I don’t know of any intensive efforts specifically on prime searching in Julia, and of course there’s not a lot of motivation to simply re-implement special-purpose algorithms that have already been intensively optimized in other languages (such as counting primes or searching for Mersenne primes).
As @Nathan_Boyer mentioned, there is a native Julia package Primes.jl that provides basic functionality in this regard, but it’s not currently designed to compete with heavily optimized libraries for well known functions like prime-counting (ala the primecount C++ program). See also the discussions in:
- Fast Generation of Arrays of Prime Numbers
- Number of primes below a given number - #37 by miguelraz
- Improving performance in checking prime numbers - #11 by Oscar_Smith
- How to reduce memory allocations in the Sieve of Eratosthenes?
However, there are reportedly people working on new research problems in computational number theory using Julia, e.g. via the Hecke.jl package.