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Max Wenqiang Xu

Publications and source records attributed to Max Wenqiang Xu.

At least 19 recordsLinked to original sources
arXiv2026

Large fluctuations of extended Rademacher random multiplicative functions

Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\log\log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.

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arXiv2026

The typical size of Hecke eigenvalue sums is $o(\sqrt{x})$

Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $λ_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds for \[\sum_{f \in \mathcal{H}_k}ω_f \left|\sum_{n \leq x} λ_f(n)\right|^{2q},\] where $k$ is a positive even integer, $1\leqslant x\leqslant k$, $0\leqslant q \leqslant 1$ and $ ω_f = \frac{Γ(k-1)} {(4π)^{k-1}\langle f,f\rangle} = \frac{2π^2} {(k-1)L(1,\mathrm{Sym}^2 f)}. $ Our estimates match the conjecturally sharp upper bound for $1\leqslant x\leqslant k^{0.99}$. In particular, whenever both $x$ and $k/x$ tend to infinity with $k$, we obtain \[\sum_{f \in \mathcal{H}_k}ω_f \left|\sum_{n \leq x} λ_f(n)\right|=o(\sqrt{x}).\] We also prove upper bounds for low moments of sums of Hecke eigenvalues associated with Hecke--Maass cusp forms for $\mathrm{SL}_2(\mathbb Z)$. We study these sums through the probabilistic model proposed by Cogdell--Michel. We also determine the order of magnitude of low moments of this probabilistic model, which is equivalent to determining the order of magnitude of low moments of Hecke eigenvalue sums in the limit.

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arXiv2026

Maximising the number of solutions to linear equations

We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $γ_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $γ_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le γ_{1,1,-3}\le 1/2-δ$ for some constant $δ>0$. In addition we show that the limit as $a \rightarrow \infty$ of $γ_{1,1,-a}$ is equal to precisely $1/5$.

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arXiv2026

Oscillations of random multiplicative functions under initial bias

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-ε}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow \infty$. This solves a conjecture of Kucheriaviy, who has a complementary result showing this exponent is sharp. We also prove that almost surely the partial sums of $\sum\frac{f(n)}{\sqrt{n}}$ change signs infinitely many times, solving a problem of Aymone.

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arXiv2026

Random Multiplicative Functions and Making Squares from Polynomial Values

For a large family of polynomials $P(X)\in \mathbb{Z}[X]$, we prove central limit theorems for $\sum_{n\le N} f(P(n))$ for both Rademacher and extended Rademacher multiplicative functions $f$. To achieve this, we establish a paucity phenomenon in counting solutions to \[P(n_1)P(n_2)P(n_3)P(n_4) = \square, \quad 1\le n_1, n_2, n_3, n_4 \le N.\] Results of Hooley, Evertse--Silverman, and Reuss play an important role in the proof. Our estimates are sharpest for $°P = 2$, thanks to the rich theory of Pell--Fermat equations.

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arXiv2026

Character sums over smooth numbers

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 \[ \frac{1}{φ(q)} \sum_{χ\bmod q} \Bigl| \sum_{\substack{n \leq x \\ P(n) \leq y}} χ(n) \Bigr| = o \Bigl( \sqrt{Ψ(x,y)} \Bigr), \] whenever $(\log x)^6 \leq y \leq x^{\frac{1}{32 \log \log x}}$ and $q \geq x^{1 + \varepsilon}$ for some small quantifiable $\varepsilon > 0$. The saving is substantial when $\varepsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums.

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arXiv2026

Distribution of random multiplicative functions in short intervals, with proper normalization

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y \asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.

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arXiv2026

Escaping Chaos in Random Multiplicative Functions

Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that the $o(1)$ density is sharp by showing that for most sets $A$, and thus confirm the existence, with density $ρ$ such that $(1-ρ)^{-1} =o((\log \log N)^{1/2})$, we have \[ \frac{1}{\sqrt{(1-ρ) |A|}} \sum_{n\in A} f(n) \xrightarrow{d} \mathcal{CN}(0,1). \] The extra factor $\sqrt{1-ρ}$ makes a difference as long as the density $ρ>0$.

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arXiv2026

Almost Affine Invariance Over Prime Fields: Green Problem 90

Let $A\subset \mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $ϕ(x)=ax+b$ if \[|A \triangle ϕ(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $ϕ(x)$ with $|a|, |b|\le K$ and $a\neq 0$, is $K=o(\log p)$. This solves Ben Green's Open Problem 90.

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arXiv2026

Helson's conjecture for smooth numbers

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, the partial sums over $y$-smooth numbers always enjoy better than squareroot cancellation, in the sense that $$ \mathbb{E} \Big|\sum_{\substack{1\leq n \leq x\\ P(n) \leq y}} f(n) \Big| = o\left( Ψ(x,y)^{1/2} \right),$$ uniformly on the entire range $ 2 \leq y \leq x$. The bounds are quantitative and give a large saving when $y$ isn't too close to $x$.

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arXiv2026

Average sizes of mixed character sums

We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} χ(n) e(nθ) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $θ$ satisfying a weak Diophantine condition, where $χ$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\le r$. In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $θ$. Certain quadratic Diophantine equations play a key role in the present paper.

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arXiv2026

Roth-type theorems in $K_{s,t}$-free sets

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $Ω(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets. We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lovász-Sauermann.

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arXiv2025

The distribution of prime values of random polynomials

The Bateman--Horn Conjecture predicts how often an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes prime values. We demonstrate that with sufficient averaging in the coefficients of $f$ (viz. exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values. We show that 100\% of polynomials (in an $L^k$ sense for all $k \in \mathbb{N}$) satisfy the Bateman--Horn Conjecture, and that that 100\% of polynomials (in an $L^2$ sense) satisfy an appropriate polynomial analogue of the Hardy--Littlewood Prime Tuples Conjecture. We use the latter to prove that 100\% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson. We also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of $f$; viewing $f$ as a random variable, we establish the limiting distribution for every sign pattern. The Chowla problem along random polynomials is a special case. A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and Möbius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).

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arXiv2025

Harper's beyond square-root conjecture

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-root cancellation'' for character twists of the Liouville function. As a corollary, the Liouville function would have nontrivial cancellation in arithmetic progressions of modulus just exceeding the well-known square-root barrier. Morally, the paper passes from random matrices to random multiplicative functions.

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arXiv2024

Paucity phenomena for polynomial products

Let $P(x)\in \mathbb{Z}[x]$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $(x_1, \dots, x_k, y_1, \dots, y_k)\in [N]^{2k}$ to the equation \[ \prod_{1\le i \le k} P(x_i) = \prod_{1\le j \le k} P(y_j)\neq 0 \] (for any $k\ge 1$) is asymptotically $k!N^{k}$ as $N\to +\infty$. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums $\frac{1}{\sqrt{N}}\sum_{n\le N}f(P(n))$ match standard complex Gaussian moments as $N\to +\infty$, where $f$ is the Steinhaus random multiplicative function.

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arXiv2024

Partial sums of typical multiplicative functions over short moving intervals

We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x, x+H]$ matches the corresponding Gaussian moment, as long as $H\ll x/(\log x)^{2k^2+2+o(1)}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x, x+H]$ with $H\ll X/(\log X)^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots, X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some initial progress on a recent question of Harper.

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