Steve Fan

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20232026

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6 papers · 1 filter

math.NT2026

The van der Corput property for sums of two squares

Steve Fan, Andrew Lott

Let $S_{N} = {1 \leq d \leq N : d = x^{2} + y^{2} for some x, y \in Z} .$ We prove a power-saving form of the van der Corput property for $S_{N}$ . As a consequence, we obtain a strong Sár…

math.NT2026

Strongly complete sets and a conjecture of Erdős

Steve Fan

A set $A \subseteq N$ is called $complete$ if every sufficiently large integer can be written as a sum of distinct elements of $A$ . It is $\textit{strongly complete…

math.NT2026

On the asymptotic density of the ordered pairs $(a, b)$ of positive integers such that $g cd (ab, a + b) = g cd (a, b)$

Steve Fan, László Tóth

Consider the arithmetic function of two variables $f (a, b) = g cd (ab, a + b) / g cd (a, b)$ , investigated in a recent preprint. We deduce asymptotic formulas for sums of the form $\sum_{a,b…

math.NT2026

Extensions of the Furstenberg-Sárközy theorem via the arithmetic level- $d$ inequality

Carlo Francisco E. Adajar, Rishika Agrawal, Mukul Rai Choudhuri +6

Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yiel…

math.NT2024

Shifted-prime divisors

Steve Fan, Carl Pomerance

Let $ω^{*} (n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p - 1$ , with $p$ prime. Studied by Prachar in an influe…

math.NT2023

Weighted Erdős-Kac Theorems via Computing Moments

Steve Fan

By adapting the moment method developed by Granville and Soundararajan [17], Khan, Milinovich and Subedi [24] recently obtained a weighted version of the Erdős--Kac theorem for $ω(…

The van der Corput property for sums of two squares

Strongly complete sets and a conjecture of Erdős

On the asymptotic density of the ordered pairs (a,b) of positive integers such that gcd(ab,a+b)=gcd(a,b)

Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-d inequality

Shifted-prime divisors

Weighted Erdős-Kac Theorems via Computing Moments

On the asymptotic density of the ordered pairs $(a, b)$ of positive integers such that $g cd (ab, a + b) = g cd (a, b)$

Extensions of the Furstenberg-Sárközy theorem via the arithmetic level- $d$ inequality