Short form: recent studies indicate that the habitable zones around red dwarf stars could be wider than previously thought, with the possibility of perhaps tens of billions of planets in the galaxy where the surface temperature could support liquid water.
That’s the good news.
The more disappointing bit is that even a wider habitable zone around a red dwarf is still pretty close, meaning that the tidal forces on a planet that close in would be pretty severe. Thus if a planet has a spin rate so that it has a “day’ anything like 24 hours, the tides in any oceans could be monstrous. It is unlikely that a planet could hold onto any sort of moon for geological timespans given the orbital perturbations from that close star. Thus over a few billion years, perhaps just a few million, tidal drag would slow the rotation of a livable planet down so that it would be tidally locked with the star… one face perpetually lit by the star; one in perpetual darkness. Exactly how such a planet would be a really livable place is a bit of a stumper.
Still… on a *billion* such worlds, it’s a safe bet that life would arise many, many times, and at least some of the time, make a go of it.