Andrew Granville's Home Page

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"


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 \( \Delta(A,B)\).


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 \( \zeta(4n+3) \)

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.


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