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Let $r(a,b)$ be a fraction depending on $a,b$ and nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$.
Let $C(a,b)$ be a squarefree positive integer depending on $b$ and different for every $b$.
Consider
$$ S_J = \sum_j^J \sum_i^I \arcsin( r(i,j) \sqrt C(i,j) ) $$
Where $...