Andrew Granville's Home Page

2004 Publications

The square of the Fermat quotient,
Integers: Electronic journal of combinatorial number theory, A22 (2004), 4 pages

We prove Skula's conjectured formula for the square of the Fermat quotient, mod \( p\)


The number of unsieved integers up to \( x \) (with K. Soundararajan)
Acta Arithmetica, 115 (2004), 305-328.

If we sieve the integers up to \( x \) by the primes not in some set \( P\), we expect around \( x\prod_{p\not\in P,\ p\leq x} (1-1/p)\) integers left. It is known that one cannot get more than \(e^\gamma\) times this expectation, which we improve (to a more-or-less best possible upper bound). Hildebrand showed that over all sets \( P\) for which \( \prod_{p\not\in P,\ p\leq x} (1-1/p) >1/u \), one gets the least number of integers left unsieved if \( P \) is the set of primes bigger than some \( y\). We give a rather different proof of Hildebrand's Theorem.


On the research contributions of Hugh C. Williams,
High Primes and Misdemeanours: Lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Institute Communications 41 (2004), 197-216.

We survey the life and works of Hugh Williams