### 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 $$p$$

Article

#### 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.

Article

#### 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

Article