arXiv subjects

Kevin J. McGown

Publications and source records attributed to Kevin J. McGown.

22 records · Page 2Linked to original sources

arXiv2011

Norm-Euclidean Galois fields

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified, but for $\ell\neq 2$, little else is known. We give the first upper bounds on the discriminants of such fields when $\ell>2$. 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.

math.NT↗

arXiv2011

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $Δ=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for $ζ_K(s)$. If $38(\ell-1)^2(\log f)^6\log\log f<f$, then $K$ is not norm-Euclidean.

math.NT↗

arXiv2010

On the constant in Burgess' bound for the number of consecutive residues or non-residues

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 $\left\{\frac{πe\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.

math.NT↗

arXiv2010

On the second smallest prime non-residue

Let $χ$ be a non-principal Dirichlet character modulo a prime $p$. Let $q_1<q_2$ denote the two smallest prime non-residues of $χ$. We give explicit upper bounds on $q_2$ that improve upon all known results. We also provide a good upper estimate on the product $q_1 q_2$ which has an upcoming application to the study of norm-Euclidean Galois fields.

math.NT↗