arXiv subjects

Paul Pollack

Publications and source records attributed to Paul Pollack.

At least 19 recordsLinked to original sources

arXiv2026

On the digits of the sum of proper divisors

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 respect to logarithmic density. Moreover, we show that, for every function $k(x) \rightarrow \infty$, almost all integers $n \leq x$ have every decimal digit occurring among the first $k(x)$ digits and the last $k(x)$ digits of $s(n)$. We also present an upper bound for the number of composite integers $n$ up to $x$ for which $s(n)$ is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs $n$ were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set $s^{-1}(\mathcal{A})$, where $\mathcal{A}$ is a set of integers with missing digits. Our result for composite $n$ shows that the count is much smaller when prime inputs are excluded.

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arXiv2026

Small values of Carmichael's lambda function

Let $\lambda(n)$ be the exponent of the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$, and set $L(x,y) = \#\{n\le x: \lambda(n)\le y\}$. We prove an upper bound for $\log \frac{L(x,y)}{x}$ valid for $\exp((\log_2{x})^{1+\epsilon}) \le y\le x/\exp((\log_2{x})^{1+\epsilon})$. Our bound is asymptotically sharp under a plausible hypothesis on powersmooth shifted primes. As an application, we obtain a new upper bound on the count of odd $n\le x$ for which the order of $2$ modulo $n$ is appreciably smaller than $x^{1/2}$.

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arXiv2025

The maximal order of the shifted-prime divisor function

For each positive integer $n$, we denote by $\omega^*(n)$ the number of shifted-prime divisors $p-1$ of $n$, i.e., \[\omega^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955, this function has interesting applications in primality testing and bears a strong connection with counting Carmichael numbers. Prachar showed that for a certain constant $c_0 > 0$, \[\omega^*(n)>\exp\left(c_0\frac{\log n}{(\log\log n)^2}\right)\] for infinitely many $n$. This result was later improved by Adleman, Pomerance and Rumely, who established an inequality of the same shape with $(\log\log n)^2$ replaced by $\log\log n$. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, Prachar also proved the stronger inequality \[\omega^*(n)>\exp\left(\left(\frac{1}{2}\log2+o(1)\right)\frac{\log n}{\log\log n}\right)\] for infinitely many $n$. By refining the arguments of Prachar and of Adleman, Pomerance and Rumely, we improve on their results by establishing \begin{align*} \omega^*(n)&>\exp\left(0.6736\log 2\cdot\frac{\log n}{\log\log n}\right) \quad\text{(unconditionally)},\\ \omega^*(n)&>\exp\left(\left(\log\left(\frac{1+\sqrt{5}}{2}\right)+o(1)\right)\frac{\log n}{\log\log n}\right) \quad\text{(under GRH)}, \end{align*} for infinitely many $n$.

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arXiv2025

Counting primes with a given primitive root, uniformly

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 the number $\Pi(x;g)$ of primes $p\le x$ for which $g$ is a primitive root is asymptotically $A(g)\pi(x)$ as $x\to\infty$, where $\pi(x)$ counts the number of primes not exceeding $x$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $\mathbb{Q}$. In this paper, we use GRH to establish a uniform version of the Artin--Hooley asymptotic formula. Specifically, we prove that $\Pi(x;g) \sim A(g) x/\log{x}$ whenever $\log{x}/\log\log{2|g|} \to \infty$, i.e., whenever $x$ tends to infinity faster than any power of $\log{(2|g|)}$. Under GRH, we also show that the least prime $p_g$ possessing $g$ as a primitive root satisfies the upper bound $p_g=O(\log^{19}(2|g|))$ uniformly for all non-square $g\ne-1$. We conclude with an application to the average value of $p_g$ and a discussion of an analogue concerning the least "almost-primitive'' root.

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arXiv2025

Extremal elasticity of quadratic orders

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 imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log{f})^{c_1 \log\log\log{f}}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log{f})^{c_2\log\log\log{f}}$ for infinitely many $f$. Here $c_1, c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^{\times}$.

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arXiv2024

The typical elasticity of a quadratic order

For an atomic domain $D$, the $elasticity$ $\rho(D)$ of $D$ is defined as $\sup\{r/s: \pi_1\cdots \pi_r = \rho_1 \cdots \rho_s,~ \text{where each $\pi_i, \rho_j$ is irreducible}\}$; the elasticity provides a concrete measure of the failure of unique factorization in $D$. Fix a quadratic number field $K$ with discriminant $\Delta_K$, and for each positive integer $f$, let $\mathcal{O}_f = \mathbb{Z} + f\mathcal{O}_K$ denote the order of conductor $f$ in $K$. Results of Halter-Koch imply that $\mathcal{O}_f$ has finite elasticity precisely when $f$ is $\textit{split-free}$, meaning not divisible by any rational prime $p$ with $(\Delta_K/p)=1$. When $K$ is imaginary, we show that for almost all split-free $f$, \[ \rho(\mathcal{O}_f) = f/(\log{f})^{\frac{1}{2}\log\log\log{f} + \frac{1}{2}C_K+o(1)}, \] for a constant $C_K$ depending on $K$. When $K$ is real, we prove under the assumption of the Generalized Riemann Hypothesis that \[ \rho(\mathcal{O}_f)= (\log{f})^{\frac12 +o(1)} \] for almost all split-free $f$. Underlying these estimates are new statistical theorems about class groups of orders in quadratic fields, whose proofs borrow ideas from investigations of Erd\H{o}s, Hooley, Li, Pomerance, Schmutz, and others into the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^\times$. One novelty of the argument is the development of a weighted version of the Tur\'{a}n--Kubilius inequality to handle a variety of sums over split-free integers.

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arXiv2024

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

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 $n \le x$ for which the Alladi-Erd\H{o}s function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the "correct" moduli $q$ that vary uniformly in a similar range, and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the "incorrect" moduli.

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arXiv2023

The distribution of intermediate prime factors

Let $P^{\left(\frac 12\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $\alpha \in (0,1)$, the $\alpha$-positioned prime factor of $n$, $P^{(\alpha)}(n)$. It has previously been shown that $\log \log P^{(\alpha)}(n)$ has normal order $\alpha \log \log x$, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of $n\leq x$ for which $P^{(\alpha)}(n)=p$, for primes $p$ in a wide range up to $x$. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that $\frac 1x \sum_{1<n \le x} \log P^{\left(\frac 12 \right)}(n) \sim A(\log x)^{\varphi-1}$, where $\varphi$ is the golden ratio, and $A$ is an explicit constant. Along the way, we obtain an extension of Lichtman's recent work on the ``dissected'' Mertens' theorem sums $\sum_{\substack{P^+(n) \le y \\ \Omega(n)=k}} \frac{1}{n}$ for large values of $k$.

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arXiv2023

Densities of integer sets represented by quadratic forms

Let $f(t_1,\ldots,t_n)$ 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 $X$ represented by $f$. When $f$ is isotropic or $n$ is at least $3$, we show that there is a $\delta(f) \in \mathbb{Q} \cap (0,1)$ such that $D_f(X) \sim \delta(f) X$ and call $\delta(f)$ the density of $f$. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density $0$ within the set of locally represented integers.

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arXiv2023

Distribution in coprime residue classes of polynomially-defined multiplicative functions

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus $q$. For example, we show that the values $\phi(n)$, sampled over integers $n \le x$ with $\phi(n)$ coprime to $q$, are asymptotically equidistributed among the coprime classes modulo $q$, uniformly for moduli $q$ coprime to $6$ that are bounded by a fixed power of $\log{x}$.

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arXiv2023

Two problems on the distribution of Carmichael's lambda function

Let $\lambda(n)$ 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 $\lambda(n)$ as $n$ varies. Under the stronger assumption that $\gcd(q,6)=1$, we prove that equidistribution persists throughout a Siegel--Walfisz-type range of uniformity. By similar methods we show that $\lambda(n)$ obeys Benford's leading digit law with respect to natural density. Moreover, if we assume GRH, then Benford's law holds for the order of $a$ mod $n$, for any fixed integer $a\notin \{0,\pm 1\}$.

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arXiv2022

On Benford's Law for multiplicative functions

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 $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Hal\'{a}sz's Theorem.

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arXiv2021

A problem in comparative order theory

Write $\mathrm{ord}_p(\cdot)$ for the multiplicative order in $\mathbb{F}_p^{\times}$. Recently, Matthew Just and the second author investigated the problem of classifying pairs $\alpha, \beta \in \mathbb{Q}^{\times}\setminus\{\pm 1\}$ for which $\mathrm{ord}_p(\alpha) > \mathrm{ord}_p(\beta)$ holds for infinitely many primes $p$. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for $\alpha,\beta$ to be order-dominant. Via the large sieve, we show that almost all integer pairs $\alpha,\beta$ satisfy our condition, with a power savings on the size of the exceptional set.

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arXiv2021

Powerfree sums of proper divisors

Let $s(n):= \sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text{$n$ is $k$th powerfree} \Longleftrightarrow \text{$s(n)$ is $k$th powerfree} \] holds almost always (meaning, on a set of asymptotic density $1$). We prove this for $k\ge 4$.

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arXiv2021

Sums of proper divisors follow the Erd\H{o}s--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 $\omega(s(n))$ has normal order $\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erd\H{o}s--Kac theorem: $\omega(s(n))$ 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 $\beta(n):=\sum_{p\mid n} p$, $n-\phi(n)$, and $n+\tau(n)$ ($\tau$ being the divisor function).

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arXiv2021

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

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\phi(n)$ and the sum-of-proper-divisors function $s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.

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arXiv2021

Joint distribution in residue classes of polynomial-like multiplicative functions

Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$, where $p$ is a growing prime (or nearly prime) modulus. This can be seen as complementary to work of Narkiewicz, who obtained comprehensive results for fixed moduli.

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arXiv2021

Variations on a theme of Schinzel and W\'ojcik

Schinzel and W\'ojcik have shown that if $\alpha, \beta$ are rational numbers not $0$ or $\pm 1$, then $\mathrm{ord}_p(\alpha)=\mathrm{ord}_p(\beta)$ for infinitely many primes $p$, where $\mathrm{ord}_p(\cdot)$ denotes the order in $\mathbb{F}_p^{\times}$. We begin by asking: When are there infinitely many primes $p$ with $\mathrm{ord}_p(\alpha) > \mathrm{ord}_p(\beta)$? We write down several families of pairs $\alpha,\beta$ for which we can prove this to be the case. In particular, we show this happens for "100\%" of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel and W\'{o}jcik's theorem for the integers of an imaginary quadratic field $K$: If $\alpha, \beta \in \mathcal{O}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of $\mathcal{O}_K$ for which $\mathrm{ord}_P(\alpha) = \mathrm{ord}_P(\beta)$.

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