On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one

TL;DR

This paper studies how the products of a fixed complex number and prime elements in imaginary quadratic fields distribute modulo one, extending classical results using advanced sieve methods and harmonic analysis techniques.

Contribution

It extends distribution results of $eta p$ modulo one to imaginary quadratic fields with class number one, employing Harman's sieve and Poisson summation in a novel setting.

Findings

Infinitely many primes p satisfy the inequality involving $ orm{eta p}_ ext{omega}$

Established distribution bounds analogous to classical results in new number field context

Introduced smoothing techniques for effective use of Poisson summation in algebraic number fields

Abstract

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

Tables

Table 1. Table 1. Improvements on the admissible exponent θ 𝜃 \theta in  Eq.   1.2 .

Date	Author(s)	$\theta$
1978	Vaughan [20]	$1/4-ϵ$	$=0.25-ϵ$
1983	Harman [6]	$3/10$	$=0.3$
1993	Jia [12]	$4/13$	$=0.3076\dots$
1996	Harman [7]	$7/22$	$=0.31\overline{81}$
2000	Jia [13]	$9/28$	$=0.3214\dots$
2002	Heath-Brown and Jia [10]	$16/49$	$=0.3265\dots$
2009	Matomäki [15]	$1/3-ϵ$	$=0.3\overline{3}-ϵ$

Equations

$$\left|\alpha -a/q\right|<q^{-2},$$

$$\Vert q\alpha \Vert <q^{-1}\text{for infinitely many }q.$$

$$\Vert p\alpha \Vert <p^{-\theta }\text{for infinitely many rational primes }p.$$

$$\varrho ={\Re }_{\omega }\varrho +({\Im }_{\omega }\varrho )\omega .$$

$$\Vert \varrho {\Vert }_{\omega }=\max \{\Vert {\Re }_{\omega }\varrho \Vert ,\Vert {\Im }_{\omega }\varrho \Vert \}.$$

$$\Vert p\alpha {\Vert }_{\omega }<\mathrm{N}(p{)}^{-\theta }$$

$$\Vert p\alpha {\Vert }_{\omega }\le \mathrm{N}(p{)}^{-1/8+ϵ}.$$

$$q\ne 0,\frac{a}{q}\notin \mathcal{O},\text{and}\gamma =\alpha -\frac{a}{q}\text{satisfies}|\gamma |\le \frac{C}{\mathrm{N}(q)}$$

$$x^{-1/28+ϵ}\le \delta <\frac{1}{2},$$

$$\bigg{\lvert}\sum_{\begin{subarray}{c}x/2\leq\mathrm{N}(p)<x\\ \lVert p\alpha\rVert_{\omega}<\delta\end{subarray}}1-4\delta^{2}\sum_{x/2\leq\mathrm{N}(p)<x}1\bigg{\rvert}\ll_{\epsilon}C^{2}\mathrm{N}(\omega)^{7}\delta^{2}x^{1-\epsilon},$$

$$\sum _{x/2\le \mathrm{N}(p)<x}1={\delta }_{\mathcal{O}}\frac{x}{\log x}(1+o_{\mathcal{O}}(1))\text{as }x⟶\infty ,$$

$$\Lambda =\{{\lambda }_1+{\lambda }_2\omega :{\lambda }_1,{\lambda }_2\in [0,1)\}.$$

$$\underset{R\to \infty }{\lim }\sum _{\begin{array}{c}r\in \mathcal{O}\\ \mathrm{N}(r)<R\end{array}}{d}_4(r\mathcal{O})\omega (r)\le X$$

$$\displaystyle\bigg{\lvert}\mathop{\sum\!\sum}_{\begin{subarray}{c}m,n\in\mathcal{O}\setminus\{0\}\\ \mathclap{0<\mathrm{N}(m)<M}\end{subarray}}a_{m}(w(mn)-\tilde{w}(mn))\bigg{\rvert}\leq Y,$$

$$\displaystyle\bigg{\lvert}\mathop{\sum\!\sum}_{\begin{subarray}{c}m,n\in\mathcal{O}\setminus\{0\}\\ \mathclap{x^{\mu}<\mathrm{N}(m)<x^{\mu+\kappa}}\end{subarray}}a_{m}b_{n}(w(mn)-\tilde{w}(mn))\bigg{\rvert}\leq Y.$$

$$|S(w,x^{\kappa })-S(\tilde{w},x^{\kappa })|\ll Y(\log (xX){)}^3,$$

$$S(\omega ,x^{\kappa })=\sum _{\begin{array}{c}r\in \mathcal{O}\setminus \{0\}\\ \text{prime }p\mid r\Rightarrow \mathrm{N}(p)\ge x^{\kappa }\end{array}}\omega (r),$$

$$S(\omega ,\sqrt{x})\approx \sum _{\begin{array}{c}\text{prime }p\in \mathcal{O}\\ \sqrt{x}\le \mathrm{N}(p)<x\end{array}}\omega (p).$$

$$\sum _n\sum _{m}e({\Im }_{\omega }(mn\alpha )),$$

$$\bigg{\lvert}\sum_{x\leq\mathrm{N}(m)\leq y}\operatorname{e}\lparen\Im_{\omega}(m\alpha)\rparen\bigg{\rvert}\ll\mathrm{N}(\omega)\sqrt{y}\min\left\{\sqrt{y},\frac{1}{\lVert\Re_{\omega}\alpha\rVert},\frac{1}{\lVert\Im_{\omega}\alpha\rVert}\right\}.$$

$${\omega }^2={\xi }_1+{\xi }_2\omega ({\xi }_1={\Re }_{\omega }{\omega }^2,{\xi }_2={\Im }_{\omega }{\omega }^2),$$

$${\Im }_{\omega }(m\alpha )$$

$$\mathrm{Lin}(0,y)={\sum _{m_1}\sum _{m_2}}_{0\le \mathrm{N}(m_1+m_2\omega )\le y}e(m_2({\Re }_{\omega }\alpha +{\xi }_2{\Im }_{\omega }\alpha ))e(m_1{\Im }_{\omega }\alpha )$$

$$\mathrm{Lin}(0,y)={\sum _{{\ell }_1}\sum _{m_2}}_{0\le \mathrm{N}({\ell }_1-m_2{\xi }_2+m_2\omega )\le y}e({\ell }_1{\Im }_{\omega }\alpha )e(m_2{\Re }_{\omega }\alpha ).$$

$$\bigg{\lvert}\sum_{a\leq j\leq b}\operatorname{e}\lparen j\rho\rparen\bigg{\rvert}\leq\frac{1}{2\lVert\rho\rVert}\quad(a,b,\rho\in\mathbb{R}),$$

$$|\mathrm{Lin}(0,y)|\le \frac{\#\{m_2:\exists m_1\text{ s.t. }0\le \mathrm{N}(m_1+m_2\omega )\le y\}}{2\Vert {\Im }_{\omega }\alpha \Vert },$$

$$|\mathrm{Lin}(0,y)|\le \frac{\#\{{\ell }_1:\exists m_2\text{ s.t. }0\le \mathrm{N}({\ell }_1-m_2{\xi }_2+m_2\omega )\le y\}}{2\Vert {\Re }_{\omega }\alpha \Vert }.$$

$$\mathrm{N}(m_1+m_2\omega )\ge m_2^2(\Im \omega {)}^2,$$

$$y$$

$$\ge \max \{|{\ell }_1-m_2{\xi }_2+m_2\Re \omega {|}^2,m_2^2(\Im \omega {)}^2\},$$

$$\sqrt{y}>c\sqrt{y}-|m_2|\cdot |{\xi }_2-\Re \omega |$$

$$|m_2\Im \omega |$$

Full text

The proof is based on Harman’s sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

1. Introduction

Dirichlet’s classical approximation theorem asserts that, given some real irrational $\alpha$, there are infinitely many rational integers $a,q$ ($q\neq 0$) with

[TABLE]

or—equivalently—on writing $\lVert\rho\rVert=\min_{x\in\mathbb{Z}}\lvert\rho-x\rvert$ for the distance to a nearest integer,

[TABLE]

Albeit individual values of $\alpha$ may allow for significantly sharper approximation by rational numbers, Hurwitz’s approximation theorem implies that the exponent $-1$ in Eq. 1.1 is optimal in the sense that it cannot be decreased without the resulting new inequality failing to admit infinitely many solutions for some real irrational $\alpha$ (see, e.g., [5, Theorems 193 and 194]).

A natural variation on the question about the solubility of Eq. 1.1 is to impose the additional restriction that $q$ be a rational prime and ask for which exponent $\theta$ one is able to establish that, for any real irrational $\alpha$,

[TABLE]

In this direction I. M. Vinogradov [21] obtained Eq. 1.2 with $\theta=\frac{1}{5}-\epsilon$, a result which has since then been improved by a number of researchers (see Table 1) culminating in the work of Matomäki [15] who obtained $\theta=1/3-\epsilon$. This exponent is considered to be the limit of the current technology (see the comments in [10]).

In view of the above, the first named author [1] proposed to study the analogue of Eq. 1.2 for the Gaussian integers. The approach in [1] rests upon Harman’s sieve method [6, 7, 8] and the required ‘arithmetical input’ is obtained using novel Gaussian integer analogues of classical ideas due to Vinogradov [21, Lemma 8a].

In this paper, we consider the more general problem of proving analogues of Eq. 1.2 for imaginary quadratic number fields. It turns out that, in our opinion, this setting also has the pleasant side effect of painting a clearer picture of the Diophantine arguments that underpin the aforementioned arithmetical information. Preliminary results in this direction were obtained in the second author’s doctoral dissertation [19]. A novel aspect of the present work is our additional use of smoothing directly incorporated into Harman’s sieve method.

2. Main results

Before stating our results, we shall introduce some notation which is used throughout the rest of the article. We fix some imaginary quadratic number field $\mathbb{K}$ with distinguished embedding into the complex numbers $\mathbb{C}$ by means of which we shall regard $\mathbb{K}$ as a subfield of $\mathbb{C}$. By $\mathcal{O}$ we denote the ring of integers of $\mathbb{K}$, i.e., the integral closure of $\mathbb{Z}$ in $\mathbb{K}$. As $\mathbb{K}$ is a quadratic extension of $\mathbb{Q}$, it follows from well-known results from elementary algebraic number theory that $\mathcal{O}$ is a free $\mathbb{Z}$-module of rank $2$ and there is some $\omega\in\mathcal{O}$ such that $\{1,\omega\}$ is a $\mathbb{Z}$-basis of $\mathcal{O}$. Since, by assumption, $\mathbb{K}\nsubseteq\mathbb{R}$, and $\mathbb{K}$ being the field of fractions of $\mathcal{O}$, it follows that $\Im\omega\neq 0$. In particular, $\{1,\omega\}$ turns out to be an $\mathbb{R}$-basis of $\mathbb{C}$ and, given some $\varrho\in\mathbb{C}$, we write $\Re_{\omega}\varrho$ and $\Im_{\omega}\varrho$ for the unique real numbers satisfying

[TABLE]

With this notation, we put

[TABLE]

The natural notion of ‘size’ of an element $m\in\mathcal{O}$ is furnished by its norm $\mathrm{N}(m)$, that is, the number of elements in the factor ring $\mathcal{O}/m\mathcal{O}$. It can be shown that $\mathrm{N}(m)=\lvert m\rvert^{2}$, where $\lvert m\rvert$ is the usual absolute value of $m$ considered as a complex number.

The question we ask may now be enunciated as follows:

Given some imaginary quadratic number field $\mathbb{K}\subset\mathbb{C}$ with ring of integers $\mathcal{O}$, a choice $\{1,\omega\}$ of $\mathbb{Z}$-basis of $\mathcal{O}$, and given some $\alpha\in\mathbb{C}\setminus\mathbb{K}$, for which $\theta>0$, does one have

[TABLE]

for infinitely many irreducible (or prime) elements $p\in\mathcal{O}$?

As unique factorisation underpins the sieve method we employ to tackle the above question, we are forced to restrict our considerations to only those $\mathbb{K}$ with class number $1$ (which, in this setting, is equivalent to $\mathcal{O}$ being a unique factorisation domain). The full determination of all such $\mathbb{K}$ is provided by the celebrated Baker–Heegner–Stark theorem [11, 2, 16, 17]:

Theorem** (Baker–Heegner–Stark).**

*The imaginary quadratic number fields $\mathbb{K}$ with class number $1$ are (up to isomorphism) precisely those $\mathbb{Q}(\sqrt{d})$ with $-d$ from the finite list $1$, $2$, $3$, $7$, $11$, $19$, $43$, $67$, $163$. *

Our main result states that in the above question any $\theta<1/8$ is admissible, provided that $\mathbb{K}$ has class number $1$. This is the precise analogue of Vaughan’s exponent $1/4-\epsilon$ for the classical case, obtained in [20]. Note that in the class number $1$ setting, the notions of prime and irreducible coincide.

Theorem 2.1**.**

Let $\mathbb{K}\subset\mathbb{C}$ be an imaginary quadratic number field with class number $1$ and let $\mathcal{O}$ be its ring of integers with $\mathbb{Z}$-basis $\{1,\omega\}$. Suppose that $\alpha$ is a complex number such that $\alpha\notin\mathbb{K}$. Then, for any $\epsilon>0$, there exists an infinite sequence of distinct prime elements $p\in\mathcal{O}$ such that

[TABLE]

Our approach to proving 2.1 involves counting prime elements with a certain smooth weight attached to them. On the other hand, one can also use sharp cut-offs and obtain a less fuzzy quantitative result at the cost of having to restrict to smaller values of $\theta$ in the above question. We prove the following:

Theorem 2.2**.**

Let $\mathbb{K}\subset\mathbb{C}$ be an imaginary quadratic number field with class number $1$ and let $\mathcal{O}$ be its ring of integers with $\mathbb{Z}$-basis $\{1,\omega\}$. Suppose that $\epsilon>0$ is sufficiently small. Let $\alpha$ be a complex number not contained in $\mathbb{K}$. Furthermore, suppose that one has coprime $a,q\in\mathcal{O}$ such that

[TABLE]

for some constant $C>0$ and put $x=\mathrm{N}(q)^{28/5}$. Then, for any $\delta$ such that

[TABLE]

we have

[TABLE]

where the summation variable $p$ (as throughout) only assumes prime elements of $\mathcal{O}$ and the implied constant depends on $\epsilon$ alone.

Remark**.**

(1) Instead of considering the homogeneous condition $\lVert p\alpha\rVert_{\omega}<\delta$ in the above theorem, one can also consider a shifted version, namely $\lVert p\alpha+\beta\rVert_{\omega}<\delta$, where $\beta$ is an arbitrary complex number. In fact, the authors [20, 6, 12, 7, 13] listed in Table 1 also consider the shifted analogue of Eq. 1.2, but the innovation introduced by Heath-Brown and Jia [10] has, as they remark, the defect of entailing the restriction to $\beta=0$. Regardless of this, the methods pursued by us in the present paper are perfectly capable of handling shifts $\beta$ and we merely chose not to implement this for cosmetic reasons.

(2) In his recent preprint [9] on the case $\mathbb{K}=\mathbb{Q}(i)$, Harman achieved the result in 2.2 with the exponent $7/44$, which corresponds to the exponent 7/22 in his classical result [7] mentioned above. To this end, he didn’t use a smoothing but introduced a number of novelties to overcome obstacles that are present in the non-smoothed approach. In particular, he was able to handle linear exponential sums over certain regions in $\mathbb{C}$ in an efficient way. It is likely that these novelties can be carried over to all imaginary-quadratic fields of class number 1, but we here confine ourselves to the simplest possible treatment, thus obtaining the exponent 1/28 for all fields of this kind. The way we overcome the said obstacles is to introduce a smoothing which allows us to use the Poisson summation formula conveniently. This leads us to 2.1, where we achieve the exponent 1/8 corresponding to Vaughan’s result for the classical case [20]. To achieve the larger exponent $7/44$ in full generality for the said fields, it would be required to carry over Harman’s lower bound sieve, established in [7], to them. This is a task we aim to undertake in a separate paper since the proof of the aforementioned sieve result is technically complicated and therefore requires a large amount of extra work. In particular, it involves a number of numerical calculations which are not required for the proof of the basic version which we are using here.

Still assuming $\mathbb{K}$ to have class number $1$, and appealing to Landau’s prime ideal theorem one easily deduces that

[TABLE]

where $\delta_{\mathcal{O}}>0$ is some constant only depending on $\mathcal{O}$.

Therefore, 2.2 implies 2.1 with the exponent $-1/8$ in Eq. 2.1 replaced with $-1/28$ provided one is able to verify the existence of infinitely many $a$ and $q$ as required by the theorem; however, the latter problem is already solved by Hilde Gintner [4]. In this regard, let $\Lambda$ be the fundamental parallelogram spanned by $1$ and $\omega$,

[TABLE]

Lemma** (Gintner).**

Let $\alpha$ be a complex number not contained in $\mathbb{K}$. Then there are infinitely many $a,q\in\mathcal{O}$ satisfying Eq. 2.2 with $C=\pi^{-1}\sqrt{6}\operatorname{area}\Lambda$ and $\Lambda$ given by Eq. 2.5.

Albeit the above lemma does not assert that $a,q$ be coprime, if $\mathbb{K}$ has class number $1$, then one can appeal to unique factorisation and cancel any potential non-trivial common factors from $a$ and $q$. So, indeed, one has the aforementioned relation between 2.2 and 2.1.

3. Outline of the method

3.1. The sieve method

For the detection of the prime elements in 2.1 and 2.2 we use a sieve result due to Harman—with additional smoothing and adapted to our number field setting—which has the pleasant feature of keeping our exposition reasonably tidy. The underlying sieve method itself and its various refinements are also capable of yielding lower bounds instead of asymptotic formulae, in exchange for the prospect of increasing the admissible range for $\delta$ in Eq. 2.3 (or $\theta$ in the main question), but we do not implement this here. The interested reader is referred to Harman’s exposition of his method [8]. The following special case suffices for our purposes; we write $d_{k}(\mathfrak{a})$ for the number of ways in which an ideal $\mathfrak{a}\subseteq\mathcal{O}$ can be written as a product of $k$ ideals of $\mathcal{O}$, and we write $d(\mathfrak{a})=d_{2}(\mathfrak{a})$.

Theorem 3.1** (Weighted version of Harman’s sieve for $\mathcal{O}$).**

Suppose that $\mathcal{O}$ has class number $1$ and let $x\geq 3$ be real. Let $w,\tilde{w}\colon\mathcal{O}\to[0,1]$ be two functions such that, for both $\omega=w$ and $\omega=\tilde{w}$,

[TABLE]

*for some $X\geq 1$ and assume that $\omega(r)=\omega(\tilde{r})$ for any pair of associate elements $r,\tilde{r}\in\mathcal{O}$. Suppose further that one has numbers $Y>1$, $0<\mu<1$, $0<\kappa\leq\tfrac{1}{2}$, and $M\in(x^{\mu},x)$ with the following property:

For any sequences $(a_{m})_{m\in\mathcal{O}}$, $(b_{n})_{n\in\mathcal{O}}$ of complex numbers with $\lvert a_{m}\rvert\leq 1$ and $\lvert b_{n}\rvert\leq d\lparen n\mathcal{O}\rparen$, one has*

[TABLE]

Then

[TABLE]

where

[TABLE]

and the implied constant is absolute. (Here all infinite series appearing in Eqs. 3.2, 3.3 and 3.4 are guaranteed to be absolutely convergent by Eq. 3.1.)

Remark 3.2**.**

We comment briefly on how we apply the above theorem: informally speaking, the goal is to choose the weight functions $w$ and $\tilde{w}$ in such a way that essentially only elements $r$ with $\mathrm{N}(r)<x$ contribute in the definition of $S(w,x^{\kappa})$ and $S(\tilde{w},x^{\kappa})$. Then, choosing $\kappa=\frac{1}{2}$, these $r$ are guaranteed to be prime, and, for $\omega\in\{w,\tilde{w}\}$,

[TABLE]

The choice of $w$ is made as to guarantee that $S(w,\sqrt{x})$ is essentially a known quantity by an appeal to Eq. 2.4, and $\tilde{w}$ is tailored to enforce a restriction such as $\lVert p\alpha\rVert_{\omega}\leq\delta$ as in 2.2. Finally, assuming that one proves suitably strong versions of Eq. 3.2 and Eq. 3.3, 3.1 asserts that $S(\tilde{w},\sqrt{x})$ must be of similar magnitude as $S(w,\sqrt{x})$, and, therefore, one ascertains information about the abundance of prime elements with the desired properties as encoded in the weight function $\tilde{w}$.

The proof of 3.1 is essentially identical to the usual proof of Harman’s sieve in the setting of $\mathbb{Z}$ [8, §§ 3.2–3.3]; it uses the sieve of Eratosthenes–Legendre to relate $S(w,x^{\kappa})-S(\tilde{w},x^{\kappa})$ to certain sums with more variables and applies Buchstab’s identity multiple times to produce variables in the correct ranges for Eqs. 3.2 and 3.3 to become applicable. In the process of doing so, one has to remove certain cross-conditions between summation variables—a feat which is accomplished using a variant of Perron’s formula (see 7.1 below). In our number field setting, this results in a slight complication which is not present when working over the rational integers: at some point one is faced with having to remove a condition of the type $\mathrm{N}(p)\leq\mathrm{N}(\tilde{p})$ from a double sum with summation variables $p$ and $\tilde{p}$ assuming only non-associate prime elements as values (see the arguments around Eq. 7.12 below). However, this problem can be overcome. For the convenience of the reader we provide a detailed proof of 3.1 in Section 7 below.

3.2. Outline of the rest of the paper

Apart from proving 3.1 in Section 7, we proceed as follows: by the outline given in 3.2, the bulk of the remaining work lies in the verification of Eq. 3.2 and Eq. 3.3 for the two choices of $w$ and $\tilde{w}$ that we use below (non-smoothed and smoothed). Both arguments ultimately hinge on distribution results related to the sequence $\lVert n\alpha\rVert_{\omega}$ ($n\in\mathcal{O}$). We devote our attentions to establishing such results first. This is done in Section 4, with its principal results being stated in 4.3 and 4.4.

Section 5 is devoted to proving 2.2, with this goal being achieved in Section 5.7. The results concerning Eq. 3.2 and Eq. 3.3 are recorded in 5.4 and 5.2 respectively.

In Section 6 we undertake proving 2.1. The analogues of the aforementioned propositions are 6.6 and 6.7 and their proof is largely parallel to the proof of their non-smoothed counterparts. The main innovation here is contained in 6.4, which takes advantage of the smooth weights by means of Poisson’s summation formula.

4. Exponential sum estimates

In Section 5 we need estimates for sums of the shape

[TABLE]

where the summation over $m\in\mathcal{O}$ is restricted to some annulus $x(n)\leq\mathrm{N}(m)\leq y(n)$ with bounds $x(n)$ and $y(n)$ depending on $n\in\mathcal{O}$. In Section 4.2 we estimate the inner summation and in Section 4.3 we deal with the additional summation over $n$. The corresponding proof is then carried out in the subsequent sections. Moreover, using the tools developed in Section 4.4, we also establish a closely related result which is useful in Section 6 (see 4.4).

In what follows, we sometimes have expressions like $1/\lVert\Re_{\omega}\alpha\rVert$; if a division by zero occurs there, then the result is understood to mean $+\infty$.

4.1. Some facts about quadratic extensions

Before being able to tackle the problem outlined above, we take the opportunity to record here some basic facts about quadratic extensions which we use throughout.

Lemma 4.1**.**

Let $\mathbb{K}\subset\mathbb{C}$ be an imaginary quadratic number field with ring of integers $\mathcal{O}$ and generator $\omega$ of $\mathcal{O}$, that is, $\mathcal{O}=\mathbb{Z}[\omega]$. Then the following statements hold:

1. (1)

The number of units in $\mathcal{O}$ is bounded by six. 2. (2)

$\lvert\Im\omega\rvert\geq\sqrt{3}/2$. 3. (3)

$2\Re\omega$* is an integer.* 4. (4)

For $y\geq 1$, $\#\{n\in\mathcal{O}:\mathrm{N}(n)\leq y\}\ll y$, where the implied constant is absolute.

Proof.

The first assertion may be found in [5].

For the second assertion just observe that there is some negative square-free rational integer $d$ such that $\mathbb{K}=\mathbb{Q}(\sqrt{d})$. Letting $D=4d$ if $d\not\equiv 1\bmod 4$, and $D=d$ otherwise, we have $\mathcal{O}=\mathbb{Z}[\tilde{\omega}]$, where $\tilde{\omega}=\frac{1}{2}(D+\sqrt{D})$. An elementary calculation shows that $\omega$ is of the form $k\pm{\tilde{\omega}}$ for some rational integer $k$. This already proves Item 3. Moreover, from this and a short computation one immediately obtains Item 2 (with equality being attained for $d=-3$).

Concerning the last assertion, we note that the quantity bounded therein, $\#\{(n_{1},n_{2})\in\mathbb{Z}^{2}:\lvert n_{1}+n_{2}\omega\rvert^{2}\leq y\}$, counts points inside some ellipse. Elementary arguments suffice to show that this is asymptotically equal to $\pi y/\lvert\Im\omega\rvert$ and the uniform lower bound for $1/\lvert\Im\omega\rvert$ furnished by the second assertion finishes the proof. ∎

4.2. Basic estimates for linear exponential sums

Lemma 4.2**.**

Let $\alpha$ be a complex number and suppose that one has numbers $x,y$ such that $0\leq x\leq y$. Then

[TABLE]

Proof.

We may assume $y\geq 1$, for Eq. 4.1 is trivial otherwise. We denote the sum on the left hand side of Eq. 4.1 by $\operatorname{Lin}(x,y)$. On writing

[TABLE]

$m=m_{1}+m_{2}\omega$ and $\ell_{1}=m_{1}+m_{2}\xi_{2}$, we obtain the following two expressions for $\Im_{\omega}\lparen m\alpha\rparen$:

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

So, on recalling the well-known bound

[TABLE]

and using the triangle inequality on the outer summations in Eq. 4.3 and Eq. 4.4, we obtain

[TABLE]

The numerators here are bounded easily: indeed, since

[TABLE]

using $y\geq 1$ and 4.1 Item 2, one easily bounds the numerator in Eq. 4.5 by $4\sqrt{y}$. On the other hand for $\lvert\ell_{1}\rvert>c\sqrt{y}$ with $c=\lparen 2+\frac{2}{\sqrt{3}}\rparen\mathrm{N}(\omega)$ there is no $m_{2}$ such that

[TABLE]

for otherwise it would follow that

[TABLE]

so that $\lvert m_{2}\rvert\cdot\lvert\xi_{2}-\Re\omega\rvert>\lparen c-1\rparen\sqrt{y}$, but then

[TABLE]

in contradiction to Eq. 4.7. Hence, the numerator in Eq. 4.6 is bounded by $2c\sqrt{y}+1\leq 8\lvert\omega\rvert^{2}\sqrt{y}$. Thus,

[TABLE]

and, together with the trivial bound $\lvert\operatorname{Lin}(x,y)\rvert\ll y$ from 4.1 Item 4, this is clearly satisfactory to establish Eq. 4.1 for $x=0$, and the case $x>0$ follows from this bound and

[TABLE]

4.3. Distribution of fractional parts

In view of 4.2 and recalling the goal stated at the beginning of Section 4, we are faced with the problem of estimating sums of the shape

[TABLE]

where $M\geq 2$, $\mathscr{X}$ is some subset of $\{n\in\mathcal{O}:1\leq\mathrm{N}(n)<x\}$ with $x\geq 1$, and

[TABLE]

The usual attack against such a problem is to replace $\alpha$ by some Diophantine approximation $a/q$. Subsequently, after bounding the error introduced from the approximation, one is able to control averages of $E(n,M)$ with $n$ constrained to boxes (say) not too large in terms of $\lvert q\rvert$. By splitting the full range of $n$ in Eq. 4.8 into such boxes, one derives a bound for Eq. 4.8 of the shape seen in 4.3 below.

Theorem 4.3**.**

Let $\mathscr{X}$ be a subset of all $n\in\mathcal{O}$ with $1\leq\mathrm{N}(n)<x$ and suppose that one has coprime $a,q\in\mathcal{O}$ satisfying Eq. 2.2. Put

[TABLE]

Then, assuming $M\geq 2$,

[TABLE]

Furthermore, if $x\leq\mathrm{N}(q)/\lparen 12C\mathrm{N}(\omega)\rparen^{2}$, then

[TABLE]

The proof of this result follows the outline given above and is undertaken in the next two subsections. Similarly, we also obtain the following result:

Theorem 4.4**.**

Suppose that one has coprime $a,q\in\mathcal{O}$ satisfying Eq. 2.2. For $0<\Delta\leq\tfrac{1}{2}$ let

[TABLE]

Then

[TABLE]

Moreover, $H_{\alpha}(x,\Delta)$ vanishes if $\Delta<1/(4\lvert q\omega\rvert)$ and $x\leq\lvert q\rvert^{2}/(12C\lvert\omega\rvert^{2})^{2}$.

4.4. Diophantine lemmas

As a first step, we show that

[TABLE]

cannot be too small if $\alpha\in\mathbb{K}$ is not an algebraic integer. The results in this section are probably already known in one form or another. However, we were unable to find a suitable reference and, therefore, provide full proofs for the reader’s convenience.

Lemma 4.5**.**

*For non-zero $a,q\in\mathcal{O}$ such that $\alpha=a/q\notin\mathcal{O}$, it holds that $\lVert\alpha\rVert_{\omega}\geq 1/(2\lvert q\omega\rvert)$. *

Proof.

Pick $m=m_{1}+m_{2}\omega\in\mathcal{O}$ such that

[TABLE]

Now certainly it holds that

[TABLE]

Therefore, to prove the lemma, it suffices to give a suitable lower bound for $\lvert\alpha-m\rvert$, which, upon noting that $\alpha\notin\mathcal{O}$, is quite easy:

[TABLE]

Next, we intend to derive a result similar to 4.5, when $\alpha$ is slightly perturbed:

Lemma 4.6**.**

*Let $\alpha$ be a complex number and $a,q$ be such that Eq. 2.2 holds. Furthermore, suppose that $n\in\mathcal{O}$ satisfies $\lvert n\rvert\leq\lvert q\rvert/(12C\lvert\omega\rvert^{2})$ and $na$ be indivisible by $q$. Then $\lVert n\alpha\rVert_{\omega}\geq 1/(4\lvert q\omega\rvert)$. *

Proof.

First, we separate the perturbation $\gamma$ from the rest: we have

[TABLE]

and the same holds when one replaces $\Im_{\omega}$ by $\Re_{\omega}$. The last term therein is bounded easily: using $\lvert\xi\rvert\geq\lvert\Im_{\omega}\xi\rvert\cdot\lvert\Im\omega\rvert$, 4.1 Item 2 and writing $N=C\lvert n\rvert/\lvert q\rvert^{2}$ for the moment, we have

[TABLE]

A similar calculation also bounds the corresponding $\Re_{\omega}$-term:

[TABLE]

Thus, using 4.5,

[TABLE]

Now, by assumption, the term in the parentheses is $\geq(4\lvert\omega\rvert)^{-1}$, and the assertion of the lemma follows. ∎

4.5. Proof of 4.3

Assume the hypotheses of 4.3 and let $n$ and $\tilde{n}$ be two distinct algebraic integers in $\mathcal{O}$ which coincide modulo $q$. Then there is some non-zero $m\in\mathcal{O}$ such that $n-\tilde{n}=mq$ and, hence, $\lvert n-\tilde{n}\rvert=\lvert m\rvert\lvert q\rvert\geq\lvert q\rvert$. Assuming $a$ and $q$ to be coprime and $\lvert n-\tilde{n}\rvert<\lvert q\rvert$, we conclude that $(n-\tilde{n})a$ is divisible by $q$ if and only if $n=\tilde{n}$. Consequently, if $\mathscr{R}\subseteq\mathbb{C}$ is some set with

[TABLE]

then, according to 4.6, any two distinct points $n\alpha$, $\tilde{n}\alpha$ ($n,\tilde{n}\in\mathscr{X}\cap\mathscr{R}$) satisfy the spacing condition

[TABLE]

Therefore, for $0<\Delta_{1},\Delta_{2}\leq\frac{1}{2}$, the sum

[TABLE]

is bounded by

[TABLE]

which in turn is bounded by four times the maximum number of points of pairwise maximum norm distance $\geq(4\lvert\omega\rvert\lvert q\rvert)^{-1}$ that can be put in a rectangle with side lengths $\Delta_{1}$ and $\Delta_{2}$, i.e.,

[TABLE]

Moving on, let $L\in\mathbb{N}$ be a parameter at our disposal. Then the sum

[TABLE]

with $E(n,M)$ as defined in Eq. 4.9 admits a decomposition

[TABLE]

By Eq. 4.13 and using $(a+b)^{2}\leq 2a^{2}+2b^{2}$ ($a,b\geq 1$),

[TABLE]

Moreover, using $\min\{2^{k_{1}},2^{k_{2}}\}\leq\sqrt{2^{k_{1}+k_{2}}}$ and Eq. 4.13,

[TABLE]

Similarly,

[TABLE]

Assuming $M\geq 2$, we take $L=\lceil\frac{1}{2\log 2}\log(2M)\rceil$ to obtain

[TABLE]

Additionally, if $x\leq\lvert q\rvert^{2}/\lparen 12C\lvert\omega\rvert^{2}\rparen^{2}$, then we take $L=\lceil\frac{1}{\log 2}\log\lparen 4\lvert q\omega\rvert\rparen\rceil$. In this case 4.6 shows that $S_{1}(\mathscr{R})$ vanishes and, consequently, we have

[TABLE]

Finally, we note that the set $\mathscr{X}$ can be covered by fewer than

[TABLE]

squares $\mathscr{R}$ with diameter Eq. 4.12. Together with Eq. 4.14 this proves Eq. 4.10, and together with Eq. 4.15 we obtain Eq. 4.11. This proves 4.3.

Proof of 4.4.

The assertion concerning the vanishing of $H_{\alpha}(x,\Delta)$ is contained in 4.6. As for the bound for $H_{\alpha}(x,\Delta)$, cover the set $\mathscr{X}=\{n\in\mathcal{O}:0<\mathrm{N}(n)\leq x\}$ with rectangles as above and employ Eq. 4.13. ∎

5. The non-smoothed version

Here we tackle the problem of verifying the assumptions of 3.1 in a setting suitable for proving 2.2. Throughout, we assume the hypotheses of 2.2, although $x\geq 3$ may be considered arbitrary until Section 5.7, where we take $x=\mathrm{N}(q)^{28/5}$.

5.1. Setting up linear and bilinear forms

Let $\mathscr{B}=\{n\in\mathcal{O}:x/2\leq\mathrm{N}(n)<x\}$ and $\mathscr{A}=\{n\in\mathscr{B}:\lVert n\alpha\rVert_{\omega}<\delta\}$. Concerning 3.1, we choose $\tilde{w}$ to be $\boldsymbol{1}_{\mathscr{A}}$, the characteristic function of the set $\mathscr{A}$, and $w=4\delta^{2}\boldsymbol{1}_{\mathscr{B}}$. Given these definitions, the limit in (3.1) is actually attained for every $R\geq x$ and trivial estimates suffice to show that $X$ therein may be taken $\ll x^{5}$.111Of course, for such divisor sums much better estimates are available (see, e.g., 5.3 below for $d_{4}$ replaced with $d_{2}$). However, since in 3.1, only the logarithm of $X$ enters in the final error term, we can be very sloppy here.

Moving on, we shall want to compare sums of the type

[TABLE]

where the summation indices $m,n$ vary through $\mathcal{O}$ and the coefficient sequences $(a_{m})_{m}$ and $(b_{n})_{n}$ consist of complex numbers and satisfy $\lvert a_{n}\rvert\leq 1$ and $\lvert b_{n}\rvert\leq d(n\mathcal{O})$.

To be more specific, for parameters $\mu>0$ and $0<\kappa\leq\frac{1}{2}$, there are two types of sums we would like to estimate

• •

Type I: $b_{n}=1$ in the above and $(a_{m})_{m}$ is supported only on $m$ with $0<\mathrm{N}(m)<M$ for some $M$ with $x^{\mu}<M\leq x$ (see Eq. 3.2).

• •

Type II: $(a_{m})_{m}$ is supported only on $m$ with $x^{\mu}\leq\mathrm{N}(m)<x^{\mu+\kappa}$ (see Eq. 3.3).

Each type requires a different treatment, but for now it is convenient to start by transforming Eq. 5.1 without restricting to either of the above types. We start with the following result which furnishes a finite Fourier approximation to the saw-tooth function $\psi$ given by

[TABLE]

Lemma 5.1**.**

For all real $x$ and $J\geq 1$, we have

[TABLE]

Proof.

This is Lemma 4.1.2 in [3]. ∎

We now derive a useful expansion of the characteristic function $\boldsymbol{1}_{\mathscr{A}}$ of $\mathscr{A}$ evaluated at algebraic integers. For an element $y$ of $\mathcal{O}$ we put

[TABLE]

Furthermore, let

[TABLE]

We now consider the characteristic function $\boldsymbol{1}_{\mathscr{A}}\colon\mathcal{O}\to\{0,1\}$ of the set $\mathscr{A}$. For any $y\in\mathscr{B}\setminus\mathcal{O}_{\delta,\alpha}$ we have the expansion

[TABLE]

Note here that the first equality in Eq. 5.2 may not hold for $y\in\mathcal{O}_{\delta,\alpha}$. Nevertheless, the last line in Eq. 5.2 remains bounded even in that case. Therefore one can, as we do below, also use the last line of Eq. 5.2 as a substitute for $\boldsymbol{1}_{\mathscr{A}}(y)$ even when $y\in\mathcal{O}_{\delta,\alpha}$. This only introduces an error bounded by a constant times for how many $y\in\mathcal{O}_{\delta,\alpha}$ is this applied.

For $k=1,2,3$ we consider the sums

[TABLE]

For $k=1,2$, on applying 5.1 with some $J\geq 1$ to be specified later (see Eq. 5.26 below), for any choice of summation ranges for $m,n$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

Similarly, we obtain

[TABLE]

where we have used the trivial estimates

[TABLE]

and

[TABLE]

Now consider

[TABLE]

where the star in the summation indicates that the range of $m$ is to be restricted to a Type I or Type II range. The first sum may be written as

[TABLE]

and we can apply Eq. 5.2 to all those terms where $mn\notin\mathcal{O}_{\delta,\alpha}$. On the other hand, we may use the last line of Eq. 5.2 as a substitute for $\boldsymbol{1}_{\mathscr{A}}(mn)$ for all terms $mn$ in the above at the cost of an error $O(G)$ (see the comment just below Eq. 5.2). Then, combining this with our analysis of the sums $\varSigma_{k}$ from above, we find that

[TABLE]

5.2. Removing the weights: dyadic intervals

Here we shall remove the weights Eq. 5.3 attached to the sums in Eq. 5.6. This may be achieved by splitting the summation over $j$ (or $j_{1}$, $j_{2}$) into dyadic intervals: indeed, for any non-negative $f:\mathbb{Z}^{2}\to\mathbb{R}$, letting

[TABLE]

we find that

[TABLE]

Of course, we shall apply this with

[TABLE]

Now assume for the moment that we have bounds

[TABLE]

where the right-hand side is symmetric in both arguments and does not depend on the particular choice of the coefficients in Eq. 5.8 (but, of course, still subject to the Type I/II conditions presented in Section 5.1); the reader may wish to glance at 5.2 and 5.4 below, where we furnish such bounds for the Type II and Type I sums respectively.

Then, using Eq. 5.6 we have

[TABLE]

We shall return to this in Section 5.7 and now focus on establishing the aforementioned bounds of the shape Eq. 5.9.

5.3. Transforming the argument in the exponential term

In the proof of the bounds for the Type I and Type II sums we need to combine variables in $\mathcal{O}$ (see Sections 5.4 and 5.5 below). Having this goal in mind, the shape of the argument of the exponential in Eq. 5.8 appears to be, at a superficial glance, a technical obstruction.

However, this putative problem vanishes after a simple variable transformation that we shall now describe: by definition of $x_{k,mn}$,

[TABLE]

Letting $\xi_{2}$ be given as in Eq. 4.2 and writing $\ell=\ell_{1}+\ell_{2}\omega$, a short computation yields

[TABLE]

Then, via the equivalence

[TABLE]

and assuming $(j_{1},j_{2})\neq(0,0)$, we observe that Eq. 5.11 equals $\Im_{\omega}\lparen\ell mn\alpha\rparen$ when $(\ell_{1},\ell_{2})$ is calculated via the above formula. We let $\mathscr{L}(J_{1},J_{2})$ be the set of algebraic integers $\ell\in\mathcal{O}$ arising from $(j_{1},j_{2})$ via the above formula, that is, $\mathscr{L}\lparen J_{1},J_{2}\rparen$ is the set

[TABLE]

Consequently, if $F(J_{1},J_{2})$ is given by Eq. 5.7 with $f$ given by Eq. 5.8, then

[TABLE]

For a later extension of the summation over $\ell$, we note that, using 4.1 Item 2, $\mathscr{L}(J_{1},J_{2})$ can be seen to be contained in the set of all $\ell$ satisfying

[TABLE]

The reader will note that this set potentially contains many more elements than $\mathscr{L}\lparen J_{1},J_{2}\rparen$, for we obviously have

[TABLE]

In any case, we require both Eq. 5.14 and Eq. 5.15.

5.4. The Type II sums

In this section, we establish the following.

Proposition 5.2** (Type II bound).**

Consider $F$ from Eq. 5.7 with $f$ given by Eq. 5.8 subject to

[TABLE]

where $\mu\in(0,1]$, $\kappa\in(0,\frac{1}{2}]$ and $x\geq 3$. For the coefficients in Eq. 5.8 assume that $\lvert a_{m}\rvert\leq 1$ and $\lvert b_{n}\rvert\leq d(n\mathcal{O})$. Moreover, suppose that $a$, $q$, $\gamma$ and $C$ are as in Eq. 2.2. Then, for any $\epsilon\in(0,\frac{1}{2}]$,

[TABLE]

In the course of the proof of 5.2 and at other places we need the following lemma to control trivial sums over $m$ and $n$.

Lemma 5.3**.**

Let $\mathbb{K}$ be a fixed quadratic number field and $\mathcal{O}$ its ring of integers. For an ideal $\mathfrak{a}\subseteq\mathcal{O}$ let $d(\mathfrak{a})$ denote the number of ideals $\mathfrak{b}\supseteq\mathfrak{a}$, and fix $\epsilon>0$ and some integer $\ell\geq 2$. Then, for $x\geq 2$,

1. (1)

$\displaystyle\sum_{N\mathfrak{a}\leq x}d(\mathfrak{a})^{\ell}\ll_{\mathcal{O},\ell}x\lparen\log x\rparen^{2^{2\ell}-1}$, 2. (2)

$d(\mathfrak{a})\ll_{\mathcal{O},\epsilon}(N\mathfrak{a})^{\epsilon}$,

where the implied constants depend at most on $\mathcal{O}$, $\ell$ and $\epsilon$.

Proof.

The first assertion is a direct consequence of [14]. On the other hand, the second assertion is immediate from the first. ∎

Using 4.1 Item 1 and 5.3 Item 1, we have

[TABLE]

(Note that here the dependence on $\mathcal{O}$ in 5.3 can be neglected, as we are dealing only with the finitely many imaginary quadratic number fields $\mathbb{K}$ with class number $1$.)

Proof of 5.2.

Looking at Eq. 5.13, we may split the summation over $m$ into ‘dyadic annuli,’ getting

[TABLE]

where, upon employing the transformation described in Section 5.3 along the way, $\hat{F}=F(J_{1},J_{2},K,K^{\prime})$ may be taken to be

[TABLE]

(Here and in the following we are always assuming $J_{1}$, $J_{2}$, $K$ and $K^{\prime}$ to be positive integers such that $K\leq K^{\prime}<2K$.) By Eq. 5.15 and 4.1 Item 4,

[TABLE]

Hence, letting

[TABLE]

Cauchy’s inequality gives

[TABLE]

which, upon expanding the square and rearranging, yields

[TABLE]

where ${\sideset{}{{}_{m}^{*}}{\sum}}$ restricts the summation to those $m$ with

[TABLE]

Next, we isolate the ‘diagonal contribution’ $\Delta$, that is, those terms where $n=\tilde{n}$, for in this case the sum over $m$ can only be bounded trivially. Using 5.3 Item 1, Eq. 5.15 and 4.1 Item 4, this is found to be

[TABLE]

Moreover, using Eq. 5.14, 5.3, Item 2, 4.2 and Eq. 5.14, we have

[TABLE]

where

[TABLE]

and $E$ is given by Eq. 4.9. Thus, using Eq. 5.19 and 4.3, and recalling Eq. 5.18,

[TABLE]

Upon taking the square root, and simplifying the resulting expressions,

[TABLE]

Recalling Eq. 5.17, we infer Eq. 5.16 after adjusting $\epsilon$. ∎

5.5. The Type I sums

The next step is to estimate the Type I sums. We establish the following.

Proposition 5.4** (Type I bound).**

Consider $F$ from Eq. 5.7 with $f$ given by Eq. 5.8 subject to

[TABLE]

where $M\leq x$ and $x\geq 3$. For the coefficients in Eq. 5.8 assume that $\lvert a_{m}\rvert\leq 1$ and $b_{n}=\boldsymbol{1}_{\{1\leq\mathrm{N}(n)<x\}}$. Moreover, suppose that $a$, $q$, $\gamma$ and $C$ are as in Eq. 2.2. Then, for any $\epsilon\in\lparen 0,\frac{1}{2}]$,

[TABLE]

Proof.

As we did with the Type II sums in the proof of 5.2, we may split the summation over $m$ into dyadic annuli, getting

[TABLE]

where $\tilde{F}=\tilde{F}(J_{1},J_{2},K,K^{\prime})$ is given by

[TABLE]

Letting $R=\sqrt{x/K}$ and employing 4.2 as well as 5.3 Item 2, we infer

[TABLE]

with $E$ given by Eq. 4.9 and $U$ by Eq. 5.20 with $K^{\prime}$ in place of $x/K$. 4.3 now shows that

[TABLE]

Herein, for very small $K$, the term $xK^{-1}$ becomes problematic. To circumvent this, we note that 4.3 also furnishes the bound

[TABLE]

provided that

[TABLE]

On the other hand, if Eq. 5.23 fails to hold, then, recalling Eq. 5.20, we have

[TABLE]

Therefore, after joining both bounds,

[TABLE]

Upon plugging this into Eq. 5.22, we obtain Eq. 5.21 after adjusting $\epsilon$. ∎

5.6. Estimation of $G$

The final task is to bound the error term $G$, defined in Eq. 5.4. We shall establish the following.

Proposition 5.5**.**

Consider $G$ from Eq. 5.4, and suppose that $a$, $q$, $\gamma$ and $C$ are as in Eq. 2.2. Then we have

[TABLE]

Proof.

Using the definition of $x_{k,mn}$, writing $r=mn$ and using 5.3, we obtain

[TABLE]

We shall bound

[TABLE]

and treat the remaining three sums of this type similarly. To this end, similarly as in Section 4.5, we cover the set of $r$’s in question,

[TABLE]

by $O(1+C^{2}N(\omega)^{2}x/N(q))$ many rectangles $\mathscr{R}$ with diameter satisfying Eq. 4.12 (see also Eq. 4.16), so that

[TABLE]

Furthermore, we write

[TABLE]

Similarly as in Section 4.5 (see Eq. 4.13), we establish that

[TABLE]

It follows that

[TABLE]

Treating the remaining three sums of this type similarly, we obtain Eq. 5.24 after adjusting $\epsilon$. ∎

5.7. Assembling the parts

Finally, we are in a position to use Eq. 5.10. Assume the hypotheses of 5.2. Recall Eq. 5.3. Set

[TABLE]

the term in the brackets on the right-hand side of Eq. 5.24. Then, looking at Eq. 5.16, we use Eq. 5.10 and Eq. 5.24 together with the inequalities

[TABLE]

and

[TABLE]

if $H,H_{1},H_{2}\leq J$, to bound the error in the Type II sums (see Eq. 5.5) as

[TABLE]

$\epsilon$ being sufficiently small.

Moving on to the Type I sums, accordingly assuming the hypotheses of 5.4 and looking at Eq. 5.21, we use Eq. 5.10 and Eq. 5.24 together with the inequalities

[TABLE]

to infer the estimate

[TABLE]

for the error in the Type I sums.

On recalling Eq. 5.5 and plugging the above bounds into 3.1, we find that the error

[TABLE]

satisfies the bound

[TABLE]

Evidently, this bound is increasing with $\kappa$ and to detect primes, we must take $\kappa=\frac{1}{2}$. In view of Eq. 2.4, we shall aim for a bound of the type

[TABLE]

with $\delta$ in some range (w.r.t. $x$) as large as possible. With this constraint in mind, and given $q$, we take $x=\mathrm{N}(q)^{28/5}$ (as was stated in 2.2) so that $\mathrm{N}(q)=x^{5/28}$ and, moreover,

[TABLE]

Then, under the additional assumption that $J\leq x$, we obtain

[TABLE]

provided $\epsilon$ is sufficiently small. This implies that Eq. 5.25 holds for sufficiently small $\epsilon$ and

[TABLE]

which concludes the proof of 2.2 after adjusting $\epsilon$.

6. The smoothed version

Here, in a similar vein to Section 5, we work on providing the details for what was outlined in 3.2. However, this time the aim is to prove 2.1.

6.1. The modified setup

Throughout the rest of Section 6 we make the following assumptions: $\epsilon>0$ is supposed to be sufficiently small and fixed. $\mathbb{K}$ is an imaginary quadratic number field with class number $1$. The number $x\geq 3$ is assumed to be sufficiently large and $\alpha,a,q,C$ are as in 2.2. Moreover, we suppose that

[TABLE]

The exact lower bound here is of no particular consequence, as our final results even fall short of being non-trivial for $\delta\leq x^{-1/16}$. However, in the course of getting there, we need to have bounds of the shape $\delta^{-1}e^{-x^{\epsilon}}\ll_{A}x^{-A}$ for any $A>0$ as $x\to\infty$. Such bounds are used—often tacitly—throughout.

Furthermore, we write

[TABLE]

and define the weight function $w$ to be used in conjunction with 3.1 by

[TABLE]

To define $\tilde{w}$, we let

[TABLE]

which by Poisson summation formula implies

[TABLE]

Then let

[TABLE]

with $\xi_{2}$ from Eq. 4.2.

6.2. Removing the weights

Our next immediate goal is to see that $w$ and $\tilde{w}$ are actually suitable weights for the type of argument outlined in 3.2. This is contained in 6.3 below, but first we need two lemmas. We use the notation $S(\,\cdot\,,\sqrt{x})$ from Eq. 3.4.

Lemma 6.1**.**

$\displaystyle S(w,\sqrt{x})\gg_{\epsilon}\delta^{2}\frac{N}{\log N}$.

Proof.

The claim follows at once from

[TABLE]

after applying the prime number theorem for $\mathcal{O}$ (see Eq. 2.4). ∎

Lemma 6.2**.**

$\displaystyle S(\tilde{w},\sqrt{x})\ll_{\epsilon,\omega}\sum_{\begin{subarray}{c}\text{\rm prime }p\in\mathcal{O}\\ \mathrm{N}(p)<x\\ \lVert p\alpha\rVert_{\omega}<\delta x^{\epsilon}\end{subarray}}e^{-\pi\mathrm{N}(p)/N}+1$.

Proof.

We may split up $S(\tilde{w},\sqrt{x})$ as follows:

[TABLE]

where the terms being summed are

[TABLE]

and

[TABLE]

The first sum is bounded using the trivial estimate $W_{\delta}(\,\cdot\,)\ll 1$.

Now if $\lVert p\alpha\rVert_{\omega}\geq\delta x^{\epsilon}$, then $\lVert\Im_{\omega}(p\alpha)\rVert\geq\delta x^{\epsilon}/2\xi_{2}$, or $\lVert\Im_{\omega}(p\alpha)\rVert\leq\delta x^{\epsilon}/2\xi_{2}$ and $\lVert\Re_{\omega}(p\alpha)\rVert\geq\delta x^{\epsilon}$. In the first case, using the inequality

[TABLE]

we have

[TABLE]

where for the last estimate we employ Eq. 6.1. In the second case, the assumptions ensure that $\lVert\Re_{\omega}(p\alpha)+\xi_{2}\Im_{\omega}(p\alpha)\rVert\geq\delta x^{\epsilon}/2$. Therefore, by arguing as before, we have

[TABLE]

Thus, altogether we have

[TABLE]

This proves the lemma. ∎

Corollary 6.3**.**

Still assuming the hypotheses from Section 6.1, suppose that one knows that

[TABLE]

where $S(\,\cdot\,,\sqrt{x})$ is defined as in Eq. 3.4. Then

[TABLE]

In particular, for any sufficiently large $x$, there is a prime element $p\in\mathcal{O}$ such that $\mathrm{N}(p)<x$ and $\lVert p\alpha\rVert_{\omega}<\delta x^{\epsilon}$.

6.3. Estimation of smoothed sums

The next result is a smoothed analogue of 4.2. Later, this is used in combination with 4.4. (The reader may contrast this with our use of 4.3 as the underlying tool for proving 5.4 and 5.2.)

Lemma 6.4**.**

Let $R\geq 1$ and $\gamma\in\mathbb{R}$. Then, for every $\epsilon>0$ and $f_{R}$ defined as in Eq. 6.2,

[TABLE]

Proof.

Let $A_{\omega}$ be the invertible $(2\times 2)$-matrix $\left\lparen\!\begin{smallmatrix}1&\Re\omega\\ 0&\Im\omega\end{smallmatrix}\!\right\rparen$ and write $A_{\omega}^{-\top}=(A_{\omega}^{-1})^{\top}$. Moreover, recall that $\omega^{2}=\xi_{1}+\xi_{2}\omega$ and let $v_{\vartheta}=(\Im_{\omega}\vartheta,\xi_{2}\Im_{\omega}\vartheta+\Re_{\omega}\vartheta)$. Then, writing $g(u,v)=e^{-\pi(u^{2}+v^{2})}$, we have

[TABLE]

By the Poisson summation formula (see, e.g., [18]) and a change of variables,

[TABLE]

Using the fact that $g$ is its own Fourier transform, we have

[TABLE]

A quick computation shows that

[TABLE]

Consequently, after a linear change of summation variables,

[TABLE]

where

[TABLE]

As the bound $\Sigma\ll R$ is trivial, we now assume that

[TABLE]

In particular, we have $R\geq 4x^{2\epsilon}$. To bound $\Sigma$ we split off the term for $k_{1}=k_{2}=0$, namely

[TABLE]

from the rest, that is,

[TABLE]

Using Eq. 6.3, we have

[TABLE]

To bound $\Sigma_{0}$, we put $r=\max\{1,2\lvert\Re\omega\rvert\}$. By assumption, $\lVert\vartheta\rVert_{\omega}\geq x^{\epsilon}/\sqrt{R}$, so that we either have $\lVert\Im_{\omega}\vartheta\rVert\geq x^{\epsilon}/(r\sqrt{R})$, or $\lVert\Im_{\omega}\vartheta\rVert<x^{\epsilon}/(r\sqrt{R})$ and $\lVert\Re_{\omega}\vartheta\rVert\geq x^{\epsilon}/\sqrt{R}$. In the former case,

[TABLE]

In the latter case, we first note that $2\Re\omega$ is an integer (see 4.1 Item 3), so that

[TABLE]

and, hence,

[TABLE]

Recalling that the original sum under consideration, $\Sigma$, is $\Sigma_{0}+\Sigma_{*}$, the assertion of the lemma follows. ∎

6.4. Cutting off

Lemma 6.5**.**

Consider the sum

[TABLE]

where $a_{m}$ and $b_{n}$ are arbitrary complex coefficients satisfying $\lvert a_{m}\rvert\leq 1$, $a_{0}=b_{0}=0$, $\lvert b_{n}\rvert\leq\#\{\text{divisors of }n\}$, and $\sum_{m}^{*}$ means some arbitrary restriction on the summation over $m\in\mathcal{O}$. Then, for every $A,\epsilon>0$,

[TABLE]

Proof.

Recalling Eq. 5.12, we have

[TABLE]

Consequently,

[TABLE]

The inner-most sum over $n$ is bounded by $e^{-\pi\mathrm{N}(m)/N}$ multiplied by

[TABLE]

Thus,

[TABLE]

A short computation shows that

[TABLE]

where $c=(1+\lvert\Re\omega\rvert)/\lvert\Im\omega\rvert$. Consequently,

[TABLE]

Hence, we obtain the claimed result from Eq. 6.4 after estimating the contribution from all terms with $\mathrm{N}(j)>\delta^{-2}x^{\epsilon}$ via the above and using the trivial inequality $e^{-\pi\delta^{2}(j_{1}^{2}+j_{2}^{2})}\leq 1$ on the remaining terms. ∎

6.5. Type I estimates

Proposition 6.6** (Type I estimate).**

Consider the sum $\Sigma$ from 6.5 with $b_{n}=1$ for all $n\in\mathcal{O}\setminus\{0\}$ and

[TABLE]

for some positive $M$. Then

[TABLE]

Proof.

By 6.5 we have

[TABLE]

Therefore, by 6.4,

[TABLE]

Moreover, by the second part of 4.4, we find that no terms with $\mathrm{N}(m)<L_{0}$ contribute to the above sum if we set

[TABLE]

Then, upon splitting the summation over $m$ into dyadic annuli, we obtain

[TABLE]

Finally, using 4.4 and $N=x^{1-\varepsilon}$,

[TABLE]

6.6. Type II estimates

Proposition 6.7** (Type II estimate).**

Consider the sum $\Sigma$ from 6.5 with

[TABLE]

for some $\mu,\kappa\in(0,1)$. Then, for any $\epsilon\in(0,\mu)$,

[TABLE]

Proof.

By 6.5 we have

[TABLE]

Upon splitting the summation over $m$ into dyadic annuli, we obtain

[TABLE]

By similar arguments as in Section 6.2, we see that one can restrict the summation over $n$ to $\mathrm{N}(n)\leq x/L$ at the cost of an error $\ll_{A,\epsilon,\omega}\delta^{2}x^{-A}$. Thus,

[TABLE]

where

[TABLE]

We write

[TABLE]

with $c_{j}$ being suitable complex coefficients satisfying $\lvert c_{j}\rvert=1$.

Next, we remove the factor $f_{N}(mn)$ by writing the Gaussian as an inverse Mellin transform in the form

[TABLE]

where $c>0$. This implies

[TABLE]

and hence

[TABLE]

where

[TABLE]

with $a_{m}(s)\coloneqq a_{m}\lvert m\rvert^{-s}$ and $b_{n}(s)\coloneqq b_{n}\lvert n\rvert^{-s}$. We set $c\coloneqq 1/\log x$. Then $N^{s/2}=O(1)$ and

[TABLE]

for all $s$ with $\Re s=c$ and $m$ and $n$ in the relevant summation ranges.

Now we estimate $\Sigma_{L}(s)$. In the following we tacitly assume that $x^{\mu}\leq L\leq x^{\mu+\kappa}$. By Cauchy’s inequality,

[TABLE]

which implies

[TABLE]

Next, we use the uniform bound $\boldsymbol{1}_{\{\mathrm{N}(k)\leq 2L^{\prime}\}}$ ($k\in\mathcal{O}$) for $L^{\prime}\coloneqq x^{\epsilon}L\delta^{-2}$ to extend the summation over $k$, getting

[TABLE]

Upon expanding the square in $\lvert\sum_{n}\ldots\rvert^{2}$,

[TABLE]

The subsum with $n_{1}=n_{2}$ is $\ll\delta^{-4}x^{1+4\epsilon}L$. Thus, on writing $\ell=j(n_{1}-n_{2})$,

[TABLE]

where

[TABLE]

By 6.4, it follows that

[TABLE]

By 4.4,

[TABLE]

Combining this with Eq. 6.8 and taking square root, we obtain the estimate

[TABLE]

We may reverse the roles of $m$ and $n$ and estimate $\lvert\Sigma_{L}(s)\rvert^{2}$ by

[TABLE]

in place of (6.7). Then we can continue in a similar way as above. We arrive at the same estimate as in (6.9) but with $L$ replaced by $x/L$, i.e.

[TABLE]

Using (6.9) if $L\leq x^{1/2}$ and (6.10) if $L>x^{1/2}$, and recalling that $x^{\mu}\leq L\leq x^{\mu+\kappa}$, we deduce that

[TABLE]

Using (6.6) together with Stirling’s approximation for the Gamma function, the same bound, up to a factor of $\log x$, holds for $\Sigma_{L}$. Plugging this into Eq. 6.5, we obtain the assertion of the proposition. ∎

6.7. Conclusion

Proof of 2.1.

We note that

[TABLE]

for all sufficiently large $x$ depending on $\epsilon$. (Mind though that the first two quantities depend implicitly on $x$ by means of the definitions of $\tilde{w}$, $w$, and $N$.) Then, for $\mu\in(0,\frac{1}{2})$, choosing $\kappa=\frac{1}{2}$ and $M=2x^{\mu}$ in accordance with 3.1, Propositions 6.6 and 6.7 give

[TABLE]

Upon taking $x=\lvert q\rvert^{3}$ and $\mu=\frac{1}{4}$, we find

[TABLE]

provided that $\tfrac{1}{2}\geq\delta\geq x^{-1/8+10\epsilon}$. By using 6.3, the theorem follows. ∎

7. Proof of the weighted version of Harman’s sieve for $\mathcal{O}$

Before embarking on the proof of 3.1, we record the following useful lemma:

Lemma 7.1**.**

For any two distinct real numbers $\rho,\gamma>0$ and $T\geq 1$ one has

[TABLE]

where the implied constant is absolute.

Proof.

See, for instance, [8, Lemma 2.2]. ∎

Proof of 3.1.

We follow [8] quite closely. We assume that $\omega$ takes the values $w$ and $\tilde{w}$ and put $z=x^{\kappa}$.

We start by introducing some notation. For each prime ideal of $\mathcal{O}$ choose a generator $p\in\mathcal{O}$ and let $\mathbb{P}_{\mathcal{O}}$ be the set of all those $p$. For $Z\geq 0$ write

[TABLE]

and

[TABLE]

We also need to introduce some $\mathcal{O}$-version of the Möbius $\mu$ function: for a non-unit $d$, let $\mu(d)$ be defined as $(-1)^{r}$ if $d$ is the product of precisely $r$ non-associate prime elements and $\mu(d)=0$ otherwise. If $d$ is a unit, then put $\mu(d)=1$. Then, by inclusion–exclusion, we have

[TABLE]

On writing

[TABLE]

applying Eq. 7.1 for $\omega=w$ and $\omega=\tilde{w}$ yields

[TABLE]

By Eq. 3.2 with $a_{m}=\mu(m){\boldsymbol{1}}_{\{m\mid P_{\mathcal{O}}(z)\}}$ we infer $\lvert S_{\mathrm{I}}\rvert\leq Y$. Therefore, to prove the theorem, it remains to establish that

[TABLE]

The next step is to arrange $S_{\mathrm{II}}$ into subsums according to the ‘size’ of the prime factors in $m$ (where $m$ is the summation variable from Eq. 7.3). To have some such notion of size, fix some total order $\prec$ on $\mathbb{P}_{\mathcal{O}}(z)$ such that $\mathrm{N}(p_{2})\leq\mathrm{N}(p_{1})$ whenever $p_{2}\prec p_{1}$. (Clearly many such orders exist, but the precise choice must not concern us.) Moreover, for $p\in\mathbb{P}_{\mathcal{O}}(z)$, let

[TABLE]

Now take $g\colon\mathcal{O}\to\mathbb{C}$ to be any function with $g(m)=g(\tilde{m})$ whenever $m$ and $\tilde{m}$ are associates. Then, we may group the terms of the sum

[TABLE]

according to the largest prime factor $p_{1}$ of $m$ (w.r.t. $\prec$):

[TABLE]

Evidently, the process giving Eq. 7.5 also works if $P_{\mathcal{O}}(z)$ is replaced by $\Pi(p)$; for any $r\in\mathcal{O}$ one has

[TABLE]

Minding the inner most sum on the right hand side above, it is obvious that the above identity can be iterated if so desired. To describe for which sub-sums iteration is beneficial, we let

[TABLE]

and, inductively for $s=2,3,\ldots$,

[TABLE]

where

[TABLE]

Assuming that $g$ vanishes on arguments $r$ with $\mathrm{N}(r)\leq x^{\mu}$, and on applying Eq. 7.5 and Eq. 7.6,

[TABLE]

On iterating this process—always applying Eq. 7.6 to the $\mathscr{Q}$-part—it transpires that

[TABLE]

for any $t\in\mathbb{N}$. Since the product of $t$ prime elements has norm $\geq 2^{t}$, we have

[TABLE]

Hence,

[TABLE]

for (say)

[TABLE]

We apply this to $S_{\mathrm{II}}$ with $g(m)=\Delta(m){\boldsymbol{1}}_{\{\mathrm{N}(m)\geq M\}}$. Note that, since $M>x^{\mu}$, we have $g(r)=0$ for all $r$ with $\mathrm{N}(r)\leq x^{\mu}$, as was assumed in the above arguments. Thus,

[TABLE]

where

[TABLE]

Another application of Eq. 7.6 gives

[TABLE]

Given $m=p_{1}\cdots p_{s-1}p_{s}$ with

[TABLE]

and noting that $\mathrm{N}(p_{s})\leq\mathrm{N}(p_{1})<z=x^{\kappa}$, we have

[TABLE]

Using this, we find that $S_{\mathrm{II},1}(s)$ can be expressed as

[TABLE]

where the coefficients

[TABLE]

are only supported on $m$ with $x^{\mu}<\mathrm{N}(m)<x^{\mu+\kappa}$. Hence, by Eq. 3.3,

[TABLE]

Moving on to $S_{\mathrm{II},2}(s)$, we expand the definition Eq. 7.2 of $\Delta$, getting

[TABLE]

where

[TABLE]

In order to apply Eq. 3.3, we must disentangle the variables $m$ and $n$ in the above summation. To this end, split

[TABLE]

to obtain a decomposition

[TABLE]

For $S_{\mathrm{II},2}^{<}(s,\omega)$ we have

[TABLE]

where

[TABLE]

and the sum $S_{\mathrm{II},2}^{=}(s,\omega)$ can be expressed similarly, but needs a little more care: by basic ramification theory, the first summation on the right hand side of Eq. 7.11 contains at most one term and we shall write $\mathscr{P}_{s}^{\prime}$ for the set of $(p_{1},\ldots,p_{s})\in\mathscr{P}_{s}$ for which there is such a term, that is, some $p\prec p_{s}$ with $\mathrm{N}(p)=\mathrm{N}(p_{s})$. Furthermore, let $\mathbb{P}_{\mathcal{O}}(z)^{\prime}$ denote the set of the $p$’s just mentioned, i.e.,

[TABLE]

Thus,

[TABLE]

where

[TABLE]

To disentangle $m$-dependent quantities ($\mathrm{N}(m)$ and $\mathrm{N}(p_{s})$) from $n$-dependent quantities ($\mathrm{N}(pd)$ and $\mathrm{N}(p)$) in the above, we employ 7.1. We pick some real number $\varrho$ (depending only on $M$) with $\lvert\varrho\rvert\leq\frac{1}{2}$ such that $\{M+\varrho\}=\frac{1}{2}$ and for $m,p,d\in\mathcal{O}$ the condition $\mathrm{N}(mpd)\geq M$ is equivalent to $\log N\lparen mpd\rparen\geq\log\lparen M+\varrho\rparen$. Then

[TABLE]

Therefore, 7.1 shows that

[TABLE]

for every $T\geq 1$. Similarly,

[TABLE]

Thus,

[TABLE]

with coefficients

[TABLE]

as well as

[TABLE]

We proceed by gathering some intermediate information before applying Eq. 3.3: in the definition of the coefficients $b_{n}$, neither of the summations over $p$ and $d$ includes associates. Thus,

[TABLE]

For the other coefficients we always have

[TABLE]

yet if $t$ and $\tau$ are small, one can (and must) do better: indeed,

[TABLE]

where

[TABLE]

In view of this, we must deal with functions $f\colon\mathbb{R}\times(0,1)\to\mathbb{R}$ of the shape

[TABLE]

and their integrals

[TABLE]

Lastly, we note that, by 4.1 and Eq. 3.1,

[TABLE]

Collecting what we have gathered so far, we may derive a bound for

[TABLE]

as follows: after applying Eq. 7.14 with $\omega=w$ and $\omega=\tilde{w}$, the $O(\ldots)$-terms are treated directly with Eq. 7.19 and Eq. 7.18, whereas for the rest one may apply Eq. 3.3. Here it is important to use Eq. 7.17 for small $\lvert t\rvert$ respectively $\lvert\tau\rvert$ first—prior to applying Eq. 3.3—and Eq. 7.18 then bounds the integrals. Therefore, after some computations, we infer

[TABLE]

Of course, the same arguments also apply to $S_{\mathrm{II},2}^{=}(s,\omega)$; in view of Eq. 7.13 we have to apply them twice, but in both cases the coefficients corresponding to Eq. 7.15 and Eq. 7.16 obey the same bounds we used to derive Eq. 7.20. Consequently, Eq. 7.20 also holds with $S_{\mathrm{II},2}^{=}$ in place of $S_{\mathrm{II},2}^{<}$. In total, recalling Eq. 7.9, Eq. 7.10 and Eq. 7.12, we have

[TABLE]

and it transpires that choosing $T=xX$ suffices to yield a bound $\ll Y(\log(xX))^{2}$. On plugging this into Eq. 7.8 and recalling Eq. 7.7, we infer Eq. 7.4. Hence, the theorem is proved. ∎