Is Zakat obligatory on an inherited sum from a father, and is it permissible to give it to needy student siblings to cover their university expenses?

If each of your brothers' shares of their father's wealth reaches the nisab (minimum threshold) and a hawl (one lunar year) passes, then zakat becomes obligatory upon them. It is not permissible for you to pay zakat on behalf of your adult, sane brothers unless they grant you permission. As for those who are not adults or not sane, you should pay zakat on their behalf if you are their legal guardian. It is not permissible for any of the brothers to pay his zakat to himself.

Paying zakat to your poor student brothers is permissible if they are studying a knowledge that Muslims need, and their benefit is hoped for, and they are unable to combine seeking knowledge with earning a living. It is also permissible to give them zakat if their knowledge is among the fard al-kifaya (communal obligations) that benefit Muslims, such as medicine and engineering, and they are unable to earn a living or combine it with studying.