S. Fan

collaborators

6 papers

math.NT2026

SÃ¡rkÃ¶zy's theorem in $F_{q} [t]$ via the van der Corput property

Steve Fan, Andrew Lott

Fix a positive prime power $q$ , and let $F_{q} [t]$ be the ring of polynomials over the finite field $F_{q}$ with $char (F_{q}) > 2$ . Suppose $A \subseteq \…

math.NT2025

The Hardy--Ramanujan inequality for sifted sets and its applications

Steve Fan

The well-known Hardy--Ramanujan inequality states that if $I ¨ (n)$ denotes the number of distinct prime factors of a positive integer $n$ , then there is an absolute constant $C > 0$ s…

math.NT2025

A family of analogues to the Robin criterion

Steve Fan, Mits Kobayashi, Grant Molnar

The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $I ¨ (n) < e^{3} n lo g lo g n$ for all $n > 5040$ , where $I ¨ (n)$ is the sum of divisors of $n$ , an…

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

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…

SÃ¡rkÃ¶zy's theorem in Fq​[t] via the van der Corput property

The Hardy--Ramanujan inequality for sifted sets and its applications

A family of analogues to the Robin criterion

The maximal order of the shifted-prime divisor function

The typical elasticity of a quadratic order

Extremal elasticity of quadratic orders

SÃ¡rkÃ¶zy's theorem in $F_{q} [t]$ via the van der Corput property