Max Xu

activity

20212026

most citedPaucity phenomena for polynomial products

3 citations · 4 across the 8 of their papers we have counts for

collaborators

9 papers

math.NT2026

The typical size of Hecke eigenvalue sums is $o (x)$

Max Wenqiang Xu, Junren Zheng

Let $H_{k}$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $SL_{2} (Z)$ . For each $f \in H_{k}$ , le…

math.NT2026

Random Multiplicative Functions and Making Squares from Polynomial Values

Régis de la Bretèche, 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 $Ψ (x, y)$ denote the count of $y$ -smooth numbers below $x$ and $P (n)$ denote the largest prime factor of $n$ . We show that \[ \frac{1}{φ(q)} \sum_{χ\bmod q} \Bigl| \sum_{\substa…

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.NT2025

Helson's conjecture for smooth numbers

Seth Hardy, Max Wenqiang Xu

Let $Ψ (x, y)$ 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, t…

math.NT2024

Average sizes of mixed character sums

Victor Y. Wang, Max Wenqiang Xu

We prove that the average size of a mixed character sum $1 \leq n \leq x \sum χ (n) e (n θ) w (n / x)$ (for a suitable smooth function $w$ ) is on the order of $x$ for all irratio…

math.NT2024★ 1 cited

Harper's beyond square-root conjecture

Victor Y. Wang, Max Wenqiang Xu

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

math.NT2022★ 3 cited

Paucity phenomena for polynomial products

Victor Y. Wang, Max Wenqiang Xu

Let $P (x) \in Z [x]$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $(x_{1}, \dots, x_{k}, y_{1}, \dots, y_{k}) \in [N]^{2 k}$ to the…

math.CO2021

Discrepancy of arithmetic progressions in grids

Jacob Fox, Max Wenqiang Xu, Yunkun Zhou

We prove that the the discrepancy of arithmetic progressions in the $d$ -dimensional grid ${1, \dots, N}^{d}$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}…

The typical size of Hecke eigenvalue sums is o(x​)

Random Multiplicative Functions and Making Squares from Polynomial Values

Character sums over smooth numbers

Escaping Chaos in Random Multiplicative Functions

Helson's conjecture for smooth numbers

Average sizes of mixed character sums

Harper's beyond square-root conjecture

Paucity phenomena for polynomial products

Discrepancy of arithmetic progressions in grids

The typical size of Hecke eigenvalue sums is $o (x)$