The Liouville function in short intervals [after Matomaki and Radziwill]

On Furstenberg systems of aperiodic multiplicative functions of Matomaki, Radziwill and Tao

It is shown that in a class of counterexamples to Elliott's conjecture by Matomaki, Radziwill and Tao, the Chowla conjecture holds along a subsequence.

Additive functions in short intervals, gaps and a conjecture of Erdős

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki-Radziwill theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of its long average. As part of this treatment, we use a variant of the Matomäki-Radziwill theorem for divisor-bounded multiplicative functions recently proven by the author. We consider two sets of applications of these methods. Our first application shows that for an additive function $g: \mathbb{N} \rightarrow \mathbb{C}$ any non-trivial savings in the size of …

几乎所有短区间内乘法函数的改进界

Improved bounds for multiplicative functions in almost all short intervals

研究改进乘法函数短平均值的Matomäki-Radziwiłł方法，对特定函数证明最优衰减界，关键是精确处理筛法误差，还给出其他函数的改进界及局限性，应用于平均Chowla猜想获改进界。

Monotone chains of Fourier coefficients of Hecke cusp forms

We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex multiplication, of equal weight, (possibly different) squarefree level and trivial nebentypus. As a first application, we show that for the Ramanujan $τ$ function and any admissible $k$-tuple of distinct non-negative integers $a_1,\ldots,a_k$ the set $$ \{n \in \mathbb{N} : |τ(n+a_1)| < \cdots < |τ(n+a_k)|\} $$ has positive natural density. This result improves upon recent work of Bilu, …

函数域上 Liouville 函数的短和

Short Sums of the Liouville Function over Function Fields

本文研究函数域上 Liouville 函数的短和，证明了固定 $q$、$h\ll\sqrt N$ 且 $h(N)\to\infty$ 时的二阶均方估计，方法改编自整数情形及 Matomäki–Radziwiłł 的技术。

The logarithmically averaged Chowla and Elliott conjectures for two-point correlations

Let $λ$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$ \sum_{n \leq x} λ(a_1 n + b_1) λ(a_2 n+b_2) = o(x) $$ as $x \to \infty$, for any fixed natural numbers $a_1,a_2,b_1,b_2$ with $a_1b_2-a_2b_1 \neq 0$. In this paper we establish the logarithmically averaged version $$ \sum_{x/ω(x) < n \leq x} \frac{λ(a_1 n + b_1) λ(a_2 n+b_2)}{n} = o(\log ω(x)) $$ of the Chowla conjecture as $x \to \infty$, where $1 \leq ω(x) \leq x$ is an arbitrary function of $x$ that goes to infinity as $x \to \infty$, thus breaking the "parity barrier" for this p…