Lee Troupe

activity

20142021

collaborators

8 papers

math.NT2021

Sums of proper divisors follow the Erdős--Kac law

Paul Pollack, Lee Troupe

Let $s (n) = \sum_{d ∣ n, d < n} d$ denote the sum of the proper divisors of $n$ . The second-named author proved that $ω (s (n))$ has normal order $lo g lo g n$ , the analogue for $s$ -va…

math.NT2019

Divisor sums representable as the sum of two squares

Lee Troupe

Let $s (n)$ denote the sum of the proper divisors of the natural number $n$ . We show that the number of $n \leq x$ such that $s (n)$ is a sum of two squares has order of magnitude $x…

math.NT2018

The distribution of sums and products of additive functions

Greg Martin, Lee Troupe

The celebrated Erdős--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening year…

math.NT2017

The distribution of the number of subgroups of the multiplicative group

Greg Martin, Lee Troupe

Let $I (n)$ denote the number of isomorphism classes of subgroups of $(Z / n Z)^{\times}$ , and let $G (n)$ denote the number of subgroups of $(Z / n Z)^{\times}$ counted as…

math.NT2016

Two problems concerning irreducible elements in rings of integers of number fields

Paul Pollack, Lee Troupe

Let $K$ be a number field with ring of integers $Z_{K}$ . We prove two asymptotic formulas connected with the distribution of irreducible elements in $Z_{K}$ . First, w…

math.NT2015

Orders of reductions of elliptic curves with many and few prime factors

Lee Troupe

In this paper, we investigate extreme values of $ω (E (F_{p}))$ , where $E / Q$ is an elliptic curve with complex multiplication and $ω$ is the number-of-distinct-prime-…

math.NT2015

Bounded gaps between prime polynomials with a given primitive root

Lee Troupe

A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq = - 1$ and $g$ is not a perfect sq…

math.NT2014

On the number of prime factors of values of the sum-of-proper-divisors function

Lee Troupe

Let $ω (n)$ (resp. $Ω (n)$ ) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$ . In 1917, Hardy and Ramanujan proved that the normal order of $ω (n)$ …