I would like to know if there a closed form solution for the sum:
$$ S_n(t) = \sum_{k=0}^{n} \cos( t \sqrt{k} ) $$
There is obviously an easy answer when the sum is replaced by an integral so this question is really asking for the exact form that $S_n(t)$ takes.
Any hints or references on how to approach this problem would be welcome.