1/4 log p, where the o(1) term is given explicitly.
Let ℓ be a rational prime. Previously, abelian ℓ-towers of multigraphs were introduced which are analogous to Z ℓ -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the ℓ-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for Z ℓ -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian ℓ-towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in Z ℓ and then study the special value at s = 1 of the Artin-Ihara L-function ℓ-adically.
A formula for the sum of any positive-integral power of the first N positive integers was published by Johann Faulhaber in the 1600s. In this paper, we generalize Faulhaber's formula to non-integral complex powers with real part greater than −1.
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