The work of Green and Tao shows that there are infinitely many arbitrarily long arithmetic progressions of primes. Recently, Maynard and Tao independently proved that for any [Formula: see text], there exists [Formula: see text] (depending on [Formula: see text]) so that for any admissible set [Formula: see text], there are infinitely many [Formula: see text] such that at least [Formula: see text] of [Formula: see text] are prime. We obtain a common generalization of both these results for primes satisfying Chebotarev conditions. We also give an improvement of the known bound for gaps between primes in any given Chebotarev set.

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In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, and the non-existence of simple zeros or poles in such quotients.

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In this article, we prove a new bound for the least prime ideal in the Chebotarev density theorem, which improves the main theorem of Zaman [13] by a factor of 5/2. Our main improvement comes from a new version of Turán's power sum method. The key new idea is to use Harnack's inequality for harmonic functions to derive a superior lower bound for the generalised Fejér kernel.