Related papers: Power-free values of the polynomial t_1...t_r - 1

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers.…

Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that…

Let $k\in \mathbb{N}\setminus\{0\}$. For a commutative ring $R$, the ring of dual numbers of $k$ variables over $R$ is the quotient ring $R[x_1,\ldots,x_k]/ I $, where $I$ is the ideal generated by the set $\{x_ix_j\mid i,j=1,\ldots,k\}$.…

Although we expect to find many smooth numbers (i.e., numbers with no large prime factors) among the values taken by a polynomial with integer coefficients, it is unclear what the asymptotic number of such smooth values should be; this is…

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…

We develop a general approach for showing when a set of integers $\mathscr{A}$ has infinitely many $k^{th}$ powerfree numbers without relying on equidistribution estimates for $\mathscr{A}$. In particular, we show that if the Fourier…

Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we compute unconditionally the density of x in Z^n such that gcd(f(x),g(x))=1. Function…

We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of…

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…

For $k \geq 2$, we consider the number $A_k(Z)$ of positive integers $n \leq Z$ such that both $n$ and $n+1$ are $k$-free. We prove an asymptotic formula $A_k(Z) = c_k Z + O(Z^{14/(9k)+\epsilon})$, where the error term improves upon…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

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…

Define a ``truncation'' $r_{t}(p)$ of a polynomial $p$ in $\{x_1,x_2,x_3,...\}$ as the polynomial with all but the first $t$ variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be…

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…

For $k\geq 2$, we derive an asymptotic formula for the number of zeros of the forms $\prod_{i=1}^{k}(x_{2i-1}^2+x_{2i}^2)+\prod_{i=1}^{k}(x_{2k+2i-1}^2+x_{2k+2i}^2)-x_{4k+1}^{2k}$ and…

Let $s(n):= \sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text{$n$ is $k$th powerfree} \Longleftrightarrow \text{$s(n)$ is $k$th…

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions $a, b \in \mathbb N$ to the equations $a+b=n$ and $a-b=n$, where $a$ is $k$-free and $b$ is $l$-free. This…

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only…

We study few properties of square-free integers in certain equations. Using this property, we derive some infinite products in powers of square free numbers. Also, we present a method, to convert power series and trigonometric series to…