arXiv subjects

Milton Espinoza

Publications and source records attributed to Milton Espinoza.

4 recordsLinked to original sources

arXiv2023

The Barnes-Hurwitz zeta cocycle at $s=0$ and Ehrhart quasi-polynomials of triangles

Following a theorem of David R. Hayes, we give a geometric interpretation of the special value at $s=0$ of certain $1$-cocycle on $\mathrm{PGL}_2(\mathbb{Q})$ previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at $s=0$, a generalization and a new proof of Hayes' theorem, and an elegant summation formula for the $0$th coefficient of the Ehrhart quasi-polynomial of certain triangles in $\mathbb{R}^2$.

math.NT↗

arXiv2020

On certain zeta integral: Transformation formula

We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $ζ$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$. Although these two functions differ from each other, we can use $\mathcal{L}$ to study $ζ$. In fact, the transformation formula for $\mathcal{L}$ under $G$-transformations provides us with a new perspective on the special values of both $ζ$ and its $s$-derivative. In particular, we obtain Kronecker limit formulas for $ζ$ when restricted to points fixed by elements of $G$. As an illustration of this principle, we evaluate certain generalized Lambert series at roots of unity, establishing pertinent algebraicity results. Also, we express the Barnes' multiple gamma function at roots of unity as a certain infinite product. It should be mentioned that this work also considers twisted versions of $ζ$.

math.NT↗

arXiv2019

Twisters and signed fundamental domains for number fields

We give a signed fundamental domain for the action on $\mathbb{R}^{r_1}_+\times{\mathbb{C}^*}^{r_2}$ of the totally positive units $E_+$ of a number field $k$ of degree $n=r_1+2r_2$ which we assume is not totally complex. Here $r_1$ and $r_2$ denote the number of real and complex places of $k$ and $\mathbb{R}_+$ denotes the positive real numbers. The signed fundamental domain consists of $n$-dimensional $k$-rational cones $C_α$, each equipped with a sign $μ_α=\pm1$, with the property that the net number of intersections of the cones with any $E_+$-orbit is 1. The cones $C_α$ and the signs $μ_α$ are explicitly constructed from any set of fundamental totally positive units and a set of $3^{r_2}$ "twisters", i.e. elements of $k$ whose arguments at the $r_2$ complex places of $k$ are sufficiently varied. Introducing twisters gives us the right number of generators for the cones $C_α$ and allows us to make the $C_α$ turn in a controlled way around the origin at each complex embedding.

math.NT↗

arXiv2019

Signed Shintani cones for number fields with one complex place

We give a signed fundamental domain for the action on $\mathbb{C}^*\times \mathbb{R}_+^{n-2}$ of the totally positive units $E(k)_+$ of a number field $k$ of degree $n$ and having exactly one pair of complex embeddings. This signed fundamental domain, built of $k$-rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of $k$.

math.NT↗