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There's some debate on this topic. It says you can fire a bow "without penalty". Some take "penalty" in a strict sense, meaning only the -1 to hit that's associated with using a weapon with your buckler arm. Others take "penalty" to mean "any drawback", and believe the archer retains the buckler's AC bonus.
Personally, I'm somewhat torn, but lean toward "no bonus".

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In any case, if you use a weapon in your off hand, you lose the buckler's Armor Class bonus until your next turn. You can cast a spell with somatic components using your shield arm, but you lose the buckler's Armor Class bonus until your next turn.
I think the intent is if you use the buckler hand for anything, casting a spell, attacking with a melee or ranged weapon, then you lose the shield bonus until your next turn.

Dolanar |
While typically D&D/PF bucklers can be strapped to your forearm instead which allows the hand to be used. This being the case based on the fact that your hand is always in a sense in a "blocking position" (pretend to fire a bow. You draw an arrow & fire with the hand not holding the bow which would be the hand without the buckler on it.) you're only holding the bow with one hand essentially, I don't see why you couldn't get AC bonus.