K. Fan

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20232026

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4 papers · 1 filter

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.NT2024

The shifted prime-divisor function over shifted primes

Steve Fan

Let $a, b \in Z ∖ {0}$ . For every $n \in N$ , denote by $ω_{a}^{*} (n)$ the number of shifted-prime divisors $p - a$ of $n$ , where $p > a$ is prime. In this paper, we st…

math.NT2023

Numerically explicit estimates for the distribution of rough numbers

Steve Fan

For $x \geq y > 1$ and $u := lo g x / lo g y$ , let $Φ (x, y)$ denote the number of positive integers up to $x$ free of prime divisors less than or equal to $y$ . In 1950 de Bruijn [1] studie…

math.NT2023

An inequality related to the sieve of Eratosthenes

Steve Fan, Carl Pomerance

Let $Φ (x, y)$ denote the number of integers $n \in [1, x]$ free of prime factors $\leq y$ . We show that but for a few small cases, $Φ (x, y) < .6 x / lo g y$ when $y \leq x$ .