What is the most correct way to generate random numbers in C with pthread

On Linux you can use the rand_r() for a mediocre generator or the drand48_r() function for a much better one. Both are thread safe replacements for rand() and drand48(), by taking a single argument consisting of the current state, instead of using global state.

With regard to your question on initialization, both of the generators above allow you to seed at whatever point you desire, so you are not forced to seed them before spawning your threads.

When working with threads and doing e.g simulations or so it is very important that you have the random generators independent. First, dependencies between them can really bias your results and then mechanisms for access control to the state of the random generator will most likely slow down execution.

On POSIX systems (where you seem to be) there is the *rand48 family of functions where erand48, nrand48 and jrand48 take the state of the random generator as input values. So you can easily have independent states in each thread. Those you can initialize with a known number (e.g the number of your thread) and you'd have a reproducible sequence of random numbers. Or you initialize it with something non-predicable such as the current time & the number to have sequences that vary for each execution.

On Windows you can use the rand_s() function, which is thread safe. If you are already using Boost, then boost::random is competent (although I appreciate this is tagged C, not C++).