Multiple Dirichlet Series for Affine Weyl Groups

Let $W$ be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting $\tilde{A}_n$ for $n$ even. We construct a multiple Dirichlet series $Z(x_1, \ldots x_{n+1})$, meromorphic in a half-space, satisfying a group $W$ of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this way is optimal for computing moments of character sums and L-functions, including the fourth moment of quadratic L-functions at the central point via $\tilde{D}_4$ and the second moment weighted by the number of divisors of the conductor via $\tilde{A}_3$. We also give evidence to suggest that this series appears as a first Fourier-Whittaker coefficient in an Eisenstein series on the twofold metaplectic cover of the relevant Kac-Moody group. The construction is limited to the rational function field $\mathbb{F}_q(t)$, but it also describes the $p$-part of the multiple Dirichlet series over an arbitrary global field

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Perfect powers in polynomial power sums

We prove that a non-degenerate simple linear recurrence sequence $(G_n(x))_{n=0}^{\infty}$ of polynomials satisfying some further conditions cannot contain arbitrary large powers of polynomials if the order of the sequence is at least two. In other words we will show that for $m$ large enough there is no polynomial $h(x)$ of degree $\geq 2$ such that $(h(x))^m$ is an element of $(G_n(x))_{n=0}^{\infty}$. The bound for $m$ depends here only on the sequence $(G_n(x))_{n=0}^{\infty}$. In the binary case we prove even more. We show that then there is a bound $C$ on the index $n$ of the sequence $(G_n(x))_{n=0}^{\infty}$ such that only elements with index $n \leq C$ can be a proper power.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1810.1214

Horadam sequences: A survey update and extension.

We give an update on work relating to Horadam sequences that are generated by a general linear recurrence formula of order two. This article extends a first ever survey published in early 2013 in this Bulletin, and includes coverage of a new research area opened up in recent times.N/