R. de la Bretèche

Séries trigonométriques à coefficients arithmétiques

By exploiting the recent analytic tool of friable summation, this work describes a new approach to a class of problems in multiplicative number theory and Fourier series theory originated by H. Davenport. A definite answer to the last original question of Davenport of this type, which was still open, as well as a number of other applications, is given.

Sums of arithmetic functions over values of binary forms

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explicit.

Manin’s conjecture for quartic Del Pezzo surfaces with a conic fibration

— An asymptotic formula is established for the number of Q-rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.

Binary linear forms as sums of two squares

We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in Heath-Browns estimate we generalise his result quite extensively.

On Manin's conjecture for singular del Pezzo surfaces of degree four, II

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 e(s) > 17/20.

Density of Châtelet surfaces failing the Hasse principle

Ch\^atelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. Restricting attention to a special class of Ch\^atelet surfaces, we investigate the frequency that such counter-examples arise over the rationals.

Le problème des diviseurs pour des formes binaires de degré 4

Abstract We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over ℚ.

Oscillations localisées sur les diviseurs

Let f be a real arithmetic function and ∆(n, f) denote the corresponding generalization of Hooleys Delta-function. We investigate weighted moments of ∆(n; f) for oscillating functions f, typical cases being those of a non principal Dirichlet character or of the Mobius function. We obtain, in particular, sharp bounds up to factors (log x) o(1) for all weighted finite integral, even moments computed on the integers not exceeding x. This is the key step to the proof, given in a subsequent work, of Manins conjecture, in the strong form conjectured by Peyre and with an e↵ective remainder term, for all Châtelet surfaces. The proof of the main results rest upon a genuinely new approach for Hooley-type functions.

Sur la concentration de certaines fonctions additives

Improving on estimates of Erdős, Halasz 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 Katai. 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.

Sur les processus arithmétiques liés aux diviseurs

Abstract 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 ).