A supplement to Chebotarev's density theorem
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913578270851072 |
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| author | Harcos, Gergely Soundararajan, Kannan |
| author_facet | Harcos, Gergely Soundararajan, Kannan |
| contents | 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 $ζ_L(s)/ζ_K(s)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_13412 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A supplement to Chebotarev's density theorem Harcos, Gergely Soundararajan, Kannan Number Theory Primary 11R42, Secondary 11M41 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 $ζ_L(s)/ζ_K(s)$. |
| title | A supplement to Chebotarev's density theorem |
| topic | Number Theory Primary 11R42, Secondary 11M41 |
| url | https://arxiv.org/abs/2210.13412 |