So back when I was running numerical simulations of pop III stellar evolution (which is rather a long time ago now), the interesting thing is that, in compariso

So back when I was running numerical simulations of pop III stellar evolution (which is rather a long time ago now), the interesting thing is that, in comparison to pop II and I, the extreme metal paucity means that you can have (a) stable and extremely long-lived very dim, very small stars, verging on being warmed up gas giants, and (b) very large and extremely short-lived stars, which will supernova or otherwise give way to instability on the order of a few million years after core ignition. You can then work back to speculative mass functions (distribution of mass in the population) by asking (a) how many large, short-lived stars you need to generate the observed metallicity of pop II, and (b) how many extremely small dim objects you'd need to fill out the missing mass without having so many that someone would have seen them by now. The latter is an interesting line of thinking; what is the upper density bound on (for example) 0.1M objects to make both the solar system's continued existence and an inability to see these things plausible. You can also look at the paucity of candidates for ~1M extremely low metallicity stars (there are just a couple out there known that could be pop III candidates) as another constraint on the pop III mass function. Any pop III ~1M stars would have a lifetime longer than that of the universe to date, or at least such was the case in my models. Research on missing mass and pop III has moved on considerably from those days of course.