Cleric archers become better the more statpoints are allowed. A lot of the builds listed here will only work on high point buy character (The Divine Might, Divine Favor Cleric archer, for instance requires high wisdom, high dexterity, decent strength and good charisma--on 28 points, you can manage Str 12, Dex 16, Con 12, Int 10, Wis 14, Cha 12 as an elf but at that point, Divine Might is pointless. You might as well take a 14 strength and 10 charisma and get your extra damage all the time without spending turn attempts. 36 points will let you get str 14, dex 18, con 12, Int 10, Wis 16, cha 10 (you could squeeze a bit of damage out of Divine Might by taking dex 16 and charisma 14 but I don't think that's a winning proposition).
Divine Metamagic: Persistent Spell is something no sane DM would allow. It is also a choice that would rapidly suck the fun out of the game for both you and other players (since it would force all other players into supporting roles or force them to find similar cheese and would either make combats non-challenging or force the DM to include tougher monsters who would stand a much better chance ot killing the whole party with a few lucky rolls). You can only power-game so far before it becomes counter-productive. It's probably not worth discussing in build options.
I don't see the attraction of Eldritch Knight for an archer myself. You run into a number of problems immediately. First, you sacrifice 3 points of BAB to qualify as an Eldritch Knight. Second, you pass up any opportunities for Weapon Specialization. So, what can you do to make up for it. You can cast true strike and then use manyshot. That's not a winning combination until you quicken the True Strike and even then it's not likely to be significantly more impressive than a fighter archer with greater weapon specialization doing full attacks (particularly if hasted). You can partially make up for the lost BAB with heroism. But that doesn't stack with bless, hero's feast or bardsong and doesn't counteract the 3 points of BAB anyway. You can cast Flame Arrow on your arrows for more damage and Greater Magic Weapon on your bow. Still, a normal archer at least stands a chance of getting someone else to cast them for him. And Flame Arrow does less damge than Greater Weapon Specialization anyway. You can cast Blink and get +2 to hit and deny your foes their dex bonus. Since you don't have sneak attack, however, that has limited utility. And you pick up a 20% miss chance. Enlarge Person is a dubious choice for an archer due to the dex nerf and someone else could cast it anyway. It seems to me that the primary attractiveness of the Eldritch Knight archer would be "I'm a slightly subpar archer who can also cast spells" but those spells don't actually make you a better archer. A melee Eldritch Knight, OTOH, can benefit from Shield, mirror image, blink, Alter Self, False Life (melee Eldritch Knights are actually in a position to benefit from defensive spells; if an archer is getting hit, there's already a problem), Polymorph, Power Attack (with True Strike--it's not usually a good idea but I think there's better synergy than there is with manyshot), Combat Reflexes (with reach from Polymorph or Enlarge Person), Improved Trip (again, see Polymorph and Enlarge Person), Expert Tactician (with Blink and/or Greater Invisibility), and Arcane Strike.
As for the other prestige classes, it really depends upon what version of them you're using. If you're using the 3.5 versions, Order of the Bow Initiate is a pretty poor choice. My take on a good fighter type archer build would be:
Fighter 12/Exotic Weapon Master 1/Occult Slayer 5/Ranger, Barbarian, Paladin, or Monk 2. This character uses a greatbow (Complete Warrior) with greater weapon specialization, gets close combat shot via Exotic Weapon Master, and has bonusses to disrupt enemy spellcasting and incredible magical defenses from Occult Slayer.
Ranger 20 or something like Ranger 10/Exotic Weapon Master 1/Pious Templar 3/Occult Slayer 5/Deepwood Sniper 1 would be a good choice too if you prefer favored enemy bonusses to Greater Weapon Specialization.
There are other builds that could be made more specialized and effective with higher stats (Ranger 2/Pal 6/Halfling Outrider 10/Order of the Bow Initiate 1/Pious Templar 1 would make an interesting mounted archer and could crank up his damage using Divine Might and (a little bit) with Divine Favor; If there were a halfling deity with bow as a favored weapon, the character could forgo the ranger levels and take 3 Pious Templar levels for weapon specialization).