arXiv subjects

Alexander Dahl

Publications and source records attributed to Alexander Dahl.

3 recordsLinked to original sources

arXiv2016

The distribution of class numbers in a special family of real quadratic fields

We investigate the distribution of class numbers in the family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ corresponding to fundamental discriminants of the form $d=4m^2+1$, which we refer to as Chowla's family. Our results show a strong similarity between the distribution of class numbers in this family and that of class numbers of imaginary quadratic fields. As an application of our results, we prove that the average order of the number of quadratic fields in Chowla's family with class number $h$ is $(\log h)/2G$, where $G$ is Catalan's constant. With minor modifications, one can obtain similar results for Yokoi's family of real quadratic fields $\mathbb{Q}(\sqrt{d})$, which correspond to fundamental discriminants of the form $d=m^2+4$.

math.NT↗

arXiv2016

Distribution of class numbers in continued fraction families of real quadratic fields

We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued fraction expansions of the form $\sqrt {D(n)}=[f(n), [u_1, u_2, \dots, u_{s-1}, 2f(n)]]$ with fixed coefficients $u_1, \dots, u_{s-1}$ and generalize well-known families such as Chowla's $4n^2+1$, for which analogous results were recently proved by Dahl and Lamzouri.

math.NT↗

arXiv2016

Subconvexity for a double Dirichlet series and non-vanishing of $L$-functions

We study a double Dirichlet series of the form $\sum_{d}L(s,χ_{d}χ)χ'(d)d^{-w}$, where $χ$ and $χ'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to $\mathbb{C}^{2}$. A convexity bound at the central point is established to be $(MN)^{3/8+\varepsilon}$ and a subconvexity bound of $(MN(M+N))^{1/6+\varepsilon}$ is proven. The developed theory is used to prove an upper bound for the smallest positive integer $d$ such that $L(1/2,χ_{dN})$ does not vanish, and further applications of subconvexity bounds to this problem are presented.

math.NT↗