A Bombieri-Vinogradov theorem for number fields. (English) Zbl 1533.11192
Let \(K/M\) be a Galois extension of number fields with Galois group \(G\) and let \(C\) be a conjugacy class in \(G\). For rational integers \(q\) and \(a\) prime to \(q\) let \(\pi(x,C,q,a)\) be the number of prime ideals \(\mathfrak p\) of \(M\) unramified in \(K\) with \(N_{M/\mathbb Q}(\mathfrak p)\le x\), \(N_{M/\mathbb Q}(\mathfrak p)\equiv a\) mod \(q\) with its Frobenius class \(\sigma_\mathfrak p\) lying in \(C\). The authors apply an analogue of a method used by R. C. Vaughan in the rational case [Acta Arith. 37, 111–115 (1980; Zbl 0448.10037)] to show that if \(H\) is the largest abelian subgroup of \(G\) with \(C\cap H\ne\emptyset\), and \(\theta=\min\{2/n_E,1/2\}\), where \(n_E\) is the degree of the fixed field \(E\) of \(H\), then for any \(A>0\) there exists \(B(A)>0\) such that one has
\[
\sum_{q\le Q} {'}\max_a\max_{y\le x}\left|\pi(y,C,q,a)-\frac{|C|}{|G|\varphi(q)}\pi(y)\right|\ll \frac x{\log^Ax},
\]
where \(Q=x^\theta\log^{-B}x\) and the sum runs over numbers \(q\) with \(K\cap \mathbb Q(\zeta_q)=\mathbb Q\).
This improves in case \(n_E\ge4\) the earlier result of M. R. Murty and K. L. Petersen [Trans. Am. Math. Soc. 365, No. 9, 4987–5032 (2013; Zbl 1336.11072)], where \(q=x^{\theta-\varepsilon}\) with \(\theta =\min\{2/(n_E-2),1/2\}\).
The main result is applied to improve J. Thorner’s bound for gaps of primes in Chebotarev sets in the case of non-abelian fields of degree \(\ge8\) [Res. Math. Sci. 1, Paper No. 4, 16 p. (2014; Zbl 1362.11082)] and the result of R. Gupta et al. [in: Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 189–201 (1987; Zbl 0618.12006)] on the Euclideanity of rings of \(S\)-integers.
This improves in case \(n_E\ge4\) the earlier result of M. R. Murty and K. L. Petersen [Trans. Am. Math. Soc. 365, No. 9, 4987–5032 (2013; Zbl 1336.11072)], where \(q=x^{\theta-\varepsilon}\) with \(\theta =\min\{2/(n_E-2),1/2\}\).
The main result is applied to improve J. Thorner’s bound for gaps of primes in Chebotarev sets in the case of non-abelian fields of degree \(\ge8\) [Res. Math. Sci. 1, Paper No. 4, 16 p. (2014; Zbl 1362.11082)] and the result of R. Gupta et al. [in: Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 189–201 (1987; Zbl 0618.12006)] on the Euclideanity of rings of \(S\)-integers.
Reviewer: Władysław Narkiewicz (Wrocław)
MSC:
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11M41 | Other Dirichlet series and zeta functions |
| 11R44 | Distribution of prime ideals |
| 11R45 | Density theorems |
Keywords:
Bombieri-Vinogradov theorem; Chebotarev density theorem; Euclidean algorithm; \(S\)-integers; Chebotarev setsReferences:
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