Ernie Croot
For completely multiplicative functions f(n) taking values 1 and -1, under certain conditions on f(n) we show that f(n) changes sign at least x exp(-7(log log x)sqrt(log x)) times as n runs through the integers <= x.
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Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $λ_{A}(n) =(-1)^{Ω_A(n)}$ where $Ω_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\sum_{k\leq n}λ_A(n)\quad{and}\quad R_A:=\lim_{n\to\infty}\frac{L_A(n)}{n}.$$ We show that for every $α\in[0,1]$ there is an $A\subset P$ such that $R_A=α$. Given certain restrictions on $A$, asymptotic estimates for $\sum_{k\leq n}λ_A(k)$ are also given. With further restrictions, more can be said. For {\em character…
Oscillations of random multiplicative functions under initial bias
研究在初始偏置下随机乘法函数的部分和性质，证明其非负性概率趋近于零，解决Kucheriaviy的猜想，并证明加权部分和符号变化无限次，解决Aymone的问题。
Consider a multiplicative function f(n) taking values on the unit circle. Is it possible that the partial sums of this function are bounded? We show that if we weaken the notion of multiplicativity so that f(pn)=f(p)f(n) for all primes p in some finite set P, then the answer is yes. We also discuss a result of Bronstein that shows that functions modified from characters at a finite number of places.
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.
In this paper, we give a new proof and an extension of the following result of Bézivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. Suppose that $F(z)$ is algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, $F(z)$ is either transcendental or rational. For $K=\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating seri…
We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent random variables taking values $\pm 1$ with probability $1/2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper bounds for $Ψ_f(x,y)$, the summatory function of $f$ on $y$-friable integers $\leq x$. We obtain estimations of the type $Ψ_f(x,y) \ll Ψ(x,y)^{1/2+ε}$, more precise formulas being given in suitable regions for $x,y$. In the special case $y=x$, this leads to the estimate $M_f(x) = \sum_{n \leq x} f(n) \ll \sqrt{x}\, (\log \log x)^{2+ε}$, which improves on pr…
Irreducibility of polynomials with random multiplicative coefficients revisited
本文针对取值于{±1}的随机完全乘法函数构成的多项式，证明其不可约概率的上极限随次数趋于无穷时等于1，推进了随机乘法系数多项式不可约性的研究。
Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Halász' mean value theorem, we compute multidimensional averages of the shape $$x^{-l} \sum_{\mathbf{n} \in [x]^l} \prod_{1 \leq j \leq k} f_j(L_j(\mathbf{n}))$$ as $x \rightarrow \infty$, where $[x] := [1,x]$ and $L_1,\ldots, L_k$ are affine linear forms that satisfy some natural conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is distinct from theirs, with \emph{explicit} main and error te…
Mean value theorems in multiplicative systems and joint ergodicity of additive and multiplicative actions
本文研究乘法系统中的加法遍历平均，推广 Halász 均值定理，刻画有限生成乘法系统的伪装动力学行为，并证明加法与乘法作用在无局部障碍时联合遍历，进而得到 Szemerédi 定理特殊情形的精细化组合结果。
We call a set of positive integers closed under taking unitary divisors a unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial complex with the property that the value on a face is the product of the values at the vertices of that face. We use this observation to solve the following problems: 1) Let r be a positive integer and c a real number. What is the maximum value that \sum_{s \in S}g(s) can obtain when S is a unitary ideal containing precisely r prime powers, and g is the mult…
Oscillation results for the summatory functions of fake mu's
本文针对Mossinghoff等提出的伪μ函数，建立了所有非平凡伪μ函数和函数在x^{1/2ℓ}尺度下的振荡结果，推广了无幂数和有幂数指示函数的技术，且当ℓ=1时与已有结果一致。
Pythagorean triples in level sets of completely multiplicative functions
研究给定取值于单位圆的完全积性函数\(f_1,\dots,f_d\)，证明存在勾股数\((x,y,z)\)使\(f_i(x),f_i(y),f_i(z)\)对所有\(i\)都任意接近\(1\)，结合非周期函数消失平均与pretentious函数集中估计来证明。
Partition regularity of Pythagorean pairs
研究勾股数对划分正则性问题，证明正整数有限染色含单色勾股数对，还表明特定乘法函数水平集生成的划分含勾股三元组，方法是结合非周期乘法函数高尔斯均匀性与乘法函数浓度估计。
On variants of Chowla's conjecture
该研究针对取值为±1的完全积性函数的移位卷积和展开，用组合方法证明了Chowla猜想方向的两项新成果，并确定了相应的谱。
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