Olivier Robert

Three-dimensional exponential sums with monomials

Abstract Fouvry and Iwaniecs theorem concerning three-dimensional exponential sums with monomials relies on a spacing lemma whose optimal form is yet unproved. We bypass their spacing lemma via a diophantine problem in four variables and we obtain the expected bound in their theorem. In the problem of abelian groups of a given order, this yields the exponent 1/4 + ε, a result close to a conjecture of H. E. Richert (1952).

A Fourth Derivative Test for Exponential Sums

We give an upper bound for the exponential sum ∑Mm=1 e2iπf(m) in terms of M and λ, where λ is a small positive number which denotes the size of the fourth derivative of the real valued function f. The classical van der Corputs exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums.

Rational points on linear slices of diagonal hypersurfaces

An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest

Zero-free regions for Dirichlet series

In this paper, we continue some work devoted to explicit zero-free discs for a large class of Dirichlet series. In a previous article, such zero-free regions were described using some spaces of functions which were defined with some technical conditions. Here we give two different natural ways in order to remove those technical conditions. In particular this allows to right down explicit zero-free regions differently and to obtain for them an easier description useful for direct applications.

A paucity estimate related to newton sums of odd degree

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order. §

Exponential Sums and Correctly-Rounded Functions

The 2008 revision of the IEEE-754 standard, which governs floating-point arithmetic, recommends that a certain set of elementary functions should be correctly rounded. Successful attempts for solving the Table Makers Dilemma in binary64 made it possible to design CRlibm, a library which offers correctly rounded evaluation in binary64 of some functions of the usual libm. It evaluates functions using a two step strategy, which relies on a folklore heuristic that is well spread in the community of mathematical functions designers. Under this heuristic, one can compute the distribution of the lengths of runs of zeros/ones after the rounding bit of the value of the function at a given floating-point number. The goal of this paper is to change, whenever possible, this heuristic into a rigorous statement. The underlying mathematical problem amounts to counting integer points in the neighborhood of a curve, which we tackle using so-called exponential sums techniques, a tool from analytic number theory.