Abstract. We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.

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For (M, a) = 1, put G(X; M, a) = supwhere p ′ n denotes the n-th prime that is congruent to a (mod M ). We show that for any positive C, provided X is large enough in terms of C, there holdsuniformly for all M ≤ κ(log X) 1/5 that satisfy2010 Mathematics Subject Classification. 11N05, 11N13.

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