Max Xu

collaborators

6 papers

math.NT2026

Random Multiplicative Functions and Making Squares from Polynomial Values

RÃ©gis de la BretÃ¨che, Victor Y. Wang, Max Wenqiang Xu

For a large family of polynomials $P (X) \in Z [X]$ , we prove central limit theorems for $\sum_{n \leq N} f (P (n))$ for both Rademacher and extended Rademacher multiplicative fu…

math.NT2026

Character sums over smooth numbers

Seth Hardy, Max Wenqiang Xu

Let $I^¨ (x, y)$ denote the count of $y$ -smooth numbers below $x$ and $P (n)$ denote the largest prime factor of $n$ . We show that \[ \frac{1}{Ï(q)} \sum_{Ï\bmod q} \Bigl| \sum_{\sub…

math.NT2026

Escaping Chaos in Random Multiplicative Functions

Max Wenqiang Xu

Let $f (n)$ be a Steinhaus random multiplicative function. Let $A \subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d…

math.NT2026

Helson's conjecture for smooth numbers

Seth Hardy, Max Wenqiang Xu

Let $I^¨ (x, y)$ denote the count of $y$ -smooth numbers below $x$ and $P (n)$ denote the largest prime factor of $n$ . We prove that for $f$ a Steinhaus random multiplicative function,…

math.NT2026

Average sizes of mixed character sums

Victor Y. Wang, Max Wenqiang Xu

We prove that the average size of a mixed character sum $1 \leq n \leq x \sum I ¨ (n) e (n I^¸) w (n / x)$ (for a suitable smooth function $w$ ) is on the order of $x$ for all irrat…

math.NT2025

Harper's beyond square-root conjecture

Victor Y. Wang, Max Wenqiang Xu

We explain how the (shifted) Ratios Conjecture for $L (s, I ¨ )$ would extend a randomization argument of Harper from a conductor-limited range to an unlimited range of ``beyond square…