Related papers: A Note on Counting Lattice Points in Bounded Domai…

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density…

We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 65$ and quantify how far we are…

We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper…

In this article we compare the set of integer points in the homothetic copy $n\Pi$ of a lattice polytope $\Pi\subseteq\R^d$ with the set of all sums $x_1+\cdots+x_n$ with $x_1,...,x_n\in \Pi\cap\Z^d$ and $n\in\N$. We give conditions on the…

A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the…

Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of $n \times n$ matrix points with finite field entries. We obtain exact formulas for matrix point counts on elliptic…

It is well known that the distribution of the prime numbers plays a central role in number theory. It has been known, since Riemann's memoir in 1860, that the distribution of prime numbers can be described by the zero-free region of the…

For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood…

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces,…

We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and…

We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving…

We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure…

We count subrings of small index of $\mathbb{Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers…

Given a unimodular lattice $\Lambda\subseteq \mathbb{R}^2$ consider the counting function $\mathcal{N}_\Lambda(T)$ counting the number of lattice points of norm less than $T$, and the remainder $\mathcal{R}_\Lambda(T)=\mathcal{N}(T)-\pi…

In this article we count tame $ SL_3 $- and $ SL_4 $-frieze patterns with width $ w $ over a finite field $ K $, as well as some tame $ SL_k $-frieze patterns for higher $ k $. Let $ n = w + k + 1 $. We consider the sets $ C_k(n) $ of…

For a locally compact second countable group G and a lattice subgroup Gamma, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and…

Lattice polyhedra $Q_1$ and $Q_2$ with the same tail cone are said to be normally located if every lattice point in the Minkowski sum $Q_1+Q_2$ is the sum of lattice points from $Q_1$ and $Q_2$, respectively. We prove that if the normal fan…

Let $k\in\mathbb{N}$ and let $f_1,\ldots,f_k$ belong to a Hardy field. We prove that under some natural conditions on the $k$-tuple $(f_1,\ldots,f_k)$ the density of the set $$ \big\{n\in \mathbb{N}: \text{gcd}(n,\lfloor…

We investigate the analytic properties of the zeta-function associated with heights on equivariant compactifications of anisotropic tori over number fields. This allows to verify conjectures about the distribution of rational points of…