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

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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 $\Delta(A,B)$.

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

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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.

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Review of this lovely book

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