  We generalize the classical Bombieri-Vinogradov theorem to a short interval,
non-abelian setting. This leads to variants of the prime number theorem for
short intervals where the primes lie in arithmetic progressions that are
"twisted" by a splitting condition in a Galois extension $L/K$ of number
fields. Using this result in conjunction with recent work of Maynard, we prove
that rational primes in short intervals with a given splitting condition in a
Galois extension $L/\mathbb{Q}$ exhibit dense clusters in short intervals. We
explore several arithmetic applications related to questions of Serre regarding
the nonvanishing Fourier coefficients of cuspidal modular forms, including
finding dense clusters of fundamental discriminants $ d $ in short intervals
for which the central values of $d$-quadratic twists of modular $L$-functions
are non-vanishing.
