In this article, we prove a new bound for the least prime ideal in the Chebotarev density theorem, which improves the main theorem of Zaman [Funct. Approx. Comment. Math. 57 (2017), no.1, 115-142] by a factor of $5/2$. Our main improvement comes from a new version of Turán's power sum method. The key new idea is to use Harnack's inequality for harmonic functions to derive a superior lower bound for the generalised Fejér kernel.
In this paper we show there exists an infinite family of number fields $L$, Galois over $\mathbb{Q}$, for which the smallest prime $p$ of $\mathbb{Q}$ which splits completely in $L$ has size at least $( \log(|D_L|) )^{2+o(1)}$. This gives a converse to various upper bounds, which shows that they are best possible.
We study the prime numbers that lie in Beatty sequences of the form $\lfloor αn + β\rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $α^{-1}\cdot\frac{|C|}{|G|}$. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri--Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bound…
We establish a joint distribution result concerning the fractional part of $αp^θ$ for $θ\in (0,1), \ α>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$. As an application, for a fixed non-CM elliptic curve $E/\mathbb{Q}$, an asymptotic formula is given for the number of primes at the extremes of the Sato-Tate measure modulo a large prime $\ell$. These are precisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. We assume a zero-free region hypothesis for Dedekin…
We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun-Titchmarsh theorem proved by the authors. We also present a new application of ou…
Théorème de Chebotarev et complexité de Littlewood
Chebotarev密度定理在广义黎曼假设和Artin猜想下的有效形式（cf. Iwaniec and Kowalski's解析数论，5.13）涉及有限群$G$的子集$D$的数值不变量，我们称之为$D$的Littlewood复杂度。我们详细研究了这个不变量。利用这项研究，和一个新的应用程序的大筛，我们给出了改进版本的两个标准问题有关Chebotarev：第一个总理的限制在一个Frobenian集，和渐近的一组素数与给定的Frobenius在一个无限的伽罗瓦扩展。然后，我们给出了具体的应用，在数论中的各种问题，如第一原根模一个素数$\ell$，因式分解的整数多项式模素数，朗-特罗特猜想及其推广，和塞尔的一致性猜想。
Lagarias, Montgomery, and Odlyzko proved that there exists an effectively computable absolute constant $A_1$ such that for every finite extension $K$ of ${\mathbb{Q}}$, every finite Galois extension $L$ of $K$ with Galois group $G$ and every conjugacy class $C$ of $G$, there exists a prime ideal $\mathfrak{p}$ of $K$ which is unramified in $L$, for which $\left[\frac{L/K}{\mathfrak{p}}\right]=C$, for which $N_{K/{\mathbb Q}}\,\mathfrak{p}$ is a rational prime, and which satisfies $N_{K/{\mathbb Q}}\,{\mathfrak{p}} \leq 2 {d_L}^{A_1}$. In this paper we show without any restriction that $N_{K/{\…
We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the Galois group of $L/K$. We show that there exists an unramified prime $\mathfrak{p}$ of $K$ such that $σ_{\mathfrak{p}}=C$ and $N \mathfrak{p} \le d_{L}^{B}$ with $B= 310$. This improves the value $B=12\,577$ as proven by Ahn and Kwon. In comparison to previous works on the subject, we modify the weights to detect the least prime, and we use a new version of…
Let $K/k$ be a finite Galois extension of number fields, and let $H_K$ be the Hilbert class field of $K$. We find a way to verify the nonsplitting of the short exact sequence $$1\to Cl_K\to \text{Gal}(H_K/k)\to\text{Gal}(K/k)\to 1$$ by finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of $k$ that factor into the product of principal prime ideals in $K$. We also find explicit equations to express the principal density in terms of the invariants of $K/k$. In particular, we prove that t…
The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new between Chebotarev's theorem and the congruences satisfied by linear sequences. This link is done into the context of number field extensions. In particular we characterize primes ideals totally decomposed by simple congruences on the terms of linear recurrent sequences. Our results are illustrated by numerous examples, including Padovan sequencesof group $S_3$ an…
We generalize the Chebotarev density formulas of Dawsey (2017) and Alladi (1977) to the setting of arbitrary finite Galois extensions of number fields $L/K$. In particular, if $C \subset G = \textrm{Gal}(L/K)$ is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of $K$: \[ -\lim_{X\rightarrow\infty} \sum_{\substack{2\leq N(I)\leq X \\ I \in S(L/K; C)}} \frac{μ_K(I)}{N(I)} = \frac{|C|}{|G|}, \] where $μ_K(I)$ denotes the generalized Möbius function and $S(L/K;C)$ is the set of ideals $I\subset \mathcal{O}_K$ such that $I$ has a uni…
We prove an effective version of the Chebotarev theorem for the density of prime ideals with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We g…
We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $δ(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{μ(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = δ(S), \] where $μ(\mathfrak{a})$ is the generalized Möbius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of min…
Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and $x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a norm form is zero, while the relative natural density of the set of prime numbers represented by a norm form exists and is positive. These results require tools from class field theory, including the Artin-Chebotarev density theorem and the Hilbert class field. We introduce these tools as we need them in the course of the main arguments. This article is expo…
Let $K$ be a number field, and let $G$ be a finitely generated subgroup of $K^\times$. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes $\mathfrak p$ of $K$ such that the order of $(G\bmod\mathfrak p)$ is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes $\mathfrak p$ for which the order is $k$-free, and those for which the order has a prescribed $\ell$-adic valuation for finitely many primes $\ell$. An additional condition on the Frobenius conjug…
We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. Interestingly, the correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly provide. Finally, we hint at possible applications of this correspondence.
We say that n ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime if any k of them are relatively r-prime. In this article, we provide an exact formula for the probability that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime.
Let $K$ be a number field with the discriminant $D_K$ and the class number $h_{K}$, which has bounded degree over $\mathbb{Q}$. By assuming GRH, we prove that every ideal class of $K$ contains a prime ideal with norm less than $h_{K}^2\log(D_K)^{2}$ and also all but $o(h_K)$ of them have a prime ideal with norm less than $h_{K}\log(D_K)^{2+ε}$. For imaginary quadratic fields $K=\mathbb{Q}(\sqrt{D})$, by assuming Conjecture~\ref{piarcor} (a weak version of the pair correlation conjecure), we improve our bounds by removing a factor of $\log(D)$ from our bounds and show that these bounds are opti…
We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.
The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias
School of Mathematics, Nanjing University（南京大学数学系）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心；中国科学院大学）；University of the Chinese Academy of Sciences
