A supplement to Chebotarev's density theorem

Harcos, Gergely and Soundararajan, K. (2023) A supplement to Chebotarev's density theorem. SCIENCE CHINA MATHEMATICS, 66 (12). pp. 2749-2753. ISSN 1674-7283

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Abstract

Let L/K be a Galois extension of number fields with Galois group G. We show that if the density of prime ideals in K that split totally in L tends to 1/|G| with a power saving error term, then the density of prime ideals in K whose Frobenius is a given conjugacy class C ⊂ G tends to |C|/|G| with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s).

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