The simple answer to your question is: The parent he lived with more .
" If a child is emancipated under state law, the child is treated as not living with either parent" is irrelevant in your case . Emancipation simple means that the "special rule" for divorced and separated parents no longer applies*. That is, he can no longer be the father's "qualifying child" and the father could only claim him under the "qualifying relative rules".
The other answer supplies the rules; but one rule is missing: The potential dependent cannot be the qualifying child of another taxpayer.
Since the child (he's under 24, so for taxes, he's still a child even if emancipated) lives with his mother, he is her "qualifying child", unless he is providing more than half his own support. Of course, if he's providing more than half his own support, you could not be proving more than half his support and therefore would not meet the support test for a qualifying relative. Scholarships are considered third party support and not support provided by the student. Student Loans are considered the child's own support, unless a parent co-signs.
* An emancipated person is not longer in anybody's "custody". But, the mother can continue to claim the child as a qualifying child, for tax purposes, because the child still resides with her. But, the father not only needs the mother to not claim the child, he needs to meet the qualifying relative dependent support and income test. It is not enough to have paid more than the other parent. You must have paid more than half his total support. The "splitting" of the tax benefits is no longer allowed.
The custodial parent can no longer release the dependency to the non-custodial parent with form 8332, if the child is emancipated. In practice, this technicality is widely ignored (or unknown) by divorced parents.
