S. Fan

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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 $σ (n) < e^{γ} n lo g lo g n$ for all $n > 5040$ , where $σ (n)$ is the sum of divisors of $n$ , and $…

math.NT2025

The maximal order of the shifted-prime divisor function

Steve Fan, Paul Pollack

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

math.NT2025

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

Steve Fan

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

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…

math.NT2024

The typical elasticity of a quadratic order

Steve Fan, Paul Pollack

For an atomic domain $D$ , the $e l a s t i c i t y$ $ρ (D)$ of $D$ is defined as $\sup\{r/s: π_1\cdots π_r = ρ_1 \cdots ρ_s,~ \text{where each $π_i, ρ_j$ is irreducible}\}$; the elasticity p…