The least unramified prime which does not split completely

TL;DR

This paper provides effective, unconditional upper bounds for the smallest prime ideal in a number field that does not ramify or split completely in an extension, improving previous estimates especially when the extension is not Galois.

Contribution

It introduces the first bounds for such primes in non-Galois extensions over arbitrary number fields, extending prior work limited to Galois cases or rational fields.

Findings

First unconditional bounds for non-Galois extensions

Improved estimates over previous results for specific cases

Applicable to general number field extensions

Abstract

Let $K/F$ be a finite extension of number fields of degree $n \geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\mathbb{Q}$ and does not ramify or split completely in $K$. We improve upon the previous best known general estimates due to X. Li when $F = \mathbb{Q}$ and Murty-Patankar when $K/F$ is Galois. Our bounds are the first when $K/F$ is not assumed to be Galois and $F \neq \mathbb{Q}$.

Figures

Tables

Table 1. Table 1. Values of A = A ​ ( n , P d ) 𝐴 𝐴 𝑛 subscript 𝑃 𝑑 A=A(n,P_{d}) when d = 100 𝑑 100 d=100 versus d = 1 𝑑 1 d=1 .

$n$	$4A(n,P_{100})\ge$	$\lambda =\lambda (n,P_{100})$	$4A(n,P_1)\ge$	$\lambda =\lambda (n,P_1)$
2	2.444	21.68	1.493	1.678
3	2.734	17.63	1.827	1.189
4	2.904	15.50	2.039	.9613
5	3.021	14.11	2.193	.8244
6	3.108	13.10	2.311	.7310
7	3.176	12.33	2.406	.6624
8	3.231	11.70	2.485	.6094
9	3.277	11.19	2.553	.5669
10	3.316	10.75	2.611	.5318
20	3.530	8.340	2.951	.3554
50	3.720	6.043	3.293	.2147
100	3.814	4.763	3.483	.1486
200	3.878	3.764	3.625	.1035
500	3.931	2.764	3.757	.0646
1000	3.956	2.191	3.826	.0454
2000	3.971	1.737	3.876	.0319
5000	3.984	1.279	3.921	.0201
10000	3.990	1.015	3.944	.0142

Equations

$$\mathcal{P}(F)$$

$$\mathsf{P}(K/F)$$

$${\mathsf{P}}_{\ast }(K/F)$$

$$\mathsf{P}(\mathbb{Q}(\sqrt{d})/\mathbb{Q}){\ll }_{\epsilon }|d{|}^{\frac{1}{4\sqrt{e}}+\epsilon }$$

$$\mathsf{P}(K/F)\ll\Big{(}\frac{1}{n}\log D_{K}\Big{)}^{2},$$

$${\mathsf{P}}_{\ast }(K/\mathbb{Q}){\ll }_{\epsilon }{D}_K^{\frac{1+\epsilon }{4A(n-1)}},$$

$$A=A(n)=\sup_{\lambda>0}\Big{(}\dfrac{1-\frac{n}{n-1}e^{-\lambda}}{\lambda}\Big{)}\geq 1-\sqrt{\frac{2}{n-1}}.$$

$$B_F=\min \{N_F\log D_F,c_1{D}_F^{1/[F:\mathbb{Q}]}\},$$

$$\mathsf{P}(K/F)\le C_F{D}_K^{\frac{4}{n-1}},$$

$$\mathrm{Re}\{P(1/z)\}\ge 0\text{for }\mathrm{Re}\{z\}\ge 1.$$

$${\mathrm{N}}_{\mathbb{Q}}^F\mathfrak{p}\le C_F{D}_K^{\frac{1+\epsilon }{4A(n-1)}},$$

$$A=A(n,P)=\sup_{\lambda>0}\left(\Big{[}P(1)-\tfrac{n}{n-1}e^{-\lambda}\sum_{k=1}^{d}a_{k}\sum_{j=0}^{k-1}\frac{\lambda^{j}}{j!}\,\Big{]}\Big{/}\lambda\right).$$

$$A(n,P)\ge 1-2n^{-2/3},$$

$$\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{p}\ll\big{(}C_{F}+n^{\frac{3P(1)}{A}}\big{)}D_{K}^{\frac{1+\varepsilon}{4A(n-1)}}.$$

$$p\ll D_K^{\frac{1+\epsilon }{4A(n-1)}},$$

$$\mathsf{P}(K/\mathbb{Q})\ll D_K^{1/8.7},$$

$${\Lambda }_F(\mathfrak{n})=\{\begin{array}{ll}\log {\mathrm{N}}_{\mathbb{Q}}^F\mathfrak{p}& \text{if n is a power of a prime ideal p,}\\ 0& \text{otherwise.}\end{array}$$

$$X_0=X_0(F,\eta ):=\exp (10n_F(\log D_F{)}^2)+\exp (B_F\log (1/\eta )),$$

$$(1-\eta)X+O\Big{(}\frac{X}{(\log X)^{2}}\Big{)}\leq\sum_{\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}\leq X}\Lambda_{F}(\mathfrak{n})\leq(1+\eta)X+O\Big{(}\frac{X}{(\log X)^{2}}\Big{)}.$$

$$\Big{|}\sum_{\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}\leq X}\Lambda_{F}(\mathfrak{n})-X\Big{|}\leq X^{\beta}+O(X\exp(-cn_{F}^{-1/2}(\log X)^{1/2})),$$

$$\beta <1-\frac{1}{B_F},$$

$$\sum _{Y<{\mathrm{N}}_{\mathbb{Q}}^F\mathfrak{n}\le X}\frac{{\Lambda }_F(\mathfrak{n})}{{\mathrm{N}}_{\mathbb{Q}}^F{\mathfrak{n}}^{\sigma }}(\log {\mathrm{N}}_{\mathbb{Q}}^F\mathfrak{n}{)}^{k-1}$$

$$\displaystyle\qquad\qquad\geq\dfrac{(k-1)!}{(\sigma-1)^{k}}\cdot(1-\eta)\big{(}Y^{1-\sigma}-X^{1-\sigma}E_{k-1}((\sigma-1)\log X)\big{)}+O_{k}\Big{(}\frac{1}{(\sigma-1)^{k-1}}\Big{)}$$

$$\displaystyle\sum_{Y<\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}\leq X}\dfrac{\Lambda_{F}(\mathfrak{n})}{\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}^{\sigma}}(\log\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n})^{k-1}=\psi_{F}(X)X^{-\sigma}(\log X)^{k-1}-\int_{Y}^{X}\psi_{F}(t)\dfrac{d}{dt}\big{[}t^{-\sigma}(\log t)^{k-1}\big{]}dt.$$

$$-\psi_{F}(t)\dfrac{d}{dt}\big{[}t^{-\sigma}(\log t)^{k-1}\big{]}\geq(1-\eta)\sigma t^{-\sigma}(\log t)^{k-1}\big{\{}1+O_{k}(\frac{1}{\log t})\big{\}}.$$

$$\sum_{Y<\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}\leq X}\dfrac{\Lambda_{F}(\mathfrak{n})}{\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}^{\sigma}}(\log\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n})^{k-1}\geq(1-\eta)\int_{Y}^{X}t^{-\sigma}(\log t)^{k-1}dt+O_{k}\Big{(}\int_{Y}^{X}t^{-\sigma}(\log t)^{k-2}dt\Big{)}.$$

$${\int }_Y^X{t}^{-\sigma }(\log t{)}^{k-1}dt$$

$$\displaystyle=\dfrac{(k-1)!}{(\sigma-1)^{k}}\big{(}Y^{1-\sigma}E_{k-1}((\sigma-1)\log Y)-X^{1-\sigma}E_{k-1}((\sigma-1)\log X)\big{)}.$$

$$\sum _{\begin{array}{c}v\in V(K/F)\end{array}}\log {\mathrm{N}}_{\mathbb{Q}}^F{\mathfrak{p}}_v\le \log D_K.$$

$$\sum _{\begin{array}{c}v\in V(K/F)\end{array}}\frac{\log {\mathrm{N}}_{\mathbb{Q}}^F{\mathfrak{p}}_v}{{\mathrm{N}}_{\mathbb{Q}}^F{\mathfrak{p}}_v}\le \sqrt{\frac{2[F:\mathbb{Q}]}{[K:F]}\log D_K}.$$

$$\log D_K=[K:F]\log D_F+\log {\mathrm{N}}_{\mathbb{Q}}^F{\mathfrak{d}}_{K/F},$$

Full text

1. Introduction

1.1. History

Let $K/F$ be a finite extension of number fields of degree $n\geq 2$. Define

[TABLE]

The focus of this paper is to establish field-uniform upper bounds for $\mathsf{P}(K/F)$ and $\mathsf{P}_{*}(K/F)$. The study of these quantities has classical origins and has been explored in a variety of cases. Indeed, when $K=\mathbb{Q}(\sqrt{d})$ is a quadratic field over $F=\mathbb{Q}$, this reduces to the problem of bounding the least quadratic nonresidue. Assuming the Generalized Riemann Hypothesis (GRH), Ankeny [Ank52] proved $\mathsf{P}(\mathbb{Q}(\sqrt{d})/\mathbb{Q})\ll(\log|d|)^{2}$. Much less is known unconditionally and progress is notoriously difficult. Namely,

[TABLE]

for $\varepsilon>0$. Aside from the factor of $\varepsilon$, this result of Burgess [Bur57, Bur62] from over fifty years ago remains essentially the best known unconditional bound.

More generally, when $K$ is Galois over $F$ of degree $n\geq 2$, V.K. Murty [Mur94] showed under the assumption of GRH for the Dedekind zeta function of $K$ that

[TABLE]

where $D_{K}=|\mathrm{disc}(K/\mathbb{Q})|$ and the implied constant is absolute. Murty remarks that the same analytic method can yield an unconditional estimate of the form $O_{F}(D_{K}^{1/2(n-1)})$. By a different approach involving geometry of numbers, Vaaler and Voloch [VV00] established an explicit variant of such an unconditional estimate for $\mathsf{P}_{*}(K/\mathbb{Q})$ when $K$ is Galois over $\mathbb{Q}$.

If $K$ is some finite extension of $\mathbb{Q}$ (not necessarily Galois) then, using an elegant argument, X. Li [Li12] superseded this prior unconditional bound for $\mathsf{P}_{*}(K/\mathbb{Q})$. Namely, he showed that

[TABLE]

where

[TABLE]

The key innovation of Li was to incorporate methods of Heath-Brown [HB92] for Dirichlet $L$-functions to obtain a stronger explicit inequality for the Dedekind zeta function.

Recently, Murty and Patankar [MP15, Theorem 4.1] adapted Li’s argument to obtain the first unconditional field-uniform estimate for $\mathsf{P}(K/F)$ when $K$ is Galois over $F$. To introduce their result, let $N_{F}=16$ if there is a sequence of fields $\mathbb{Q}=F_{0}\subset F_{1}\subset\cdots\subset F_{m}=F$ with each $F_{i+1}/F_{i}$ normal and $N_{F}=4[F:\mathbb{Q}]!$ otherwise. Define

[TABLE]

for some sufficiently small absolute constant $c_{1}>0$. Murty and Patankar proved if $K/F$ is Galois of degree $n\geq 2$ then

[TABLE]

where $C_{F}=e^{O([F:\mathbb{Q}](\log D_{F})^{2})}+e^{O(B_{F})}$ and the implied constants are absolute. Note the constant $C_{F}$ in the quoted result (1.6) differs from [MP15, Theorem 4.1] since there seems to be a typo stemming from equation (4.1) therein and its application in their proof. We remark that the dependence on $F$ in (1.6) is natural given the current status of the effective Chebotarev Density Theorem [LO77] and the Brauer–Siegel theorem [Sta74].

1.2. Results

The primary focus of this paper is to improve the exponent of $D_{K}^{1/(n-1)}$ in both (1.3) and especially in (1.6). As a secondary objective, we consider both $\mathsf{P}_{*}(K/F)$ and $\mathsf{P}(K/F)$ for any finite extension $K/F$ which, in that generality, is new. We also demonstrate that one may take the non-split prime in (1.3) to be unramified in $K$ with some minor loss.

Our approach is founded upon Li’s argument blended with ideas of Heath-Brown [HB92] for zero-free regions of Dirichlet $L$-functions and their generalization in [Zam16, Zam17] for Hecke $L$-functions. Namely, we consider more general sums over prime ideals of $F$ which depend on a choice of polynomial. To state our main result, we introduce a definition: a polynomial $P(x)\in\mathbb{R}_{\geq 0}[x]$ is admissible if $P(0)=0,P^{\prime}(0)=1,$ and

[TABLE]

Theorem 1.1**.**

Let $K/F$ be an extension of number fields of degree $n\geq 2$. Let $\varepsilon>0$ be fixed and $P(x)=\sum_{k=1}^{d}a_{k}x^{d}$ be a fixed admissible polynomial. There exists a prime ideal $\mathfrak{p}$ of $F$ such that $\mathfrak{p}$ does not split completely in $K$, $\mathfrak{p}$ is degree 1 over $\mathbb{Q}$, and

[TABLE]

where $C_{F}=e^{O([F:\mathbb{Q}](\log D_{F})^{2})}+e^{O(B_{F})}$, $B_{F}$ is given by (1.5), and

[TABLE]

If $K/F$ is Galois then one may take $\mathfrak{p}$ to also be unramified in $K$. All implied constants depend at most on $\varepsilon$ and $P$.

Remark*.*

• •

While $A$ depends on $n$, it is bounded above and below independent of $n$. In particular, if $P(x)=x+x^{2}$ then

[TABLE]

which improves over (1.4) as $n\rightarrow\infty$. Moreover, the exponent $\tfrac{1+\varepsilon}{4A}$ becomes a nearly sixteen-fold improvement over the exponent $4$ in (1.6) as $n\rightarrow\infty$. With a different choice of $P$, we have by Table 1 that $\frac{1+\varepsilon}{4A}<\frac{5}{12}$ for all $n\geq 2$. This constitutes a nine-fold improvement over (1.6) for all $n\geq 2$.

• •

If $K/F$ is not assumed to be Galois, then one may still take $\mathfrak{p}$ to be unramified in $K$ but we show it satisfies the slightly weaker bound

[TABLE]

By a classical result of Minkowski, recall that $n\leq[K:\mathbb{Q}]\ll\log D_{K}$ so, unless $n$ is unusually large, this additional factor is negligible compared to $D_{K}^{\varepsilon/(n-1)}$.

We restate Theorem 1.1 in the special case $F=\mathbb{Q}$.

Corollary 1.2**.**

Let $K$ be a number field of degree $n\geq 2$. Let $\varepsilon>0$ be fixed and $P(x)=\sum_{k=1}^{d}a_{k}x^{d}$ be a fixed admissible polynomial. The least rational prime $p$ which does not split completely in $K$ satisfies

[TABLE]

where $A=A(n,P)$ is given by (1.8). If $K/\mathbb{Q}$ is Galois then one may also take $p$ to be unramified in $K$. Furthermore, if $P(x)=x+x^{2}$ then $A\geq 1-2n^{-2/3}$. All implied constants depend at most on $\varepsilon$ and $P$.

Choosing a certain admissible polynomial $P(x)=P_{100}(x)$ of degree 100, say, Corollary 1.2 yields savings for every degree $n$ over the special case (1.3) where $P(x)=P_{1}(x)=x$. For example, if $K/\mathbb{Q}$ is an extension of degree 5 then, by Corollary 1.2 with $P=P_{100}$,

[TABLE]

whereas if $P=P_{1}$ then $1/8.7$ is replaced by $1/6.1$. See Sections 5 and 1 for further details on these computations.

Finally, we describe the organization of the paper. Section 2 collects standard estimates related to counting prime ideals in a number field $F$. Section 3 contains an explicit inequality of the Dedekind zeta function and a generalization related to admissible polynomials. Section 4 has the proof of Theorem 1.1 and Section 5 outlines the computation of admissible polynomials and Table 1.

Notation

We henceforth adhere to the convention that all implied constants in all asymptotic inequalities $f\ll g$ or $f=O(g)$ are absolute with respect to all parameters and effectively computable. If an implied constant depends on a parameter, such as $\varepsilon$, then we use $\ll_{\varepsilon}$ and $O_{\varepsilon}$ to denote that the implied constant depends at most on $\varepsilon$.

Acknowledgements

The author would like to thank John Friedlander, Kumar Murty, and Jesse Thorner for their encouragement and helpful comments.

2. Counting prime ideals

Let $F$ be a number field of degree $n_{F}=[F:\mathbb{Q}]$ with discriminant $D_{F}=|\mathrm{disc}(F/\mathbb{Q})|$ and ring of integers $\mathcal{O}_{F}$. Denote $\mathrm{N}^{F}_{\mathbb{Q}}$ to be the absolute norm of $F$ over $\mathbb{Q}$. For each integral ideal $\mathfrak{n}\subseteq\mathcal{O}_{F}$, define

[TABLE]

Lemma 2.1**.**

Let $F$ be a number field and $\eta>0$ be arbitrary. Define

[TABLE]

where $B_{F}$ is defined by (1.5). For $X\geq X_{0}$,

[TABLE]

*All implied constants are absolute. *

Proof.

The effective Chebotarev Density Theorem [LO77] implies that, for $X\geq X_{0}$

[TABLE]

where $c>0$ is some absolute constant and $\beta>1/2$ is a real zero of the Dedekind function of $K$, if it exists. By a theorem of Stark [Sta74, Theorem 1’], any real zero $\beta$ of the Dedekind zeta function $\zeta_{F}(s)$ satisfies

[TABLE]

where $B_{F}$ is given by (1.5). Hence, by (2.1), we have $X^{\beta}=X\cdot X^{\beta-1}\leq\eta X$. By Minkowski’s bound, observe that $n_{F}\ll\log D_{F}\ll\sqrt{\log X}$. It follows that $n_{F}^{-1/2}(\log X)^{1/2}\gg(\log X)^{1/4}$, so the error term in (2.2) is crudely bounded by $O(X/(\log X)^{2})$. ∎

Lemma 2.2**.**

Let $k\geq 1$ be an integer and $\eta\in(0,1/2)$ be arbitrary. Let $X\geq Y\geq X_{0}$ where $X_{0}=X_{0}(F,\eta)$ is defined by (2.1). Denote $E_{k-1}(t)=\sum_{j=0}^{k-1}t^{j}/j!$. Then

[TABLE]

uniformly for $1<\sigma<2$.

Proof.

This is a combination of partial summation and Lemma 2.1. We include the proof for sake of completeness. Define $\psi_{F}(t)=\sum_{\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{n}<t}\Lambda_{F}(\mathfrak{n})$ for $t>1$. By partial summation,

[TABLE]

By Lemma 2.1, it follows for $t\geq Y\geq X_{0}$ that

[TABLE]

Discarding the first term in the previous equation by positivity and using the above inequality, we deduce that

[TABLE]

The remaining integrals are computed by parts. One iteration yields:

[TABLE]

Proceeding by induction, we conclude that

[TABLE]

Substituting this expression in (2.3) and observing $1\leq E_{k-1}(t)\leq e^{t}$ (in order to simplify the main term and error term involving $Y$), we obtain the desired result. ∎

Lemma 2.3**.**

Let $K$ be a finite extension of $F$. Let $V(K/F)$ be the set of places $v$ of $F$ which ramify in $K$ and $\mathfrak{p}_{v}$ be the prime ideal of $F$ attached to $v$. Unconditionally,

[TABLE]

If $K/F$ is Galois then

[TABLE]

Proof.

The unconditional inequality follows from the well-known formula

[TABLE]

where $\mathfrak{d}_{K/F}=\mathrm{N}^{K}_{F}\mathfrak{D}_{K/F}$ and $\mathfrak{D}_{K/F}$ is the relative different ideal of $K/F$. If $K/F$ is Galois then, by Cauchy-Schwarz and [Ser81, Proposition 5, Section I.3],

[TABLE]

as desired. In the above, we used that there are at most $[F:\mathbb{Q}]$ prime ideals $\mathfrak{p}$ of $F$ above a given rational prime $p$ and $\sum_{p}\frac{\log p}{p^{2}}<1$. ∎

3. Polynomial explicit inequality

Let $K$ be a number field with $D_{K}=|\mathrm{disc}(K/\mathbb{Q})|$ and let $\zeta_{K}(s)$ be the Dedekind zeta function of $K$. Our starting point is a variant of the classical explicit formula.

Proposition 3.1** (Thorner–Z.).**

Let $K$ be a number field and $0<\varepsilon<1/8$ be arbitrary. There exists $\delta=\delta(\varepsilon)>0$ such that

[TABLE]

uniformly for $s=\sigma+it$ with $1<\sigma<1+\varepsilon$ and $|t|\leq 1$.

Remark*.*

The value $1/4$ is derived from the convexity bound for $\zeta_{K}(s)$ in the critical strip.

Proof.

This follows from [TZ17, Proposition 2.6]; similar variants appear in [Li12, KN12]. See [Zam17, Proposition 3.2.3] for details. ∎

We would like to analyze more general sums over prime ideals by considering higher derivatives of the logarithmic derivative $-\dfrac{\zeta_{K}^{\prime}}{\zeta_{K}}(s)$. This generalization (Proposition 3.2) is motivated by the work of Heath-Brown [HB92, Section 4].

Given a polynomial $P(x)\in\mathbb{R}_{\geq 0}[x]$ of degree $d$ with $P(0)=0$, write

[TABLE]

and define

[TABLE]

for $\sigma>1$. Recall the definition of an admissible polynomial from (1.7). Note the condition $P^{\prime}(0)=1$ is imposed for normalization purposes since it implies $a_{1}=1$.

Proposition 3.2**.**

Let $0<\varepsilon<1/8$ and $\lambda>0$ be arbitrary. If $P(x)=\displaystyle\sum_{k=1}^{d}a_{k}x^{k}$ is an admissible polynomial of degree $d$ then

[TABLE]

uniformly for

[TABLE]

Proof.

This is essentially [Zam16, Proposition 5.2] with Proposition 3.1 used in place of [Zam16, Lemma 4.3]. Our argument proceeds similarly but we exhibit a different range of $\sigma$ which is more suitable for our purposes. For simplicity, denote $\mathscr{L}=\log D_{K}$ and $n_{K}=[K:\mathbb{Q}]$. Define

[TABLE]

From the functional equation of $\zeta_{K}(s)$, it follows by [Zam16, Lemma 2.6] that

[TABLE]

for $\mathrm{Re}\{s\}>1$. On the other hand, from the Euler product of $\zeta_{K}(s)$ one can verify that

[TABLE]

for $\mathrm{Re}\{s\}>1$. Comparing these two expressions at $s=\sigma$ with (3.1) and taking real parts111This is redundant as the expression is already real, but clarifies the later use of admissibility of $P$., we deduce that

[TABLE]

for $\sigma>1$. We wish to restrict the sum over zeros $\rho$ in (3.2) to $|1-\rho|<\delta$ for $\delta=\delta(\varepsilon)>0$ given by Proposition 3.1. Observe by [LO77, Lemma 5.4] that

[TABLE]

since $k\geq 2,\sigma<1+\frac{\lambda n_{K}}{\mathscr{L}},$ and $n_{K}\ll\mathscr{L}$. Now, consider the linear polynomial $P_{1}(x)=a_{1}x=x$ as $P^{\prime}(0)=1$. By Proposition 3.1, we find that

[TABLE]

Notice $\mathcal{S}(\sigma;P)=\mathcal{S}(\sigma;P_{1})+\mathcal{S}(\sigma;P_{2})$ by linearity in the second argument. Hence, we may combine the above with (3.2) and (3.3) yielding

[TABLE]

In the last step, we noted \mathrm{Re}\big{\{}P\big{(}\tfrac{\sigma-1}{\sigma-\rho}\big{)}\big{\}}\geq 0 by admissibility of $P$. ∎

4. Proof of Theorem 1.1

We will deduce Theorem 1.1 from the following result.

Theorem 4.1**.**

Let $K/F$ be an extension of number fields of degree $n\geq 2$ and $X\geq Y$. Assume one of the following holds:

1. (A1)

Every prime ideal $\mathfrak{p}$ of $F$ which is degree 1 over $\mathbb{Q}$ with $Y<\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{p}\leq X$ splits completely in $K$. 2. (A2)

Every unramified prime ideal $\mathfrak{p}$ of $F$ which is degree 1 over $\mathbb{Q}$ with $Y<\mathrm{N}^{F}_{\mathbb{Q}}\mathfrak{p}\leq X$ splits completely in $K$. 3. (A3)

Assumption (A2) holds and $K/F$ is Galois.

Let $0<\varepsilon<\tfrac{1}{8}$ be arbitrary and $P(x)=\sum_{k=1}^{d}a_{k}x^{d}$ be an admissible polynomial. For $M=M(\varepsilon,P)$ sufficiently large, define

[TABLE]

where $X_{0}=X_{0}(F,\eta)$ is given by (2.1) and $\eta=\eta(\varepsilon,P)$ is sufficiently small. If $X\geq Y\geq Y_{0}$ then

[TABLE]

where $A=A(n,P)$ is given by (1.8).

4.1. Proof of Theorem 1.1 from Theorem 4.1

Without loss, assume $\varepsilon\in(0,\tfrac{1}{8})$. Taking $Y=Y_{0}$ and rescaling $\varepsilon>0$ appropriately in Theorem 4.1 yields

[TABLE]

By considering cases arising from (4.1) and fixing $\varepsilon$ and $P$, this yields the desired bound for $X$ in all cases. Moreover, if $P(x)=x+x^{2}$ and $\lambda>0$ then

[TABLE]

upon setting $\lambda=\sqrt[3]{6/n}$. ∎

4.2. Proof of Theorem 4.1

Let $0<\lambda<\lambda(\varepsilon,P)$ where $\lambda(\varepsilon,P)$ is some sufficiently large constant and let $\sigma=1+\frac{\lambda}{\log X}$. One can verify $A=A(n,P)\geq A(2,P)\gg_{P}1$ and $A\ll P(1)$ from (1.8). Thus, by (4.2), we may assume without loss that $X\geq e^{\lambda(\varepsilon,P)/\varepsilon}$ and $X\geq D_{K}^{1/4(n-1)}$. This implies that $1<\sigma<1+\min\{\varepsilon,\frac{4\lambda(\varepsilon,P)[K:\mathbb{Q}]}{\log D_{K}}\}$. Now, letting $\mathfrak{D}_{K/F}$ be the relative different of $K/F$, consider

[TABLE]

By the positivity of the terms and Proposition 3.2, it follows that

[TABLE]

On the other hand, by any of (A1), (A2), or (A3), each unramified prime ideal of $F$ splits completely into $[K:F]$ prime ideals. Hence, denoting $\mathfrak{d}_{K/F}=\mathrm{N}^{K}_{F}\mathfrak{D}_{K/F}$, we have that

[TABLE]

where

[TABLE]

Here $\sum^{\prime}$ indicates a restriction to ideals $\mathfrak{n}=\mathfrak{p}^{j}$ where $\mathfrak{p}$ is of degree $\geq 2$ over $\mathbb{Q}$ and $j\geq 1$. We estimate each $S_{k}$ using Lemma 2.2 with $\eta=\eta(\varepsilon,P)$ sufficiently small to deduce that

[TABLE]

Since $X\geq Y$, $\sigma=1+\frac{\lambda}{\log X}$, and $e^{-t}\geq 1-t$ for $t>0$, we have that $Y^{1-\sigma}\geq 1-(\sigma-1)\log Y$. The above equation therefore implies that

[TABLE]

To estimate $R_{k}$, we claim that

[TABLE]

We divide into cases according to assumptions (A1), (A2), and (A3).

• •

If (A1) holds then $R_{k}=0$ for all $k$ which trivially yields the claim.

• •

If (A2) holds then, as $\lambda<\lambda(\varepsilon,P)$ and $\sigma=1+\frac{\lambda}{\log X}$,

[TABLE]

Since $Y\geq Y_{0}\geq M[K:F]$ from (4.1) and $M=M(\varepsilon,P)$ is sufficiently large, it follows by Lemma 2.3 that

[TABLE]

• •

If (A3) holds then we argue as above and apply Lemma 2.3 in the $K/F$ Galois case to deduce that

[TABLE]

By AM-GM, claim (4.7) follows.

This proves the claim in all cases. Finally, to estimate $T_{k}$, we similarly observe that

[TABLE]

Combining (4.4), (4.5), (4.6), (4.7), and the above, it follows that

[TABLE]

where $n=[K:F]$,

[TABLE]

One can verify that the supremum of $b(\lambda)$ over $\lambda>0$ exists and $A=A(n,P)=\displaystyle\sup_{\lambda>0}a(\lambda)$ is bounded independent of $n$. By taking $\eta=\eta(\varepsilon,P)$ sufficiently small, we may therefore assume that $\eta b(\lambda)<\tfrac{n-1}{n}\varepsilon A$. Hence, (4.8) implies

[TABLE]

Dividing both sides by $n-1$ and taking the supremum over $0<\lambda<\lambda(\varepsilon,P)$ yields the desired result, except for the range of $\lambda$ in definition of $A$. By straightforward calculus arguments, the supremum of $a(\lambda)$ occurs at $\lambda=\lambda_{n,P}>0$ and one can verify that $\lambda_{n,P}$ is bounded above independent of $n$. Hence, for $\lambda(\varepsilon,P)$ sufficiently large,

[TABLE]

This completes the proof. ∎

5. Admissible polynomials with large values

Here we outline the computation of admissible polynomials $P(x)$ such that $P(1)$ is large which leads to large values for $A(n,P)$ in Theorem 1.1. The key lemma for our calculations follows from arguments in [HB92, Section 4] based on the maximum modulus principle.

Lemma 5.1** (Heath-Brown).**

A polynomial $P(x)\in\mathbb{R}_{\geq 0}[x]$ satisfying $P(0)=0$ and $P^{\prime}(0)=1$ is admissible provided

[TABLE]

For each integer $d\geq 1$, write $P(x)=\sum_{k=1}^{d}a_{k}x^{k}$ where $a_{k}\geq 0$ and $a_{1}=1$. We wish to determine $a_{2},\dots,a_{d}$ such that $P(1)=1+a_{2}+\dots+a_{d}$ is maximum. From Lemma 5.1, it suffices to verify that for all $y\geq 0$,

[TABLE]

Expanding the above as a polynomial in $y$, let $\mathbf{a}=(a_{1},a_{2},a_{3},\dots,a_{d})$ and $C_{2j}^{(d)}=C_{2j}^{(d)}(\mathbf{a})$ be the coefficient of $y^{2j}$ for $0\leq j\leq d-1$; all other coefficients are zero. As $a_{1}=1$, one can see that $C_{0}^{(d)}=1+a_{2}+\dots+a_{d}=P(1)$. Therefore, $P(x)$ is admissible if the remaining $d-1$ coefficients $C_{2j}^{(d)}$ for $1\leq j\leq d-1$ are non-negative. Notice $C_{2j}^{(d)}$ are linear expressions in $a_{2},\dots,a_{d}$. Thus, one may apply the simplex method to maximize the objective function $P(1)=1+a_{2}+\dots+a_{d}$ given the system of linear inequalities $\{C_{2j}^{(d)}(\mathbf{a})\geq 0\}_{j=1}^{d-1}\cup\{a_{j}\geq 0\}_{j=2}^{d-1}$. Based on computational evidence for $1\leq d\leq 100$, the maximum of this linear system occurs precisely when $C_{2j}^{(d)}(\mathbf{a})=0$ for all $1\leq j\leq d-1$. We suspect this scenario is always the case, but we did not seriously investigate it as that is not our aim.

Thus, for each integer $d\geq 1$, let $P_{d}(x)$ be the polynomial associated to the unique solution $\mathbf{a}$ (if it exists) satisfying $C_{2j}^{(d)}(\mathbf{a})=0$ for $1\leq j\leq d-1$. For example,

[TABLE]

These are the same polynomials exhibited in [HB92, Section 4]. Estimate (1.3) is based on the choice of $P_{1}(x)=x$. Setting $d=100$, we may compute $P_{100}(x)$ and subsequently $A(n,P_{100})$ in Table 1 for fixed values of $n$. One can compare $A(n,P_{100})$ with $A(n,P_{1})$ in Tables 1 and 1 above to observe the savings afforded by Corollary 1.2 over (1.3).