Related papers: Squarefree Integers in Arithmetic Progressions to …

This paper is concerned with the relationship of $y$-smooth integers and de Bruijn's approximation $\Lambda(x,y)$. Under the Riemann hypothesis, Saias proved that the count of $y$-smooth integers up to $x$, $\Psi(x,y)$, is asymptotic to…

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any $x \ge 2$, that every interval of the form $(x, x + 11x^{1/5}\log x]$ contains a squarefree…

Let $\tau$ denote the divisor function and $\mathcal{H}=\{h_{1},...,h_{k}\}$ be an admissible set. We prove that there are infinitely many $n$ for which the product $\prod_{i=1}^{k}(n+h_{i})$ is square-free and…

Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding regions, conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for regions of…

An exponent of distribution 1/16 is established for square-free palindromes. The main input is an upper bound for the number of palindromes, in arithmetic progressions to large moduli, divisible by large squares. Our argument combines a…

In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free…

In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and…

We describe the structure of a set of integers $A$ of positive density $\delta$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $\delta_0=1/4-\frac{2}{\pi^2}=0.0473...$ and…

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $\Delta=D/16$, where D is the discriminant, by T.D. Browning and…

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…

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…

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$ (computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[…

We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y.

Let \( X \geq y \geq 2 \), and let \( u = \frac{\log X}{\log y} \). We say a number is \textit{$y$-smooth} if all of its prime factors are less than or equal to \( y \). In this paper, we study the distribution of $y$-smooth numbers in…

We provide an asymptotic estimate for certain sums over k-free integers with small prime factors. These sums depend upon a complex parameter \alpha and involve a smooth cut-off f. They are a variation of several classical number-theoretical…

A positive square-free integer is called a \textit{congruent number} if it arises as the area of a right triangle with rational side lengths. Let $ n = p_1p_2 \cdots p_t q $ be a square-free integer, where each $ p_i \equiv 1 \pmod{8} $ and…

We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power…

In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of…

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and…