Orthogonality of restricted primes with nilsequences

The randomness of arithmetic functions with respect to linear correlations can be measured by Gowers uniformity norms. We show that the von Mangoldt function of primes restricted to a fixed Chebotarev class varies randomly around its average, up to structure arising from congruences to small moduli. By the inverse theory of Green-Tao-Ziegler, we can achieve this by studying correlations with nilsequences. Under GRH, we get analogous results for primes with a prescribed primitive root. This is joint work with Magdaléna Tinková.