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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We consider a certain class of multiplicative functions f : N → C. Let F (s) = ∞ n=1 f (n)n −s be the associated Dirichlet series and FN (s) = n≤N f (n)n −s be the truncated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f . More precisely, we prove estimates for the sum x n=1 f (n)/n in terms of the size of |F (1 + 1/ log x)| and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums FN (s).In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field K. More precisely, we give some improved results for the number of zeros upto height T as well as new zero density results for the number of zeros up to height T , lying to the right of Re(s) = σ, where σ ≥ 1/2.

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