Doychin Tolev
It is not difficult to find an asymptotic formula for the number of pairs of positive integers $x, y \le H$ such that $x^2 + y^2 + 1$ is squarefree. In the present paper we improve the estimate for the error term in this formula using the properties of certain exponential sums. A.Weils's estimate for the Kloosterman sum plays the major role in our analysis.
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We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Brüdern and Perelli.
We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s\textgreater{}\tfrac{1}{2}$. This exhibits a high fluctuation of the coefficients at squarefree integers.
We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.
We describe the structure of a set of integers $A$ of positive density $δ$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $δ_0=1/4-\frac{2}{π^2}=0.0473...$ and $δ_1=1/18$.
We investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound, and lying in some arithmetic progression a (mod q). In particular, we prove an upper bound for its variance as a varies over $(\mathbb{Z}/q\mathbb{Z})^{\times}$ which considerably improves upon earlier work of Blomer.
In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function valid for any integer and fully explicit.
We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.
Distribution of squarefree integers with double congruence conditions
本文研究无平方因子整数在双重同余条件下的分布，证明了等分布结果并发现偏差现象，提出有数值支持的猜想。
Explicit bounds for large gaps between squarefree integers
该研究得到无平方因子整数间隙渐近上界的显式形式，证明对任意$x≥2$，区间$(x, x + 11x^{1/5}\log x]$内必有一个无平方因子整数，且当$x$足够大时常数11可改进。
We show that the exponent of distribution of the sequence of squarefree numbers in arithmetic progressions of prime modulus is $\geq 2/3 + 1/57$, improving a result of Prachar from 1958. Our main tool is an upper bound for certain bilinear sums of exponential sums which resemble Kloosterman sums, going beyond what can be obtained by the Polya-Vinogradov completion method.
We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer polynomials, where it is conjectured that a separable polynomial f(x) with integer coefficients takes infinitely many squarefree values, barring some simple exceptional cases, in fact that the integers n for which f(n) is squarefree have a positive density. We show that if f(x) in F[t][x] is separable, with square-free content, of bounded degree and height, then…
For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\in \mathbb Z)$ are irreducible and their discriminants are squarefree.
In this article, we prove that the density of integers $a, b$ such that $a^4+b^3$ is squarefree, when ordered by $\max\{|a|^{1/3},|b|^{1/4}\}$, equals the conjectured product of the local densities. We show that the same is true for polynomials of the form $βa^4 + αb^3$ for any fixed integers $α$ and $β$. We give an exact count for the number of pairs $(a,b)$ of integers with $\max\{|a|^{1/3},|b|^{1/4}\}<X$ such that $βa^4 + αb^3$ is squarefree, with a power-saving error term.
Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $μ$ be the Möbius function and $μ_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=μ^2g$ and $f=μ_2^2g$ have unbounded partial sums. Our proofs are built upon Klurman and Mangerel's proof of Chudakov's conjecture, Klurman's work on correlations of multiplicative functions and Tao's resolution of the Erdős discrepancy problem.
Determination of Hilbert modular forms using squarefree coefficients
该研究针对窄类数为1的全实数域，推广了Kohnen关于半整权模形式由无平方因子代数整数支撑的傅里叶系数确定的结果，还给出了此类无穷多傅里叶系数非零的软证明。
Alternating state complexity of the set of primes and squarefree integers
该研究证明素数集具有指数级交替复杂度，证实了Fijalkow的猜想，同时指出无平方因子整数集的交替复杂度基本达到理论最大值。
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we may also force the discriminant to be coprime to any given integer. The number of fields produced with discriminant in the range [-N, N] is at least c N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic numbe…
On the Squarefree Values of Degree-$2q$ Polynomials
本研究证明满足特定伽罗瓦条件的2q次首一不可约整系数多项式有无穷多整数n使其值为无平方因子数，并据此构造了单位形状收敛到六边形格的非伽罗瓦全实三次域无穷子族。
On the Sum of Squarefree Integers and a Power of Two
本研究针对Erdos提出的“大于1的奇数均可表为无平方因子数与2的幂之和”的猜想，采用GPU上的高度并行化算法，将数值验证范围拓展至$2^{50}$以内的所有奇数，较此前结果提升超80万倍，同时给出了相关新的数值发现。
A study of the Dedekind psi function concludes that its extreme values are supported on the subset of primorial integers N_k = 2*3***p_k, where p_k is the kth prime. In particular, the inequality psi(N_k) > cloglogN_k, c > 0 constant, holds for all large primorial integers.
We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $ζ(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discrim…
On the variance of squarefree integers in short intervals and arithmetic progressions
本文渐近计算了短区间和等差数列中无平方因子整数数量的方差，并在假设 Lindelöf 假设及其推广下改进了适用范围，同时证明在完整范围内获得精确界等价于 Riemann 假设。
We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maass newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $Γ_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Stromberg, who proved this for $q = 9$ by different methods.
Squarefree Integers in Arithmetic Progressions to Smooth Moduli
本文证明在光滑模数下算术级数中无平方因子整数可获幂次节省误差的渐近公式，突破 $X^{3/4}$ 障碍，改进 Nunes 与 Hooley 的结果。
We prove that the rational elliptic curve y^2 = x^3 - n^2x satisfies the full Birch and Swinnerton-Dyer conjecture for at least 41.9% of positive squarefree integers n equal to 1, 2, or 3 mod 8, and that it satisfies the regular BSD conjecture for at least 55.9% of positive squarefree integers n equal to 5, 6, or 7 mod 8. In particular, at least 55.9% of positive squarefree integers equal to 5, 6, or 7 mod 8 are congruent numbers. These proofs complete an argument started by Tian, Yuan, and Zhang.
For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb Q$-rational subgroup. Let also $\mathscr C_{\mathbb Q}(N)$ be the subgroup of $\mathscr C(N)(\mathbb Q)$ generated by $\mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $\mathscr C(N)(\mathbb Q)$ and $\mathscr C_{\mathbb Q}(N)$ are equal. To achieve this, we show that…
For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific trinomial forms in the literature. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.
We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cu…
We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the discriminant polynomials of various prehomogeneous and coregular representations where generic stabilizers are finite. This has applications to a number of arithmetic distribution questions, e.g., to the density of small degree number fields having squarefree discriminant, and the density of certain unramified nonabelian extensions of quadratic fields. In separate wo…
Powerfree integers and Fourier bounds
本文提出一种不依赖等分布估计的通用方法，利用傅里叶变换的$L^{\infty}$、$L^{1}$界及递减性证明整数集含无穷多个无幂数，并应用于回文数、倒序数及缺数字数，得到新界与无穷性结果。
We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.
Squarefree numbers in short intervals
该数论研究通过引入计数曲线附近受限格点的新技术，结合Green和Tao的定量定理，将短区间含无平方因子数的区间长度指数从1/5改进至1/5 - η。
Some ergodic theorems over squarefree numbers and squarefull numbers
本文基于平方自由数和平方满数的不变平均性质，推导出若干遍历定理，与Erdős-Kac定理、Bergelson-Richter定理及Loyd定理有重要关联。
Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least $λ_0=1.176...$.
Higher Rédei reciprocity and integral points on conics
该研究推广Rédei经典互反律，结合Alexander Smith的工作，给出特定二元二次型方程整数可解性的渐近公式，推导相关域Hasse单位指数的渐近结果，并改进负Pell方程可解性的已有结论。
On the exponential sum over squarefree integers
该研究证明了无平方因子整数上指数和的新上界估计，改进了此前带$N^ε$因子的结果，采用带素数区间限制的平方筛法等技术，推导了Hardy-Littlewood剖分劣弧上的对应估计。
For a positive integer $N$, let $\mathscr{C}_N(\mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $\mathscr{C}(N)$ be the rational cuspidal divisor class group of $X_0(N)$, which are both subgroups of the rational torsion subgroup of $J_0(N)$. We prove that two groups $\mathscr{C}_N(\mathbb{Q})$ and $\mathscr{C}(N)$ are equal when $N=p^2M$ for any prime $p$ and any squarefree integer $M$. To achieve this we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m, h}$, which are constructed from generalized Dedekind eta functions. Also, we deter…
We present an algorithm, based on the explicit formula for $L$-functions and conditional on GRH, for proving that a given integer is squarefree with little or no knowledge of its factorization. We analyze the algorithm both theoretically and practically, and use it to prove that several RSA challenge numbers are not squarefull.
Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We then apply our result to prove that for a given squarefree positive odd integer $n$ there exist infinitely many $N$ such that $n$ divides both $N$ and $r(N)$, where $r(N)$ is the representation number of $N$ as sums of three squares.
Squarefrees are Gaussian in short intervals
本文证明了在特定短区间内无平方因子数的计数趋于高斯分布，回答了Hall在1989年提出的问题，并将其推广至B-free整数，展示了其缩放后趋向分数布朗运动。
Generalized Fermat equation over number fields
在两个标准数论猜想假设下，利用模方法研究数域上广义费马方程的渐近行为，建立基于$S_K'$-单位方程解的渐近判别准则，针对一类虚二次域证明方程无渐近解，且该类无平方因子整数相对密度为5/6。
A counterexample to the Pellian equation conjecture of Mordell
本文通过扩展研究满足 Pell 基本单位系数 $y$ 被 $d$ 整除的无平方因子整数列表，给出了 Mordell 相关 Pell 方程猜想的一个反例。
In this note we give a brief survey of the most elementary criteria used to determine the surjectivity of the trace operator on the ring of integers of a number field $K$. Furthermore, we introduce an easy to state yet unknown surjectivity criterion depending only on the prime factorization of the degree $n$ of $K$ and on the squarefree part of the discriminant $d_K$.
We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.
Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes where f has bad reduction. After excluding all the natural counter-examples, we define a subset A(f,a) of pairs of integers (m,n) with m nonnegative and n positive, and show that for all but finitely many (m,n) in A(f,a) there is a prime p of K which is not in S such that the p-adic valuation of f^{m+n}(a)-f^m(a) is precisely equal to 1, and moreover a has p…
In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.
In a previous article, the author proves that the value of the root number varies in a non-isotrivial family of elliptic curves indexed by one parameter $t$ running through $\mathbb{Q}$. However, a well-known example of Washington has root number $-1$ for every fiber when $t$ runs through $\mathbb{Z}$. Such examples are rare since, as proven in this paper, the root number of the integer fibers varies for a large class of families of elliptic curves. Our results depends on the squarefree conjecture and Chowla's conjecture, and are unconditional in many cases.
Squarefree values of polynomial discriminants II
本文确定具有无平方因子判别式及截出极大序的整系数二元型密度，证明算术 Bertini 猜想，并改进 $S_n$ 数域计数下界、修复 Nakagawa 的相关结果。
Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < θ< π$ has rational cosine, say $\cos (θ)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A positive integer $n$ is called a $(\mathbb K,θ)$-congruent number if there is a triangle, called the $(\mathbb K,θ, n)$-triangles, with sides in $\mathbb K$ having $θ$ as an angle and $nα_θ$ as area, where ${α_θ}=\sqrt{r^2-s^2}$. Consider the $(\mathbb K,θ)$-congruent number elliptic curve $E_{n,θ}: y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over $\mathbb K$. Denote the squarefree part of posi…
Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The present paper gives explicit construction of an integral basis of $K$ along with applications. This construction of an integral basis of $K$ extends a result proved in [J. Number Theory, {173} (2017), 129-146] regarding periodicity of integral bases of $\mathbb{Q}(\sq…
We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.
We show that for $100\%$ of the odd, squarefree integers $n > 0$ the $4$-rank of $\text{Cl}(\mathbb{Q}(i, \sqrt{n}))$ is equal to $ω_3(n) - 1$, where $ω_3$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $Γ_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $Γ_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic repre…
Given a positive integer $n$, we let ${\rm sfp}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$ by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many $n$ for which \[ \max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} < n^{1/3}. \]
We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go beyond $\sqrt{x}$-cancellation.
Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $ H_K(α, m)$ a quaternion algebra over $K$, where $α\in \mathcal{O}_K$. In this paper we give necessary and sufficient conditions for $ H_K(α, m)$ to split over $K$ for some values of $α$, and we obtain a complete characterization of division quaternion algebras $ H_K(α, m)$ over $K$ whenever $α$ and $m$ are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers.
Let $g>1$ be an integer and $f(X)\in{\mathbb Z}[X]$ a polynomial of positive degree with no multiple roots, and put $u(n)=f(g^n)$. In this note, we study the sequence of quadratic fields ${\mathbb Q}(\sqrt{u(n)}\,)$ as $n$ varies over the consecutive integers $M+1,\ldots,M+N$. Fields of this type include Shanks fields and their generalizations. Using the square sieve together with new bounds on character sums, we improve an upper bound of Luca and Shparlinski (2009) on the number of $n \in \{M+1,\ldots,M+N\}$ with ${\mathbb Q}(\sqrt{u(n)}\,) = {\mathbb Q}(\sqrt{s}\,)$ for a given squarefree in…
We give necessary and sufficient conditions on a squarefree integer $d$ for there to be non-trivial solutions to $x^{3} + y^{3} = z^{3}$ in $\Q(\sqrt{d})$, conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.
On differences of perfect powers and prime powers
本研究提出一种结合Frey-Hellegouarch曲线伽罗瓦表示、Thue-Mahler方程有效求解及改进对数下界的方法，显式确定完全幂与素数幂的差的无平方因子部分对应元组，并求解了指定参数范围的情形。
Products of shifted primes simultaneously taking perfect power values
本文研究了移位素数的乘积同时取完全幂值的问题，给出了无平方因子整数n的数量下界，并探讨了特定A集合下的素因子数量下界。
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