Related papers: Entiers ultrafriables en progressions arithm\'etiq…

An integer is said to be $y$-friable if its greatest prime factor P(n) is less than $y$. In this paper, we study numbers of the shape $n-1$ when $P(n)\leq y$ and $n\leq x$. One expects that, statistically, their multiplicative behaviour…

An integer n is said to be y-friable if its largest prime factor $P^+(n)$ is less than y. In this paper, it is shown that the y-friable integers less than x have a weak exponent of distribution at least $3/5-\varepsilon$ when $(\log…

We study the average error term in the usual approximation to the number of $y$-friable integers congruent to $a$ modulo $q$, where $a\neq 0$ is a fixed integer. We show that in the range $\exp\{(\log\log x)^{5/3+\varepsilon}\} \leq y \leq…

An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log…

A number is said to be $y$-friable if it has no prime factor greater than $y$. In this paper, we prove a central limit theorem on average for the distribution of divisors of $y$-friable numbers less than $x$, for all $(x, y)$ satisfying…

Let $\mathcal{A}$ be a finite set of positive integers and $y\geq 1$. We give an effective lower bound of the cardinality of the set $\{n\in\mathcal{A};\,p|n\Longrightarrow p\leq y\}$ under the condition of a good knowledge of the level of…

For any $\epsilon>0$, there exists $q_0(\epsilon)$ such for any $q\ge q_0(\epsilon)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three…

The standard probability law on the set $S(x,y)$ of $y$-friable integers not exceeding $x$ assigns to each friable integer $n$ a probability proportional to $1/n^\alpha$ where $\alpha=\alpha(x,y)$ is the saddle-point of the inverse Laplace…

According to a general probabilistic principle, the natural divisors of friable integers (i.e.~free of large prime factors) should normally present a Gaussian distribution. We show that this indeed is the case with conditional density…

A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a…

Given any irrational number $\alpha$, we show that for any $0<\theta<6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\|n\alpha\| < n^{-\theta},$$ where $(\log n)^C\leq y\leq n$ for some large constant $C>0$.…

A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…

Fix a modulus $q$. One would expect the number of primes in each invertible residue class mod $q$ to be multinomially distributed, i.e. for each $p \,\mathrm{mod}\, q$ to behave like an independent random variable uniform on…

This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…

A positive integer n is called r-free if n is not divisible by the r-th power of a prime. Generalizing earlier work of Orr, we provide an upper bound of Bombieri-Vinogradov type for the r-free numbers in arithmetic progressions.

We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent random variables taking values $\pm 1$ with probability $1/2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper…

Let U and V be two Bertrand numeration systems, and, a and b the two Parry numbers there are naturally associated with. Suppose they are multiplicatively independent. We prove that, if E is a subset of positive integers which is both U and…

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ represents a finite extension of degree $n$ of the finite field ${\mathbb{F}_{q}}$. In this article, we investigate the existence of $m$ elements in arithmetic…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

In this article, we study the classification of some natural numbers related to the combinatorics of congruence subgroups of the modular group. More precisely, we will focus here on the notion of minimal monomial solutions. These are the…