Stanley Yao Xiao
In this paper we prove that decomposable forms, or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer coefficients which split completely into linear factors over $\mathbb{C}$, take on infinitely many square-free values subject to simple necessary conditions and $\operatorname{deg} f \leq 2n + 2$ for all irreducible factors $f$ of $F$. This work generalizes a theorem of Greaves.
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Explicit construction of decomposable Jacobians
本文给出了特征0域上可分解超椭圆 Jacobian 簇的显式构造，并应用于构造具有无穷多二次扭转且含有有理非 Weierstrass 点的超椭圆曲线族。
This is a sequel to our paper "Permute, Graph, Map, Derange", involving decomposable combinatorial labeled structures in the exp-log class of type a=1/2, 1, 3/2, 2. As before, our approach is to establish how well existing theory matches experimental data and to raise open questions.
A set $\mathcal{A}\subset \mathbb{N}$ is called additively decomposable (resp. asymptotically additively decomposable) if there exist sets $\mathcal{B},\mathcal{C}\subset \mathbb{N}$ of cardinality at least two each such that $\mathcal{A}=\mathcal{B}+\mathcal{C}$ (resp. $\mathcal{A}Δ(\mathcal{B}+\mathcal{C})$ is finite). If none of these properties hold, the set $\mathcal{A}$ is called totally primitive. We define $\mathbb{Z}$-decomposability analogously with subsets $\mathcal{A,B,C}$ of $\mathbb{Z}$. Wirsing showed that almost all subsets of $\mathbb{N}$ are totally primitive. In this paper, …
Let $g^{\square}(p)$ denote the least square-free primitive root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$.
Counting square-free values of random polynomials
本文研究了随机多项式平方自由值的计数问题，证明了平均误差项与主项的四次方根关系。
We prove that every integer greater than two may be written as the sum of a prime and a square-free number.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive integer.
We investigate the density of square-free values of polynomials with large coefficients over the rational function field $\mathbb{F}_q[t]$. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial $N$ as a sum of a small $k$-th power and a square-free polynomial.
Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.
We study a function field version of a classical problem concerning square-free values of polynomials evaluated at primes. We show that for a square-free polynomial $f\in \mathbb{F}_q[t][x]$, there is a limiting density as $n\to \infty$ of primes $P \in \mathbb{F}_q[t]$ of degree $n$ such that $f(P)$ is square-free. Over the integers the analogous result is only known when all irreducible factors of $f$ have degree at most 3.
We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences generalizing results of Allouche - Shallit and Frid.
In the present paper we prove that for any fixed $1<c<7/6$ there exist infinitely many consecutive square-free numbers of the form $[n^c], [n^c]+1$ and we also establish an asymptotic formula in given interval.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[αn]$, $[αn]+1$, where $α>1$ is irrational number with bounded partial quotient or irrational algebraic number.
It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article, we propose a method to show that ``cyclotomic polynomials $Φ_{\ell}(X)$ take square free values with positive proportion". Following this method, conditionally, we do prove the Square-free conjecture for all cyclotomic polynomials. The method is applicable to some other class of polynomials as well. Moreover we show t…
We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erdös, Odlyzko, and Sárközy.
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
Square-free values of random polynomials
本文在平均意义上研究了任意度数整系数多项式是否取无穷多个无平方因子值的问题，突破了以往仅无条件解决度数不超过3多项式的限制。
In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[αp]$, $[αp]+1$, where $p$ is prime and $α>0$ is irrational algebraic number. We also establish an asymptotic formula for the number of such square-free pairs when $p$ does not exceed given sufficiently large positive integer.
Distribution of square-free palindromes
研究平方自由回文数的分布，通过回文数在算术进展中的上界估计，结合指数和估计、6近似质回文数研究及大筛法，建立分布指数为1/16的结论。
We prove that every integer $n \geq 10$ such that $n \not\equiv 1 \text{mod} 4$ can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author's numerical computation regarding GRH to extend the explicit bounds of Ramaré-Rumely.
The main purpose of this paper is to give an explicit formula for the volumes of indefinite ternary lattices with square free discriminant. We also consider several examples of lattices, for which the fundamental domain is well-known and make sure the results obtained by our computation coincide with the known ones.
We prove that if $f$ is a non zero cusp form of weight $k$ on $Γ_0(N)$ with character $χ$ such that $N/(\text{conductor }χ)$ square-free, then there exists a square-free $n\ll_ε k^{3+ε}N^{7/2+ε}$ such that $a(f,n)\neq 0$. This significantly improves the already known existential and quantitative result from previous works.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.
We give asymptotics for correlation sums linked with the distribution of squarefree numbers in arithmetic progressions over a fixed modulus. As a particular case we improve a result of Blomer concerning the variance.
We determine the automorphism group of the modular curve $X_0^*(N)$, obtained as the quotient of the modular curve $X_0(N)$ by the group of its Atkin-Lehner involutions, for all square-free values of $N$.
We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.
Let $S_2$ be the set of integer squares. We show that the dimension $d$ of a Hilbert cube $a_0+\{0,a_1\}+...+ \{0, a_d\}\subset S_2$ is bounded by $d=O(\log \log N)$
Principal minors of Fourier matrices of square-free order
本文研究了平方自由阶傅里叶矩阵主余子式非零的条件，并探讨了包含平方因子的复合情况。
We improve on the best available bounds for the square-free sieve and provide a general framework for its applicability. The failure of the local-to-global principle allows us to obtain results better than those reached by a classical sieve-based approach. Techniques involving sphere-packing yield upper bounds on the number of integer and rational points on curves of positive genus.
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