Are there analogous works to PPSZ algorithm for #P? The PPSZ algorithm tells us that we can do SAT-solving for
$k-$CNF in time at-most $2^{1-(1-o(1))\frac{\pi^2}{6k}}$.
My question is that do we know such results for counting problems in class #P too ? For example, for #SAT do we know any algorithm that provably runs better than $O(2^n)$ ?
There are several #k-SAT algorithms in the literature which can beat $2^n$. Here is a randomized one that gets
$2^{n(1-1/O(k))}$ time (like PPSZ):
https://cseweb.ucsd.edu/~paturi/myPapers/pubs/ImpagliazzoMatthewsPaturi_2012_soda.pdf
There is also a deterministic algorithm with $2^{n(1-1/O(k))}$ runtime behavior. Here is a link:
http://tmc.web.engr.illinois.edu/detapsp_soda.pdf
A caveat: this second algorithm uses exponential space. I believe it is open to find a deterministic algorithm with a similar running time and polynomial space. Be sure look on Google scholar for related references too.