1999 Publications

Motivating the multiplicative spectrum (with K. Soundararajan) Topics in Number Theory, (S.D. Ahlgren et al., eds.), Netherlands: Kluwer, 1999, 1-5.

We motivate the results that will appear in our 2001 Annals paper "The spectrum of multiplicative functions"

Article

The set of differences of a given set (with F. Roesler) American Mathematical Monthly, 106 (1999), 338-344.

If $$a=\prod_i p_i^{a_i},\ b=\prod_i p_i^{b_i}$$ then $$a/(a,b)=\prod_i p_i^{c_i}$$ where $$c_i=\max\{ a_i-b_i,0\}$$; we write $$\Delta(a,b)$$ for the vector $$c$$. We study here the size of the set $$\Delta(A,B)$$.

Article

Borwein and Bradley's Apéry-like formulae for $$\zeta(4n+3)$$ (with Gert Almkvist) Experimental Mathematics, 8 (1999), 197-204.

We prove Borwein and Bradley's conjectured Apéry-like formulae for $$\zeta(4n+3)$$

Article

On the scarcity of powerful binomial coefficients, Mathematika, 46 (1999), 397-410.

Assuming the $$abc$$-conjecture we show that there are only finitely many powerful binomial coefficients $$\binom nk$$ with $$3\leq k\leq n/2$$, since if $$q^2$$ divides $$\binom nk$$ then $$q\ll n^2 \binom nk^{o(1)}$$. Unconditionally we show that there are $$N^{1/2+o(1)}$$ powerful binomial coefficients in the top $$N$$ rows of Pascal's triangle.

Article

Review of Notes on Fermat's Last Theorem'" by Alf van der Poorten American Mathematical Monthly, 106, pages 177-181. 98 (1999), 5-8

Review of this lovely book

Article