We study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ y ⩽ x and on average over q ⩽ x/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.

For natural integer n, let D n denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(D n ≤n t ) (0≤t≤1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the generalized moments of arbitrary orders for the limit laws. We also evaluate the mean value of the two-dimensional distribution function ℙ(D n ≤n u , D{n/D n}≤n v ).

Suite à une remarque de Dimitris Koukoulopoulos, que nous prenons plaisir à remercier ici, nous modifions légèrement l'énoncé et la démonstration du Corollaire 1.4, de façon à justifier la remarque (ii).

Improving on estimates of Erdős, Halász and Ruzsa, we provide new upper and lower bounds for the concentration function of the limit law of certain additive arithmetic functions under hypotheses involving only their average behaviour on the primes. In particular we partially confirm a conjecture of Erdős and Kátai. The upper bound is derived via a reappraisal of the method of Diamond and Rhoads, resting upon the theory of functions with bounded mean oscillation.

We revisit recent work of Heath-Brown on the average order of the quantity r(L1(x))⋯r(L4(x)), for suitable binary linear forms L1,…,L4, as x=(x1,x2) ranges over quite general regions in ℤ2. In addition to improving the error term in Heath-Brown’s estimate, we generalise his result to cover a wider class of linear forms.

This paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four . In fact, if U ⊂ X is the open subset formed by deleting the lines from X, and H is the usual projective height function on , then the height zeta function is analytically continued to the half-plane ℜe(s) > 17/20.

Soit P un polynôme à coefficients entiers de degré 2 et Δ son discriminant. La quantité τP(n), définie par τP(n) := card {P(m) > 0: P(m)[mid ]n, m ∈ ℤ}, compte le nombre de diviseurs d de n qui s'écrivent sous la forme d = P(m). Lorsque P(X) = X(X + 1), le cardinal τX(X+1)(n) est égal au nombre de diviseurs consécutifs de n.

Let q and N be integers, let a be an integer coprime to q, and let zN be defined implicitly by q = (log N)log22−ZN √(log2N). We show that for large N, an integer n has at least one divisor d with q [les ] d [les ] N and d ≡ a(mod q) with probability approximately Φ(zN), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall.

Let $S(x,y)$ be the set $S(x,y)=\{ 1\leq n\leq x : P(n)\leq y\}$, where $P(n)$ denotes the largest prime factor of $n$. We study$E_f(x,y;\theta)=\sum_{n\inS(x,y)}f(n)e^{2\pi in\theta}$, where $f$ is a multiplicative function. When $f=1$and when $f=\mu$, we widen the domain of uniform approximation using the method of Fouvry and Tenenbaum andmaking explicit the contribution of the Siegel zero.

Soit$S(x,y)$ l'ensemble $S(x,y)=\{ 1\leq n \leq x: P(n)\leq y\}$, o\`u$P(n)$ d\'esigne le plus grand facteur premier de $n$. Nousétudions $E_f(x,y;\theta)=\sum_{n\in S(x,y)}f(n)e^{2\pi in\theta}$, lorsque $f$ est une fonctionmultiplicative. Quand $f=1$ et quand $f=\mu$, nous élargissons le domaine d'approximation uniformeenutilisant la méthode d\'eveloppée par Fouvry et Tenenbaum et en explicitant la contribution du zéro deSiegel.

1991 Mathematics Subject Classification: 11N25, 11N99.