C. Lau

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20242026

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math.NT2026

On the Number of Prime Factors of Consecutive Integers

Cheuk Fung Lau

We prove that there are infinitely many $n$ such that $ω (n + k) ≪ lo g k$ for all integers $k \geq 2$ . This improves on a result of Tao-Teräväinen (2025), who has $O (k)$ in place of…

math.NT2025

Smoothed Shifted Convolutions of Generalised Divisor Functions

Cheuk Fung Lau

We prove an asymptotic formula for the smoothed shifted convolution of the generalised divisor function $d_{k} (n)$ and the divisor function $d (n)$ for $k \geq 4$ , with a power-saving…

math.NT2024

Residue Class Patterns of Consecutive Primes

Cheuk Fung Lau

Dickson's conjecture and the Hardy--Littlewood prime tuple conjecture predict that every pattern of reduced residue classes modulo $q$ is attained by infinitely many strings of $m$ …

math.NT2024

Simultaneously Small Fractional Parts of Polynomials

Cheuk Fung Lau

Let $f_{1}, \dots, f_{k} \in R [X]$ be polynomials of degree at most $d$ with $f_{1} (0) = \dots = f_{k} (0) = 0$ . We show that there is an $n < x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o…