Andrew Granville's Home Page

2006 Publications

Estimates for the representation numbers of binary quadratic forms (with Valentin Blomer)
Duke Mathematical Journal, 135 (2006), 261-302.

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.


Aurefeuillian factorization (with Peter Pleasants)
Mathematics of Computation, 75 (2006), 497--508.

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.


Prime number races (with Greg Martin)
American Mathematical Monthly, 113 (2006), 1-33.

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.


Residue races (with Peter and Daniel Shiu)
Ramanujan Journal, 11 (2006), 67-94

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


A good new millenium for the primes,
The Madrid Intelligencer, (2006), 32-36

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

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

Cycle lengths in a permutation are typically Poisson distributed,
Electronic Journal of Combinatorics, 13 (2006), R107.

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.