Dirichlet's theorem shows that, for any fixed prime integer a, "big prime numbers mod a" are uniformly distributed between 1 and a1. If we similarly pick different prime integers b,c,..., are these uniform distributions independent of each other?
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Yes (asymptotically of course). The prime number theorem for arithmetic progressions tells us that primes are (asymptotically) uniformly distributed in the $\phi(m)$ reduced residue classes modulo $m$ for any integer $m$, even composite $m$. You can quickly convince yourself that, if $a$ and $b$ are primes, the independence of the uniform distribution of primes modulo $a$ and modulo $b$ is exactly the same thing as the uniform distribution of primes modulo $ab$. The same holds no matter how (finitely) many primes $a,b,c,...$ you use. 

