Paul Pollack

An elemental ErdÅs-Kac theorem for algebraic number fields

Paul Pollack

Fix a number field $K$ . For each nonzero $I^\pm \in Z_{K}$ , let $I^½ (I^\pm)$ denote the number of distinct, nonassociate irreducible divisors of $I^\pm$ . We show that $I^½ (I^\pm)$ is norma…

math.NT2026

On the digits of the sum of proper divisors

KÃ¼bra Benli, CÃ©cile Dartyge, Charlotte Dombrowsky +2

We study several probabilistic questions concerning the digits of $s (n)$ , the sum of proper divisors of an integer $n$ . In particular, we show that $s (n)$ obeys Benford's law with…

Clusters of primes with square-free translates

Roger C. Baker, Paul Pollack

Let $R$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p - b$ s…

math.NT2011

The average least character nonresidue and further variations on a theme of Erdos

Greg Martin, Paul Pollack

For each nonprincipal Dirichlet character $I ¨ $ , let $n_{I ¨} $ be the least $n$ with $I ¨ (n) \in / {0, 1}$ . We show that as the average of $n_{I ¨} $ over all nonprincipal characters $I ¨ $ …

Finite sets containing near-primitive roots

Komal Agrawal, Paul Pollack

Fix $a \in Z$ , $a \in / {0, \pm 1}$ . A simple argument shows that for each $I^µ > 0$ , and almost all (asymptotically 100% of) primes $p$ , the multiplicative order of $a$ mo…

A quick route to unique factorization in quadratic orders

Paul Pollack, Noah Snyder

We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.

The truth about torsion in the CM case, II

Pete L. Clark, Paul Pollack

Let $T_{CM} (d)$ be the largest size of the torsion subgroup of an elliptic curve with complex multiplication (CM) defined over a degree $d$ number field. Work of Breuer and Cla…

Bounded gaps between primes with a given primitive root, II

Roger C. Baker, Paul Pollack

Let $m$ be a natural number, and let $Q$ be a set containing at least $exp (C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each…

Distribution in coprime residue classes of polynomially-defined multiplicative functions

Paul Pollack, Akash Singha Roy

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F (T) \in Z [T]$ with $f (p) = F (p)$ for all pri…

Phi, Primorials, and Poisson

Paul Pollack, Carl Pomerance

The primorial $p #$ of a prime $p$ is the product of all primes $q \leq p$ . Let pr $(n)$ denote the largest prime $p$ with $p # ∣ I ¨ (n)$ , where $I ¨ $ is Euler's totient function. We…

Refinements of Lagrange's four-square theorem

Leo Goldmakher, Paul Pollack

A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^{2} + b^{2} + c^{2} + d^{2}$ , where $a, b, c, d \in Z$ . We characterize the values…

Joint distribution in residue classes of polynomial-like multiplicative functions

Paul Pollack, Akash Singha Roy

Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$ , wher…

The average of the first invariant factor for reductions of CM elliptic curves mod $p$

Tristan Freiberg, Paul Pollack

Let $E / Q$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E (F_{p}) ≅ Z / d_{p} Z \oplus Z / e_{p} Z$ , where…

Two problems on the distribution of Carmichael's lambda function

Paul Pollack

Let $I^» (n)$ denote the exponent of the multiplicative group modulo $n$ . We show that when $q$ is odd, each coprime residue class modulo $q$ is hit equally often by $I^» (n)$ as $n$ v…

Counting primes with a given primitive root, uniformly

Steve Fan, Paul Pollack

The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $- 1$ nor a perfect square, there is an explicit constant $A (g) > 0$ such that t…

math.NT2010

On common values of phi(n) and sigma(n), I

Kevin Ford, Paul Pollack

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n…

Typically bounding torsion

Pete L. Clark, Marko Milosevic, Paul Pollack

We formulate the notion of \emph{typical boundedness} of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in t…

A problem in comparative order theory

Sergei Konyagin, Paul Pollack

Write $ord_{p} (\cdot)$ for the multiplicative order in $F_{p}^{\times}$ . Recently, Matthew Just and the second author investigated the problem of classifying pairs $Î…

math.NT2013

Variations on a theorem of Davenport concerning abundant numbers

Emily Jennings, Paul Pollack, Lola Thompson

Let Ï(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/Ï(n) possesses a continuous distribution function. In other words, the li…

Pursuing polynomial bounds on torsion

Pete L. Clark, Paul Pollack

We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such that for all elliptic curves E defined over a number field F with j(E) in Q we have #E(F)[tors] \leq C(eps…

The distribution of intermediate prime factors

Nathan McNew, Paul Pollack, Akash Singha Roy

Let $P^{(\frac{1}{2})} (n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $I^\pm \in (0, 1)$ , the $I^\pm$ -posit…

math.NT2026

Small values of Carmichael's lambda function

Paul Pollack

Let $I^» (n)$ be the exponent of the multiplicative group $(Z / n Z)^{\times}$ , and set $L (x, y) = # {n \leq x : I^» (n) \leq y}$ . We prove an upper bound for $\log \frac{L(…

math.NT2013

Some normal numbers generated by arithmetic functions

Paul Pollack, Joseph Vandehey

Let $g \geq 2$ . A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let Ïdenote Euler's tot…

Besicovitch, Bisection, and the normality of $0. (1) (4) (9) (16) (25) \dots$

Paul Pollack, Joseph Vandehey

We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his p…

The typical elasticity of a quadratic order

Steve Fan, Paul Pollack

For an atomic domain $D$ , the $e l a s t i c i t y$ $I ¨ (D)$ of $D$ is defined as $\sup\{r/s: Ï_1\cdots Ï_r = Ï_1 \cdots Ï_s,~ \text{where each $Ï_i, Ï_j$ is irreducible}\}$; the elast…

Distribution mod $p$ of Euler's totient and the sum of proper divisors

Noah Lebowitz-Lockard, Paul Pollack, Akash Singha Roy

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $I ¨ (n)$ and the sum-of-proper-divisors function $s (n) := I ¨ (n) - n$ . We prove that the val…

Torsion subgroups of CM elliptic curves over odd degree number fields

Abbey Bourdon, Paul Pollack

Let $G_{CM} (d)$ denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. We comp…

Anatomy of torsion in the CM case

Abbey Bourdon, Pete L. Clark, Paul Pollack

Let $T_{CM} (d)$ denote the maximum size of a torsion subgroup of a CM elliptic curve over a degree $d$ number field. We initiate a systematic study of the asymptotic behav…

Numbers which are orders only of cyclic groups

Paul Pollack

We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,Ï(n)…

math.NT2019

Symmetric primes revisited

William Banks, Paul Pollack, Carl Pomerance

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an…

math.NT2012

Uncertainty principles connected with the MÃ¶bius inversion formula

Paul Pollack, Carlo Sanna

We say that two arithmetic functions f and g form a Mobius pair if f(n) = \sum_{d \mid n} g(d) for all natural numbers n. In that case, g can be expressed in terms of f by the fami…

math.GT2015

The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

Benjamin Linowitz, Jeffrey S. Meyer, Paul Pollack

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent…

math.GT2017

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 $x_{0} > 0$ , the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and…

math.NT2019

Twists of hyperelliptic curves by integers in progressions modulo $p$

David Krumm, Paul Pollack

Let $f (x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that t…

Sums of proper divisors follow the ErdÅs--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 $I ¨ (s (n))$ has normal order $lo g lo g n$ , the analogue for $s$ -v…

Small prime $k$ th power residues for $k = 2, 3, 4$ : A reciprocity laws approach

KÃ¼bra Benli, Paul Pollack

Nagell proved that for each prime $p \equiv 1 (mod 3)$ , $p > 7$ , there is a prime $q < 2 p^{1/2}$ that is a cubic residue modulo $p$ . Here we show that for each fixed $I^µ > 0$ , and each…

The reciprocal sum of divisors of Mersenne numbers

Zebediah Engberg, Paul Pollack

We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^{n} - 1$ . Conditional on the Elliott-Halberstam Conjecture and…

Bounded gaps between primes in number fields and function fields

Abel Castillo, Chris Hall, Robert J. Lemke Oliver +2

The Hardy--Littlewood prime $k$ -tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs…

Arithmetic functions at consecutive shifted primes

Paul Pollack, Lola Thompson

For each of the functions $f \in {I ¨ , I ¨ , I ¨ , I ¨ }$ and every natural number $k$ , we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \…

Digitally delicate primes

Jackson Hopper, Paul Pollack

Tao has shown that in any fixed base, a positive proportion of prime numbers cannot have any digit changed and remain prime. In other words, most primes are "digitally delicate". W…

The maximal order of the shifted-prime divisor function

Steve Fan, Paul Pollack

For each positive integer $n$ , we denote by $I ¨ ^{*} (n)$ the number of shifted-prime divisors $p - 1$ of $n$ , i.e., \[Ï^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955,…

math.NT2009

On quantitative analogues of the Goldbach and twin prime conjectures over F_q[t]

Andreas O. Bender, Paul Pollack

We study the number of ways to decompose a monic polynomial in F_q[t] of degree n as a sum of two monic irreducible polynomials in F_q[t]. Our principal result is an asymptotic for…

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.NT2010

On common values of $I ¨ (n)$ and $I ¨ (m)$ , II

Kevin Ford, Paul Pollack

Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values a…

Divisor-sum fibers

Paul Pollack, Carl Pomerance, Lola Thompson

Let $s (\cdot)$ denote the sum-of-proper-divisors function, that is, $s (n) = \sum_{d ∣ n, d < n} d$ . ErdÅs-Granville-Pomerance-Spiro conjectured that for any set $A$ of as…

math.NT2012

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…

Extremal elasticity of quadratic orders

Steve Fan, Paul Pollack

We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an ima…

math.NT2012

On the degrees of divisors of T^n-1

Paul Pollack, Lola Thompson

Fix a field $F$ . 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 $m$ in} F[T]}.] When $\D_F(n)$…

Averages of the number of points on elliptic curves

Greg Martin, Paul Pollack, Ethan Smith

If $E$ is an elliptic curve defined over $Q$ and $p$ is a prime of good reduction for $E$ , let $E (F_{p})$ denote the set of points on the reduced curve modulo $p$ . De…

math.NT2022

On Benford's Law for multiplicative functions

Vorrapan Chandee, Xiannan Li, Paul Pollack +1

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$ -divisor functions, where $k \neq = 1 0^{j}$ ,…

math.GT2018

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…

Densities of integer sets represented by quadratic forms

Pete L. Clark, Paul Pollack, Jeremy Rouse +1

Let $f (t_{1}, \dots, t_{n})$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_{f} (X)$ , the number of integers of absolute value up to $X$ …

math.AC2015

The number of atoms in an atomic domain

Pete L. Clark, Saurabh Gosavi, Paul Pollack

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all $m, n \in Z^{+}$ with $…

Powerfree sums of proper divisors

Paul Pollack, Akash Singha Roy

Let $s (n) := \sum_{d ∣ n, d < n} d$ denote the sum of the proper divisors of $n$ . It is natural to conjecture that for each integer $k \geq 2$ , the equivalence \[ \text{ $n$ is $k$ th p…

The truth about torsion in the CM case

Pete L. Clark, Paul Pollack

We show that the upper order of the size of the torsion subgroup of a CM elliptic curve over a degree d number field is d log log d.

Bounds for the first several prime character nonresidues

Paul Pollack

Let $ε > 0$ . We prove that there are constants $m_{0} = m_{0} (ε)$ and $I^º = I^º (ε) > 0$ for which the following holds: For every integer $m > m_{0}$ and every no…

Variations on a theme of Schinzel and WÃ³jcik

Matthew Just, Paul Pollack

Schinzel and WÃ³jcik have shown that if $I^\pm, I^^{2}$ are rational numbers not $0$ or $\pm 1$ , then $ord_{p} (I^\pm) = ord_{p} (I^^{2})$ for infinitely many primes $p$ , where $\mathr…

math.NT2024

Mean values of multiplicative functions and applications to residue-class distribution

Paul Pollack, Akash Singha Roy

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \…

math.GT2017

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…