Let \( f\) be a primitive positive integral binary quadratic form of discriminant \( -D\), and
let \( r_f(n)\) be the number of representations of \( n\) by \( f\) up to automorphisms of \( f\) . In this
article, we give estimates and asymptotics for the quantity
\( \sum_{n\leq x} r_f(n)^\beta\) for all \( \beta\geq 0\)
and uniformly in \( D=o(x)\). As a consequence, we get more-precise estimates for the
number of integers which can be written as the sum of two powerful numbers.Article

Aurefeuillian factorization is a technique to partially factor numbers of the form \(b^n\pm 1\) by replacing the \( b^n\) by a polynomial in such a way that the resulting polynomial can be factored. For example, taking \( y=2^k\) in \( (2y^2)^2+1=(2y^2-2y+1)(2y^2+2y+1)\) we can partially factor \( 2^{4k+2}+1\). We show here that Schinzel determined all such identities, and then attack a generalization of the problem using Faltings' Theorem.
Article

This is a survey of what was then known about "prime number races". Subsequently two of my students from that time, Youness Lamzouri and Daniel Fiorilli, have proved several extraordinary results (some with collaborators) greatly extending this theory.
Article

We prove asymptotic formulas for residue races in terms of Dedekind sumsArticle

In this article, distributed at the Madrid ICM, I discuss various recent developments on the theory of prime numbersArticle

In Spanish: La Gaceta de la RSME**12 **(2009), 547-556.

In Spanish: La Gaceta de la RSME

The set of cycle lengths of almost all permutations in \( S_n\) are "Poisson distributed":
we show that this remains true even when we restrict the number of
cycles in the permutation. The formulas we develop allow us to also show that almost
all permutations with a given number of cycles have a certain "normal order"
(in the spirit of the Erdos-Turan theorem). Our results were inspired by analogous
questions about the size of the prime divisors of "typical"S integers.Article