Sifting for small split primes
of an imaginary quadratic field
in a given ideal class

For integers $q\geqslant 2$ and $(a,q)=1$, let $p(q,a)$ denote the least prime $p\equiv a\;(q)$. In 1944, Linnik [29] showed that

(1)

$$p(q,a)\;\ll\;q^{L},$$

where both $L$ and the implied constant are absolute. Since then, there have been many improvements on this result. Building on the work of Heath-Brown [19], Xylouris [37] showed that unconditionally one can take $L=5$ in (1), which is the current record. This comes quite close to the bound

$$p(q,a)\;\ll\;(q\log q)^{2},$$

which is what follows assuming that the Riemann hypothesis holds for the Dirichlet $L$-functions $L(s,\chi)$.

Such results are difficult to establish unconditionally, and have traditionally (following Linnik) depended on deep results on the zeros of these $L$-functions, namely a log-free zero-density estimate and a quantitative version of the Deuring-Heilbronn phenomenon (exceptional zero repulsion effect).

Linnik’s theorem has been generalized in the setting of the Chebotarev density theorem: given a Galois extension of number fields $L/K$ with Galois group $G$, each prime $\mathfrak{p}$ of $K$ (unramified in $L$) can be associated to a conjugacy class $C\subset G$ by the Artin symbol $\big{[}\frac{L/K}{\mathfrak{p}}\big{]}$. The analogue of Linnik’s theorem is a bound (in terms of the various number field parameters involved) on the least norm $\text{N}_{K/\mathbf{Q}}\mathfrak{p}$ of a prime $\mathfrak{p}$ with prescribed Artin symbol. There have been many works in this direction, both conditional (see [28] and [2]) and unconditional (see [8], [27], [36], [26], [39], and [35]). These unconditional results proceed by establishing analogues of Linnik’s log-free zero-density estimate and quantitative Deuring-Heilbronn phenomenon for the Hecke $L$-functions.

In a different direction, there has been a growing interest in finding new proofs of Linnik’s theorem that avoid using input about the zeros of $L$-functions—see for instance [7], [16], [25], [32], [11], [12], [31], [30], and Chapter 24 in [10]. Often these works combine sieve theoretic techniques with “pretentious methods” and/or with techniques coming from additive combinatorics. While such proofs are not usually (as of yet) as numerically strong as those that use the zeros of $L$-functions, they are very interesting from a conceptual point of view. Similar to the “elementary proof” of the prime number theorem, such proofs show how challenging results in arithmetic can be achieved without (or with minimal) use of results on the zeros of $L$-functions.

In this article, we are interested in a particular analogue of Linnik’s theorem for primes in imaginary quadratic fields, which we prove without using zero-density theorems or exceptional zero repulsion results. Let $D>3$ be a prime, $D\equiv 3\;(4)$, so that $-D$ is a negative fundamental discriminant. (We work with prime $D$ for simplicity, though we expect that our methods could be adapted without major modifications to work for general fundamental discriminants $-D$.) For an integral ideal $\mathfrak{a}\subseteq\mathcal{O}_{K}$, we denote by $\text{N}\mathfrak{a}=|\mathcal{O}_{K}/\mathfrak{a}|$ its (absolute) norm. The class group $\mathcal{H}$ of $K$ is a finite abelian group of order $h=|\mathcal{H}|$, the class number.

In analogy to Dirichlet’s theorem on primes in arithmetic progression, one can show using class group characters $\chi\in\widehat{\mathcal{H}}$ that for any specified ideal class $\mathcal{C}\in\mathcal{H}$, there are infinitely many split primes $\mathfrak{p}$ (i.e., unramified primes $\mathfrak{p}$ with $\text{N}\mathfrak{p}=p$, a rational prime) in the class $\mathcal{C}$. Therefore, in analogy to Linnik’s theorem, we are interested in bounding

$$p(D,\mathcal{C})\;\coloneqq\;\min\{\text{N}\mathfrak{p}:\mathfrak{p}\in\mathcal{C}\text{ is a split prime of }\mathcal{O}_{K}\},$$

the least norm of a split prime in the class $\mathcal{C}$. Our main result is

This result is a special case of the results for the Chebotarev theorem, so it has been established several times before in those works listed above. In particular, Thorner and Zaman [35] showed

$$p(D,\mathcal{C})\;\ll\;D^{694}$$

for all negative fundamental discriminants $-D$. However, our work is novel in that it is the first time such a result has been established without the use of zero-density theorems and quantitative Deuring-Heilbronn results. We also handle new sieve-theoretic challenges (in comparison with works on Linnik’s theorem) in the sifting dimension aspect; see Section 3 for details.

Theorem 1.1 is the result of combining Theorems 2.2 and 2.3 below; we now establish our notations and state these theorems.

Given an ideal class $\mathcal{C}$ in the class group $\mathcal{H}$ of $K=\mathbf{Q}(\sqrt{-D})$, we put

$$\lambda_{\mathcal{C}}(n)\;=\;\#\{\mathfrak{a}\in\mathcal{C};\;\text{N}\mathfrak{a}=n\}.$$

Given a character $\chi\in\widehat{\mathcal{H}}$ of the class group, we define

(2)

$$\lambda_{\chi}(n)\;=\;\sum_{\text{N}\mathfrak{a}=n}\chi(\mathfrak{a}),$$

the sum being taken over integral ideals $\mathfrak{a}\subseteq\mathcal{O}_{K}$ of norm $n$. Then by the orthogonality of the class group characters we have

(3)

$$\lambda_{\mathcal{C}}(n)\;=\;\frac{1}{h}\sum_{\chi\in\widehat{\mathcal{H}}}\overline{\chi}(\mathcal{C})\lambda_{\chi}(n).$$

For the trivial character $\chi_{0}\in\widehat{\mathcal{H}}$ we have

(4)

$$\lambda_{\chi_{0}}(n)\;=\;(1*\chi_{D})(n)\;=\;\sum_{d\mid n}\chi_{D}(d),$$

which is the number of ideals in $\mathcal{O}_{K}$ of norm $n$, and

$$\chi_{D}(n)\;=\;\Big{(}\frac{-D}{n}\Big{)}$$

is the Kronecker symbol. Indeed, $\chi_{D}$ is a primitive real Dirichlet character of conductor $D$. By the Dirichlet class number formula, we have

$$h\;=\;\frac{1}{\pi}\sqrt{D}\;L(1,\chi_{D}).$$

Our main object of study is the sequence

(5)

$$a_{n}\;=\;\frac{1}{n}\lambda_{\mathcal{C}}(n)f\Big{(}\frac{\log n}{\log x}\Big{)},$$

$f(u)\geqslant 0$ a smooth function with $\widehat{f}(0)>0$, $\widehat{f}$ denoting the Fourier transform,

$$\widehat{f}(\xi)\;\coloneqq\;\int_{-\infty}^{\infty}f(u)\text{e}(-\xi u)\mathop{}\!\mathrm{d}u,$$

and $\text{e}(z)\coloneqq e^{2\pi iz}$. We assume that $f(u)$ is supported in the segment

$$1-\nu\leqslant u\leqslant 1,$$

where $\nu>0$ is a small concrete number whose value can be determined in the course of our arguments (though the exact value is not important to us). Here, $x$ is a large parameter going to infinity, and our goal is to estimate

$$S\;\coloneqq\;\sum_{p}a_{p}\;=\;\sum_{p}\frac{1}{p}\lambda_{\mathcal{C}}(p)f\Big{(}\frac{\log p}{\log x}\Big{)}.$$

By the prime number theorem, we have

$$\sum_{p}\frac{1}{p}f\Big{(}\frac{\log p}{\log x}\Big{)}\;\sim\;\widetilde{f}(0)\;>\;0,$$

where $\widetilde{f}$ denotes the Mellin transform of $f$,

$$\widetilde{f}(s)\;=\;\int_{0}^{\infty}u^{s-1}f(u)\mathop{}\!\mathrm{d}u.$$

One expects prime ideals $\mathfrak{p}$ to equidistribute among the $h$ ideal classes $\mathcal{C}$ in $\mathcal{H}$ even when the discriminant $D$ is comparable in size (in the logarithmic scale) to the norm $\text{N}\mathfrak{p}$. Thus we expect the asymptotic formula

$$S\;\sim\;\frac{1}{h}\;\widetilde{f}(0)$$

to hold for $x\geqslant D^{A}$ for some absolute constant $A>0$. Indeed, the Riemann hypothesis for the class group $L$-functions $L_{K}(s,\chi)$ implies that the above asymptotic formula holds with $x\geqslant D^{2}(\log D)^{4}$. Here we establish the bound

(6)

$$S\;\gg\;\frac{1}{h}\;\widetilde{f}(0)$$

uniformly for $x\gg D^{L}$ for some absolute $L>0$. (In actuality, we prove a slightly weaker lower bound in the case of an exceptional character; see the precise statement in Theorem 2.2 and the remarks that follow.) In particular, it follows from this lower bound that for all prime $D\equiv 3\;(4)$, we have

(7)

$$\lambda_{\mathcal{C}}(p)>0\quad\text{for some }p\ll D^{L}.$$

In other words, every class $\mathcal{C}$ contains a prime ideal $\mathfrak{p}$ with $p=\text{N}\mathfrak{p}\ll D^{L}$.

To establish the bound (6) unconditionally, we split our argument into two cases depending on the non/existence of real zeros of the Dirichlet $L$-function $L(s,\chi_{D})$. We will use assumptions about such zeros in several places in the work (and also for other $L$-functions), so to clarify this, we make the following

Now we state our main two theorems, which together prove Theorem 1.1.

We follow the general approach of Friedlander and Iwaniec’s proof of Linnik’s theorem in Chapter 24 of [10]—see also their recent related articles [11] and [12]. Our work differs from theirs in a few key aspects, which we now explain.

The goal is to give a positive lower bound for the sum $\sum_{p}a_{p}$, where for them $\mathcal{A}=(a_{n})$ is the characteristic function of the arithmetic progression $n\equiv a\;(q)$, $x/2<n\leqslant x$, and for us $(a_{n})$ is defined by (5). They begin with an application of Buchstab’s identity,

(11)

$$S(\mathcal{A},\sqrt{x})\;=\;S(\mathcal{A},z)\;-\sum_{z\leqslant p<\sqrt{x}}S(\mathcal{A}_{p},p),$$

where $S(\mathcal{A},w)$ denotes the sum of $(a_{n})$ over $n$ having no prime factor less than $w$, and $\mathcal{A}_{p}=(a_{pm})$ denotes the subsequence of $\mathcal{A}$ over multiples of $p$. Here we take $z=x^{1/r}$ with $r$ taken to be as large as necessary.

First they treat the case where there is an exceptional character $\chi\;(q)$. In this case, they apply (11) to the “twisted” sequence $\widetilde{a}_{n}=\lambda(n)a_{n}$, where

$$\lambda(n)=\sum_{d\mid n}\chi(d).$$

They show using the Fundamental Lemma of Sieve Theory that the two terms on the right-hand side of (11) are (asymptotically as $r$ becomes large) of the same size, up to a factor of

$$\delta(z,x)\;\coloneqq\;\sum_{z\leqslant p<x}\frac{\lambda(p)}{p}\;=\;\sum_{z\leqslant p<x}\frac{1+\chi(p)}{p}$$

present in the second term. Assuming that $\chi$ is exceptional, $\delta(z,x)$ is very small, and a positive lower bound for $S(\mathcal{A},\sqrt{x})=\sum_{p}a_{p}$ follows. Friedlander and Iwaniec also give in [11] an alternative approach via Selberg’s sieve that works on similar principles and gives comparable results.

We follow their approach in [10] to prove our Theorem 2.2, which is under the assumption of an exceptional character. This is taken up in Section 9. We require little modification of their arguments, since their method does not require very specific properties of the sequence $\mathcal{A}=(a_{n})$ beyond some basic sieve assumptions that also apply in our case. In fact, it is even simpler for us, since we have no need to twist our sequence $(a_{n})$ by the weights $\lambda(n)$ above—such a factor naturally appears in this particular sequence already; see Proposition 7.1 for a precise statement.

For the non-exceptional case, Friedlander and Iwaniec work with a combinatorial sieve identity that leads to

(12)

$$S(\mathcal{A},z)\;\geqslant\;S^{-}(\mathcal{A},z)\;+\;\frac{1}{24}Q(\mathcal{A}),$$

where

$$S^{-}(\mathcal{A},z)\;=\;\sum_{d\mid P(z)}\lambda_{d}^{-}A_{d}$$

is the lower bound coming from the beta-sieve (so that $(\lambda_{d}^{-})$ are the lower-bound beta-sieve weights, and $A_{d}$ are the congruence sums for the sequence $\mathcal{A}$—see Section 5.1 for details), and

$$Q(\mathcal{A})\;\coloneqq\;\sum_{p_{0}}\sum_{p_{1}}\sum_{p_{2}}\sum_{p_{3}}\sum_{p_{4}}a_{p_{0}p_{1}p_{2}p_{3}p_{4}},$$

where the variables $p_{j}$ run over specific segments $x^{\alpha_{j}}\leqslant p_{j}\leqslant x^{\beta_{j}}$.

Their sifting problem is linear (i.e. of sieve dimension $\kappa=1$; again, see 5.1 for details), which means that one can show that $S^{-}(\mathcal{A},z)$ is negligible (relatively very small) for $z$ close to $\sqrt{x}$. The upshot is that they show that

$$S(\mathcal{A},\sqrt{x})\;\gtrsim\;\frac{1}{24}Q(\mathcal{A})$$

up to some comparably negligible contributions. This reduces the problem of counting primes to finding a lower bound for $Q(\mathcal{A})$, which counts products of prime quintuplets in arithmetic progression. Indeed, the common parity here (products of 1 and 5 primes, respectively) is an artifact of the sieve process.

By contrast, in this work we cannot so readily work with (12). The sifting density function $g(d)$ for our sequence $\mathcal{A}=(a_{n})$ is given on primes $p$ by

$$g(p)\;=\;\frac{1+\chi_{D}(p)}{p}\;+\;O\Big{(}\frac{1}{p^{2}}\Big{)},$$

and the presence of the character $\chi_{D}(p)$ causes fluctuations that hinder one from easily claiming the one-dimensionality of the sieve problem. One possible approach would be to use the fact that we are working in the non-exceptional case (i.e., assuming Hypothesis $\text{H}(c)$ for $L(s,\chi_{D})$, say) to effectively bound the sum $\sum_{p<z}g(p)$ and hence control the sieve dimension.

However, here we choose to proceed differently: by the trivial bound $g(p)\leqslant 2/p+O(1/p^{2})$, we can work with a $\kappa=2$-dimensional sieve. We can no longer show that $S^{-}(\mathcal{A},z)$ is negligible for $z$ so close to $\sqrt{x}$ (only for $z\leqslant x^{1/\beta(2)}=x^{1/4.8339\dots}$; see Section 5.1), and so we employ a different combinatorial identity than in [12]. This identity comes from applying a second iteration of the Buchstab formula to each term $S(\mathcal{A}_{p},p)$ in (11), which gives

(13)

$$S(\mathcal{A},\sqrt{x})\;=\;S(\mathcal{A},z)-\sum_{z\leqslant p<\sqrt{x}}S(\mathcal{A}_{p},z)+\mathop{\sum\sum}_{z\leqslant p_{2}<p_{1}<\sqrt{x}}S(\mathcal{A}_{p_{1}p_{2}},p_{2}).$$

Rather than work with an inequality, we evaluate (nearly asymptotically) each of the three terms on the right-hand side of (13) and show that the result is positive. The first two terms are readily handled via the Fundamental Lemma, so we reduce the problem to analyzing the third term, which is

(14)

$$\mathop{\sum\sum}_{z\leqslant p_{2}<p_{1}<\sqrt{x}}S(\mathcal{A}_{p_{1}p_{2}},p_{2})\;=\mathop{\sum\sum\sum}_{\begin{subarray}{c}z\leqslant p_{2}<p_{1}<\sqrt{x}\\ (b,P(p_{2}))=1\end{subarray}}a_{p_{1}p_{2}b}.$$

This sum is our analogue of $Q(\mathcal{A})$—note that it is supported on integers which are products of three (almost-) primes, the same parity as in $Q(\mathcal{A})$.

Friedlander and Iwaniec handle $Q(\mathcal{A})$ via a multiplicative analogue of a additive ternary problem treated by the classical circle method. They use Dirichlet characters $\chi\;(q)$ to decouple the prime variables $p_{j}$. After removing the contribution of the principal character (the “major arc”), they use two of the prime variables and the orthogonality of the characters to recover the cost of opening the sum with the characters. The remaining three prime variables are used to obtain a nontrivial cancellation in the character sums over primes—importantly, they do not have need for any zero density bounds or repulsion properties of the exceptional zeros.

We handle the sum (14) in a similar manner, here using the class group characters $\chi\in\widehat{\mathcal{H}}$ instead of Dirichlet characters to decouple our variables. Just as above, two variables ($p_{1}$ and $b$) and the orthogonality of the class group characters are used to recover the cost in using these characters. This involves a type of large sieve inequality for these characters (over integers free from small prime factors) that we develop in Section 8. For nontrivial cancellation in a character sum over the final prime variable $p_{2}$, we apply the explicit formula and use a zero-free region for the class group $L$-functions. It is a technical reason that we do not use three prime variables for this as they do in [12]. While their sequence $(a_{n})$ is localized dyadically, $x/2<n\leqslant x$, ours is supported in a longer segment $x^{1-\nu}\leqslant n\leqslant x$. This means that we work with a longer sum over the prime variable $p_{2}$, which effectively localizes the dual sum over zeros in the explicit formula to essentially be supported on zeros within the classical zero-free region. It is in this way that we do not make use of any zero density estimates or repulsion effects of exceptional zeros.

This work was completed as part of the author’s PhD thesis. He is deeply grateful to his advisor, Henryk Iwaniec, who provided constant support and guidance throughout this project, and who provided very insightful and helpful feedback during the writing of this article.

In this section, we consider the sequence $\mathcal{A}=(a_{n})$ defined in (5) and evaluate the associated congruence sums,

$$A_{d}\;=\;\sum_{n\equiv 0\;(d)}a_{n}\;=\;\sum_{n\equiv 0\;(d)}\frac{1}{n}\lambda_{\mathcal{C}}(n)f\Big{(}\frac{\log n}{\log x}\Big{)}.$$

Expressing $\lambda_{\mathcal{C}}$ in terms of $\lambda_{\chi}$ by (3), this is accomplished via

To prove Proposition 6.1, we use Lemma 6.2, a summation formula for the harmonics $\lambda_{\chi}$ (see for instance (5.16) in [23]). We omit its proof, which is standard—it follows essentially from the functional equation (21).

Next, we establish a version of the above formula in the logarithmic scale.