Five. And you'll need to do more than scale them for party size. A number of them need considerable beefing up. Best early example: Macbannin in adventure #2. (Fights in Zeitgeist are often one-offs, so the players usually tackle them at more-or-less full strength, which turns dailies into encounter powers!)
To add to what gideon said: I strongly recommend thinking about introducing healing surge costs into skill challenges to try and strain player resources a bit more, because otherwise the amount of battle healing in 4e can be overwhelming. Maybe not from the first session, but as soon as you get a hang on general group dynamics.
I started with a party of 5, but lost a player fairly soon after starting. I have been running the adventures as written quite successfully with only 4 players. I imagine that 6 players will require a bit of tweaking.
Zeitgeist 4e is generally on the very easy side, and can be completely stomped by any group with even a modicum of mid-op characters. Not even high-op, but mid-op. Our group played through books #1 through #8 with three mid-op PCs, and books #9 through #13 with two mid-OP PCs. Despite the DM actually beefing up encounters, we have still been completely crushing them, especially at the epic tier.
The only real difficulties stem from monsters with inexplicably potent one-off mechanics, like Reed Macbannin's curse of the writhing cauldron, or Lya Jierre's sunder, or the various forms of antimagic.
Zeitgeist for Pathfinder 1e is even easier. The monsters and NPCs there are very weak.
I'm up to adventure 4 with my group of 4. We had a fifth player but he was only around for the first adventure (and without him it would have been next to impossible for my group to complete).
The second and third adventure were very hard, especially because there is no leader, so healing mid-battle is difficult. And they're not using the strongest character building options either. That said, they are very experienced (10 years playing 4e) and excellent at character optimization.
So far, there have been 2 deaths (the fifth player exited after his characters death at the end of adventure 1, and another player changed characters after dying atop cauldron hill).
So I would say the adventures play better with 4 players as long as they have a leader, but 5 players otherwise.
Lacking a leader in a 4e party, particularly at the heroic tier, is a major handicap for a party. They really have to pull out all of the stops and ensure that their healing surges can actually be used. Being experienced at character optimization helps, but it can only do so much if they are deliberately not using the strongest character-building options.
I've been running with only three party members. I've usually been dropping the equivalent of two standard monsters from each encounter (so a five-monster encounter becomes three monsters) and reducing all solo monsters to 60% of their normal hitpoints.
Depending on how optimized the party is, that may be extravagantly unnecessary.
4e Zeitgeist, from book #4 onwards, already makes many battles terribly easy by arbitrarily chopping away hit points from many monsters, making them even more susceptible to strikers' encounter novas than usual.