14 citations · 32 across the 17 of their papers we have counts for
25 papers
On the digits of the sum of proper divisors
Kübra Benli, Cécile Dartyge, Charlotte Dombrowsky +2
We study several probabilistic questions concerning the digits of , the sum of proper divisors of an integer . In particular, we show that obeys Benford's law with…
A statistical investigation of a divisor-sum function
Ivan Aidun, Lola Thompson
The sum of proper divisors function has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) \coloneqq \sum_{…
Counting Salem numbers arising from arithmetic hyperbolic orbifolds
Michelle Chu, Plinio G. P. Murillo, Otto Romero +1
The relationship between Salem numbers and short geodesics has been fruitful in quantitative studies of arithmetic hyperbolic orbifolds, particularly in dimensions 2 and 3. In this…
Sums of proper divisors with missing digits
Kübra Benli, Giulia Cesana, Cécile Dartyge +2
Let denote the sum of proper divisors of an integer . In 1992, Erdős, Granville, Pomerance, and Spiro (EGPS) conjectured that if is a set of integers with a…
Summing : a faster elementary algorithm
Harald A. Helfgott, Lola Thompson
We present a new elementary algorithm that takes \[ \mathrm{time} \ \ O_ε\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+ε} \right) \ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\fr…
Counting Salem numbers of arithmetic hyperbolic 3-orbifolds
Mikhail Belolipetsky, Matilde Lalín, Plinio G. P. Murillo +1
It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show tha…
Divisor-sum fibers
Paul Pollack, Carl Pomerance, Lola Thompson
Let denote the sum-of-proper-divisors function, that is, . Erdős-Granville-Pomerance-Spiro conjectured that for any set of asy…
Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds
Benjamin Linowitz, D. B. McReynolds, Paul Pollack +1
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-m…
Lower Bounds for Heights in Relative Galois Extensions
Shabnam Akhtari, Kevser Aktaş, Kirsti Biggs +3
The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our fir…
A generalization of the practical numbers
Nicholas Schwab, Lola Thompson
A positive integer is practical if every can be written as a sum of distinct divisors of . One can generalize the concept of practical numbers by applying an arit…
On integers for which has a divisor of every degree
Carl Pomerance, Lola Thompson, Andreas Weingartner
A positive integer is called -practical if the polynomial has a divisor in of every degree up to . In this paper, we show that the count of -pr…
Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds
Benjamin Linowitz, D. B. McReynolds, Paul Pollack +1
Our main result is that for all sufficiently large , the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field and…
Counting and effective rigidity in algebra and geometry
Benjamin Linowitz, D. B. McReynolds, Paul Pollack +1
The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic…
Arithmetic functions at consecutive shifted primes
Paul Pollack, Lola Thompson
For each of the functions and every natural number , we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots…
Bounded gaps between primes in number fields and function fields
Abel Castillo, Chris Hall, Robert J. Lemke Oliver +2
The Hardy--Littlewood prime -tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs…
Abelian surfaces over finite fields with prescribed groups
Chantal David, Derek Garton, Zachary Scherr +3
Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_…
Variations on a theorem of Davenport concerning abundant numbers
Emily Jennings, Paul Pollack, Lola Thompson
Let σ(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/σ(n) possesses a continuous distribution function. In other words, the limi…
The sign changes of Fourier coefficients of Eisenstein series
Benjamin Linowitz, Lola Thompson
In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this pr…
Distribution of squarefree values of sequences associated with elliptic curves
Shabnam Akhtari, Chantal David, Heekyoung Hahn +1
Let E be a non-CM elliptic curve defined over Q. For each prime p of good reduction, E reduces to a curve E_p over the finite field F_p. For a given squarefree polynomial f(x,y), w…
Variations on a question concerning the degrees of divisors of x^n-1
Lola Thompson
In this paper, we examine a natural question concerning the divisors of the polynomial x^n-1: "How often does x^n-1 have a divisor of every degree between 1 and n?" In a previous p…
On the degrees of divisors of T^n-1
Paul Pollack, Lola Thompson
Fix a field . In this paper, we study the sets $\D_F(n) \subset [0,n]$ defined by [\D_F(n):= {0 \leq m \leq n: T^n-1\text{has a divisor of degree in} F[T]}.] When $\D_F(n)$…
Practical pretenders
Paul Pollack, Lola Thompson
Following Srinivasan, an integer n\geq 1 is called practical if every natural number in [1,n] can be written as a sum of distinct divisors of n. This motivates us to define f(n) as…
Heights of divisors of x^n-1
Lola Thompson
The height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic po…
Polynomials with divisors of every degree
Lola Thompson
We consider polynomials of the form t^n-1 and determine when members of this family have a divisor of every degree in Z[t]. With F(x) defined to be the number of such integers up t…