Related papers: Squarefree integers in large arithmetic progressio…

Let $r \ge 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$, up to $x$, for which the products $\prod_{p \mid n} (p+a)$ (over primes $p$) are…

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…

Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman…

We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve…

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a…

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…

We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct…

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…

We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.$ This improves the work of Matom\"aki (2009) who obtained the result for $\theta=1/2-\varepsilon$ (with…

We show that there exists an upper bound for the number of squares in arithmetic progression over a number field that depends only on the degree of the field. We show that this bound is 5 for quadratic fields, and also that the result…

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…

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…

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by…

We show the primes have level of distribution $66/107\approx 0.617$ using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level $3/5=0.60$ of Maynard. We…

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out…

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which…

We calculate the mean and variance of sums of the M\"obius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$…

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and…