arXiv subjects

Lee Troupe

Publications and source records attributed to Lee Troupe.

8 recordsLinked to original sources

arXiv2021

Sums of proper divisors follow the Erdős--Kac law

Let $s(n)=\sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. The second-named author proved that $ω(s(n))$ has normal order $\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erdős--Kac theorem: $ω(s(n))$ is asymptotically normally distributed with mean and variance $\log\log{n}$. The same method applies with $s(n)$ replaced by any of several other unconventional arithmetic functions, such as $β(n):=\sum_{p\mid n} p$, $n-ϕ(n)$, and $n+τ(n)$ ($τ$ being the divisor function).

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arXiv2019

Divisor sums representable as the sum of two squares

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 $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$ which are a sum of two squares. Our result confirms a special case of a conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero (e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two squares), then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

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arXiv2018

The distribution of sums and products of additive functions

The celebrated Erdős--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to hold for certain non-additive functions and for amenable arithmetic functions over certain subsets of the natural numbers. Continuing in this vein, we show that if $g_1(n), \ldots, g_k(n)$ is a collection of functions satisfying certain mild hypotheses for which an Erdős--Kac-type normal distribution law holds, and if $Q(x_1, \ldots, x_k)$ is a polynomial with nonnegative real coefficients, then $Q(g_1(n), \ldots, g_k(n))$ also obeys a normal distribution law. We also show that a similar result can be obtained if the set of inputs $n$ is restricted to certain subsets of the natural numbers, such as shifted primes. Our proof uses the method of moments. We conclude by providing examples of our theorem in action.

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arXiv2017

The distribution of the number of subgroups of the multiplicative group

Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subgroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erdös-Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that $\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of $\log G(n)$ and $\log I(n)$.

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arXiv2016

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

Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated irreducibles dividing a nonzero element of $\mathbb{Z}_K$ of norm not exceeding $x$ (in absolute value), as $x\to\infty$. Second, we count the number of irreducible elements of $\mathbb{Z}_K$ of norm not exceeding $x$ lying in a given arithmetic progression (again, as $x\to\infty$). When $K=\mathbb{Q}$, both results are classical; a new feature in the general case is the influence of combinatorial properties of the class group of $K$.

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arXiv2015

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

In this paper, we investigate extreme values of $ω(E(\mathbb{F}_p))$, where $E/\mathbb{Q}$ is an elliptic curve with complex multiplication and $ω$ is the number-of-distinct-prime-divisors function. For fixed $γ> 1$, we prove that \[ \#\{p \leq x : ω(E(\mathbb{F}_p)) > γ\log\log x\} = \frac{x}{(\log x)^{2 + γ\logγ- γ+ o(1)}}. \] The same result holds for the quantity $\#\{p \leq x : ω(E(\mathbb{F}_p)) < γ\log\log x\}$ when $0 < γ< 1$. The argument is worked out in detail for the curve $E : y^2 = x^3 - x$, and we discuss how the method can be adapted for other CM elliptic curves.

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arXiv2015

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

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 $ω(n)$ is $\log\log n$, and the same is true of $Ω(n)$; roughly speaking, a typical natural number $n$ has about $\log\log n$ prime factors. We prove a similar result for $ω(s(n))$, where $s(n)$ denotes the sum of the proper divisors of $n$: For any $ε> 0$ and all $n \leq x$ not belonging to a set of size $o(x)$, \[ |ω(s(n)) - \log\log s(n)| < ε\log\log s(n) \] and the same is true for $Ω(s(n))$.

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arXiv2015

Bounded gaps between prime polynomials with a given primitive root

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 square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide an unconditional proof of the analogue of Pollack's work in the function field case; namely, that given a monic polynomial $g(t)$ which is not an $v$th power for any prime $v$ dividing $q-1$, there are bounded gaps between monic irreducible polynomials $P(t)$ in $\mathbb{F}_q[t]$ for which $g(t)$ is a primitive root (which is to say that $g(t)$ generates the group of units modulo $P(t)$). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice $g(t) = t$.

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