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arXiv · 2210.13412

A supplement to Chebotarev's density theorem

Gergely Harcos·Kannan Soundararajan
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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 $C\subset G$ 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 $\zeta_L(s)/\zeta_K(s)$.

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Gergely Harcos, Kannan Soundararajan. 2022-10-24. A supplement to Chebotarev's density theorem. https://doi.org/10.1007/s11425-022-2141-1

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