math.SE
Define $(x \mid y ) = \begin{cases}1 \text{ if } x \text{ divides }y \\ 0 \text{ if } x \text{ does not divide } y \end{cases}$. $$ \theta(i,k) = \sum_{n=p_i + 2k}^{p_{i+1}^2 - 2k}\left(\dfrac{1}{n -k -\sqrt{n+k} +1}\sum_{j=\sqrt{n+k}}^{n-k}\left( \ \sum_{d\ |\ j\#} (-1)^{\Omega(\cdot)}\left((-1)...