math phd : cambridge or harvard princeton

in terms of prestige id say Harvard.

if you are at Cam right now, why not try different places.

You should find out from each what they intend to offer you financially (whether you can get a scholarship/have your teaching done for you) and what research is available, and whether it interests you.

Does anyone know how hard it is to get a phd place in cambride after part III?

How long is a piece of string? This is an absurd question - it cannot be answered satisfactorily. Admission to a doctoral program is based on any number of factors personal to you on which general comments cannot be made. These include (but are by no means limited to) your previous educational attainment (obviously this would have to be excellent for Cambs) and, particularly in the case of admission to British universities (since the PhD is research only), the strength of your research proposal and whether anyone in the prospective department is willing to supervise your work. Grow up, please.

Yeah the question is vague if you say so. But knowing that phd candidates in the math department are recruited from the part III pool, someone who has completed the program could have told me, something like, in pure math we were 100 doing part III, they only admit 10 people or 20 or 30 people to a Phd or something like that. Then I might have had an idea.
Phd places are determined by part III results. Therefore my question wasn't that stupid and that vague

At Cambridge, the process is simple, if you wanna get into the phd, get a distinction in part III. However the distinction is very hard to get.

Candidates for the PhD at Cambridge are not only recruited from the Part III pool (though it would most likely help):

"Students from outside Cambridge should note that they are likely to be accepted directly for the Ph.D. degree only when they have been educated to a standard comparable with that of the Certificate of Advanced Study in Mathematics. Applicants who are not so qualified should ask in the first instance for acceptance for the Certificate. Subsequent acceptance for research usually depends on the achievement of a relatively high standard in the Certificate."

http://www.maths.cam.ac.uk/CASM/

In any event, the advice above still applies - admission (whether from the Part III pool or not) is going to depend on you providing a strong research proposal for a doctoral thesis, and a member of faculty being sufficiently interested in your work in order to take you on. Since the PhD is a research degree and not a taught degree, these factors are far more decisive when it comes to your chances of admission than your mere past grades (or for that matter quotas of people admitted), though clearly if you are on the CASM you will need to be at the top of your class (but that's not very helpful advice is it? That's just common sense).

Its very rare for someone to come in from another Undergraduate degree and go straight into a PhD (I don't actually know of it ever happening, but I'm not ruling out the possibility). It's 99.9999% certain that they'll ask you to sit Part III. And if you do well enough, they may offer you a place.

Sadly, life isn't that simple. Because of the small amount of places available, you'll need to come among the top 3 (ish) in your chosen subject - if it's pure you're going for.

So if you thought a Distinction was your only hurdle, you're quite a bit out! You need a very good distinction. And to my knowledge, around 30-40 people per year get distinctions (from the 180-200). And to top it all off, the people you are competing against will all be among the top performers from their Undergraduate Degrees from all over the world.

Top 3 in your chosen subject; as in your chosen area. May that be Algebra, Combinatorics, Number Theory etc.

It will change slightly from area to area. In some places you'll need to come top, in others you could probably get away with 5th (ish).