  We establish mean convergence for multiple ergodic averages with iterates
given by distinct fractional powers of primes and related multiple recurrence
results. A consequence of our main result is that every set of integers with
positive upper density contains patterns of the form $\{m,m+[p_n^a],
m+[p_n^b]\}$, where $a,b$ are positive non-integers and $p_n$ denotes the
$n$-th prime, a property that fails if $a$ or $b$ is a natural number. Our
approach is based on a recent criterion for joint ergodicity of collections of
sequences and the bulk of the proof is devoted to obtaining good seminorm
estimates for the related multiple ergodic averages. The input needed from
number theory are upper bounds for the number of prime $k$-tuples that follow
from elementary sieve theory estimates and equidistribution results of
fractional powers of primes in the circle.
