1 Definitions
Some definitions, local to this paper, that occur frequently.
For any finite \(A\subset \mathbb {N}\)
\[ R(A)=\sum _{n\in A}\frac{1}{n}. \]
For any finite \(A\subset \mathbb {N}\) and prime power \(q\) we define
\[ A_q = \{ n\in A : q\mid n\textrm{ and }(q,n/q)=1\} \]
and let \(\mathcal{Q}_A\) be the set of all prime powers \(q\) such that \(A_q\) is non-empty (i.e. those \(p^r\) such that \(p^r\| n\) for some \(n\in A\)).
For any finite \(A\subset \mathbb {N}\) and prime power \(q\in \mathcal{Q}_A\) we define
\[ R(A;q) = \sum _{n\in A_q}\frac{q}{n}. \]
For any finite set \(A\subset \mathbb {N}\), \(K\in \mathbb {R}\) and interval \(I\), we define \(\mathcal{D}_I(A;K)\) to be the set of those \(q\in \mathcal{Q}_A\) such that
\[ \# \{ n\in A_q: \textrm{no element of }I\textrm{ is divisible by }n\} \ {\lt}\ K / q. \]