Paul Pollack

collaborators

4 papers

math.NT2025

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.NT2025

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.NT2025

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…

math.NT2025

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…