2.1 $16$ -rank of class groups

Aside from two recent results due to the authors [Reference MilovicMil17b, Reference Koymans and MilovicKM18], density results about the $16$ -rank of class groups in one-prime-parameter families $\{\mathbb{Q}(\sqrt{dp})\}_{p}$ ( $d$ fixed and $p$ varying) have remained elusive despite a large body of work on algebraic criteria for the $16$ -rank in such families [Reference KaplanKap77, Reference OriatOri78, Reference Kaplan and WilliamsKW82, Reference Leonard and WilliamsLW82, Reference Kaplan and WilliamsKW84, Reference YamamotoYam84, Reference Kaplan, Williams and HardyKWH86, Reference StevenhagenSte93a, Reference Bruin and HemenwayBH13]. This gap between algebraic and analytic understanding of the $16$ -rank can be largely attributed to the absence of appropriate governing fields and the subsequent inability to apply the Čebotarev density theorem. More precisely, for a finite extension of number fields $E/F$ , let $\text{Art}_{E/F}$ denote the corresponding Artin map. Cohn and Lagarias [Reference Cohn and LagariasCL83, Reference Cohn and LagariasCL84] conjectured that, for each integer $k\geqslant 1$ and each integer $d\not \equiv 2\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ , the map

$$\begin{eqnarray}f_{d,k}:p\mapsto \text{rk}_{2^{k}}\text{Cl}^{+}(dp)\end{eqnarray}$$

is Frobenian, in the sense of Serre [Reference SerreSer12]. In other words, they conjectured that there exists a normal field extension $M_{d,k}/\mathbb{Q}$ for which there is a class function

$$\begin{eqnarray}\unicode[STIX]{x1D719}:\text{Gal}(M_{d,k}/\mathbb{Q})\rightarrow \mathbb{Z}_{{\geqslant}0},\end{eqnarray}$$

satisfying

$$\begin{eqnarray}f_{d,k}(p)=\unicode[STIX]{x1D719}(\text{Art}_{M_{d,k}/\mathbb{Q}}(p))\end{eqnarray}$$

for all primes $p$ unramified in $M_{d,k}/\mathbb{Q}$ ; such a field $M_{d,k}$ is called a governing field for $\{\text{rk}_{2^{k}}\text{Cl}^{+}(dp)\}_{p}$ . For $k\leqslant 3$ , Stevenhagen [Reference StevenhagenSte89] proved these conjectures for all $d\not \equiv 2\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ . Perhaps the simplest case is $d=-4$ , where one can take $M_{-4,3}$ to be the field $M=\mathbb{Q}(\unicode[STIX]{x1D701}_{8},\sqrt{1+i})$ as above and where $h(-4p)\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}8$ if and only if $p$ splits completely in $M$ . Hence, by the Čebotarev density theorem, the density of primes $p$ such that $h(-4p)\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}8$ is equal to $1/[M:\mathbb{Q}]=1/8$ .

Cohn and Lagarias [Reference Cohn and LagariasCL84] ruled out some obvious candidates for $M_{-4,4}$ , i.e., the governing field for the $16$ -rank of $\text{Cl}(-4p)$ , and to this day no governing fields for the $16$ -rank in any family have been found. Nevertheless, we are able to show, in Theorem 2, that the density of primes $p$ such that $h(-4p)\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}16$ exists and is equal to $1/16$ . It is proved unconditionally in [Reference MilovicMil17a] that there are infinitely many primes  $p$ such that $h(-4p)\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}16$ , but that result implies nothing about the density as in Theorem 1.

The key innovation that allows us to go beyond the $8$ -rank is to use Vinogradov’s method [Reference VinogradovVin47, Reference VinogradovVin54] for studying the distribution of prime numbers instead of the heretofore used Čebotarev density theorem (as in [Reference SmithSmi16], for instance). Moreover, the current state-of-the-art bounds for the error term in the Čebotarev density theorem are essentially of size $X\exp (-\sqrt{\log X})$ , far worse than the power-saving bound $X^{1-\unicode[STIX]{x1D6FF}^{\prime }}$ in Theorem 2. In fact, obtaining such a power-saving error term in the Čebotarev density theorem would be tantamount to proving a zero-free region for the associated Artin $L$ -functions of the form $\Re (s)>1-\unicode[STIX]{x1D6FF}^{\prime }$ , and this is well out of reach of current methods in analytic number theory. Nonetheless, the power-saving bound $X^{1-\unicode[STIX]{x1D6FF}^{\prime }}$ does not prove the non-existence of a governing field – it merely suggests that one is unlikely to exist. We summarize this discussion with the following immediate corollary of Theorem 2.

2.2 Real quadratic fields and the negative Pell equation

In the case $d<0$ , the narrow class group $\text{Cl}^{+}(dp)$ is the same as the ordinary class group $\text{Cl}(dp)$ . If $d>0$ , however, then $\text{Cl}^{+}(dp)$ and $\text{Cl}(dp)$ may be different; in fact, $\text{Cl}^{+}(dp)=\text{Cl}(dp)$ if and only if the fundamental unit $\unicode[STIX]{x1D700}_{dp}$ of $\mathbb{Q}(\sqrt{dp})$ has norm $-1$ . While Cohn and Lagarias stated their conjecture on the existence of governing fields only for narrow class groups, one can ask what happens for ordinary class groups. As mentioned before, Stevenhagen proved the conjecture of Cohn and Lagarias for the $8$ -rank of narrow class groups of both imaginary and real quadratic fields. Theorem 3 is the first density result for the $8$ -rank of the ordinary class group in a family of real quadratic fields. Again the power-saving error term suggests that there is no governing field for $\text{rk}_{8}\text{Cl}(8p)$ in the family $\{\text{Cl}(8p)\}_{p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4}$ . To place Theorem 3 in context, we note that the $2$ -part of $\text{Cl}^{+}(8p)$ is cyclic, and, for $p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ , one has (for instance, see [Reference StevenhagenSte93a]):

1. – $h^{+}(8p)=h(8p)\equiv 2\hspace{0.2em}{\rm mod}\hspace{0.2em}4\Leftrightarrow p\text{ splits completely in }\mathbb{Q}(i)\text{ but not in }\mathbb{Q}(\unicode[STIX]{x1D701}_{8})$ ;

2. – $h^{+}(8p)\equiv h(8p)+2\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}4\Leftrightarrow p\text{ splits completely in }\mathbb{Q}(\unicode[STIX]{x1D701}_{8})\text{ but not in }\mathbb{Q}(\unicode[STIX]{x1D701}_{8},\sqrt[4]{2})$ ;

3. – $h^{+}(8p)=h(8p)\equiv 4\hspace{0.2em}{\rm mod}\hspace{0.2em}8\Leftrightarrow p\text{ splits completely in }\mathbb{Q}(\unicode[STIX]{x1D701}_{8},\sqrt[4]{2})\text{ but not in }\mathbb{Q}(\unicode[STIX]{x1D701}_{16},\sqrt[4]{2})$ ;

4. – $h^{+}(8p)\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}8\Leftrightarrow p\text{ splits completely in }\mathbb{Q}(\unicode[STIX]{x1D701}_{16},\sqrt[4]{2})$ .

Hence, Theorem 3 in conjunction with the Čebotarev density theorem implies that

$$\begin{eqnarray}\lim _{X\rightarrow \infty }\frac{\#\{p\text{ prime}:p\leqslant X,p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4,h(8p)\equiv 4\hspace{0.2em}{\rm mod}\hspace{0.2em}8\}}{\#\{p\text{ prime}:p\leqslant X,p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4\}}=\frac{3}{16}\end{eqnarray}$$

and

$$\begin{eqnarray}\lim _{X\rightarrow \infty }\frac{\#\{p\text{ prime}:p\leqslant X,p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4,h(8p)\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}8\}}{\#\{p\text{ prime}:p\leqslant X,p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4\}}=\frac{1}{16}.\end{eqnarray}$$

The 2-torsion subgroup $\text{Cl}^{+}(8p)[2]$ is generated by the classes of the ramified ideals $\mathfrak{t}$ and $\mathfrak{p}$ lying above $2$ and $p$ , respectively. Since the $2$ -part of $\text{Cl}^{+}(8p)$ is cyclic, we have $\#\text{Cl}^{+}(8p)[2]=2$ , so exactly one of the three ideals $\mathfrak{t}$ , $\mathfrak{p}$ , and $\mathfrak{t}\mathfrak{p}$ is in the trivial class in $\text{Cl}^{+}(8p)$ , while the remaining two are both in the non-trivial class in $\text{Cl}^{+}(8p)[2]$ . Moreover, (1.4) has a solution over the integers if and only if $\mathbb{Z}[\sqrt{2p}]$ has a unit of norm $-1$ , which occurs if and only if the ideal $\mathfrak{t}\mathfrak{p}=(\sqrt{2p})$ can be generated by a totally positive element in $\mathbb{Z}[\sqrt{2p}]$ , i.e., if and only if $\mathfrak{t}\mathfrak{p}$ is in the trivial class in $\text{Cl}^{+}(8p)$ . Stevenhagen conjectured in [Reference StevenhagenSte93b] that as $p$ varies over all prime numbers, each of $\mathfrak{t}$ , $\mathfrak{p}$ , and $\mathfrak{t}\mathfrak{p}$ is in the trivial class in $\text{Cl}^{+}(8p)$ equally often, which is why we expect $\lim _{X\rightarrow \infty }\unicode[STIX]{x1D6FF}^{-}(X)$ to exist and be equal to $1/3$ ( $\unicode[STIX]{x1D6FF}^{-}(X)$ is defined following (1.4)).

Since $\mathbb{Z}[\sqrt{2p}]$ has a unit of norm $-1$ if and only if the narrow class group $\text{Cl}^{+}(8p)$ coincides with the ordinary class group $\text{Cl}(8p)$ , we can obtain successively better upper and lower bounds for the proportion of primes $p$ for which (1.4) is solvable over $\mathbb{Z}$ by comparing $h^{+}(8p)$ and $h(8p)$ modulo successively higher powers of $2$ . Note that (1.4) has no solutions (even over $\mathbb{Q}$ ) whenever $p\equiv 3\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ , since in that case $-1$ is not a quadratic residue modulo $p$ . From this, the list of splitting criteria above, and the Čebotarev density theorem, one immediately deduces that

(2.1) $$\begin{eqnarray}\frac{5}{16}\leqslant \liminf _{X\rightarrow \infty }\unicode[STIX]{x1D6FF}^{-}(X)\leqslant \limsup _{X\rightarrow \infty }\unicode[STIX]{x1D6FF}^{-}(X)\leqslant \frac{3}{8}.\end{eqnarray}$$

Hence $|\unicode[STIX]{x1D6FF}^{-}(X)-1/3|\leqslant 1/24+o(X)$ as $X\rightarrow \infty$ , i.e., at worst, the bounds above are within $4.17\%$ of Stevenhagen’s conjecture. Theorem 4 hence cuts the possible discrepancy from Stevenhagen’s conjecture in half. Although the problem of improving (2.1) may have been first explicitly stated in 1993 in [Reference StevenhagenSte93b, p. 127], in essence it has been open since the 1930s, when Rédei [Reference RédeiRed34], Reichardt [Reference ReichardtRei34], and Scholz [Reference ScholzSch35] supplied the algebraic criteria sufficient to deduce (2.1).

2.3 Other results on $2$ -parts of class groups of number fields

Finally, we would like to contrast our results concerning one-prime-parameter families with results on $2$ -parts of class groups in families parametrized by arbitrarily many primes. The first significant achievement for families with arbitrary discriminants was made by Fouvry and Klüners [Reference Fouvry and KlünersFK07], who translated Rédei’s theory on $4$ -ranks of class groups to sums of characters conducive to analytic techniques and then successfully dealt with these sums, basing some of their work on the techniques developed by Heath-Brown in [Reference Heath-BrownHea93, Reference Heath-BrownHea94]. Fouvry and Klüners subsequently developed their methods in various settings [Reference Fouvry and KlünersFK10a, Reference Fouvry and KlünersFK10b, Reference Fouvry and KlünersFK10c, Reference Fouvry and KlünersFK11], most notably obtaining impressive upper and lower bounds for the solvability of the negative Pell equation $x^{2}-dy^{2}=-1$ for general squarefree integers $d>0$ . When specialized to the one-prime-parameter family $d=2p$ with $p$ prime, their results are as strong as the bounds in (2.1), so Theorem 4 can be viewed as the next natural step in the line of work initiated by Fouvry and Klüners.

A recent paper of Smith [Reference SmithSmi17] (see also [Reference SmithSmi16]) features ground-breaking distribution theorems about $2^{k}$ -ranks of class groups of imaginary quadratic fields for all $k\geqslant 3$ . The very deep methods that underlie these theorems require the number of prime parameters on average to go to infinity and hence are unlikely to yield results in the direction of Theorems 2, 3, or 4; from the standpoint of analytic number theory, Theorem 2 is a result about the distribution of prime numbers, while the main analytic techniques underlying the results of [Reference SmithSmi17] are consequences of a very careful study of the anatomy of the prime divisors of highly composite integers.

3.1 The governing field for the $8$ -rank of $\text{Cl}(-4p)$

As in § 1, let $M=\mathbb{Q}(\unicode[STIX]{x1D701}_{8},\sqrt{1+i})$ be the (minimal) governing field for the $8$ -rank in the family $\{\mathbb{Q}(\sqrt{-4p})\}_{p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4}$ . Using a computer algebra package such as Sage, one can readily check that:

1. (P1) the ring of integers of every subfield of $M$ (including $M$ itself) is a principal ideal domain;

2. (P2) the discriminant $\unicode[STIX]{x1D6E5}_{M}$ of $M/\mathbb{Q}$ is equal to $2^{22}$ , and $2$ is totally ramified in $M/\mathbb{Q}$ ; and

3. (P3) the torsion subgroup of the group of units in ${\mathcal{O}}_{M}$ is $\langle \unicode[STIX]{x1D701}_{8}\rangle$ .

Recall that $\text{rk}_{8}\text{Cl}(-4p)=1$ if and only if $p$ splits completely in $M/\mathbb{Q}$ , that is, if and only if $p$ is odd and every prime ideal $\mathfrak{p}$ in ${\mathcal{O}}_{M}$ lying over $p$ is of degree $1$ .

As noted in § 1, $M/\mathbb{Q}$ is a normal extension with Galois group isomorphic to the dihedral group $D_{8}$ of order $8$ . We fix an automorphism $r\in \text{Gal}(M/\mathbb{Q})$ such that $r$ generates the order $4$ subgroup $\text{Gal}(M/\mathbb{Q}(\sqrt{-2}))$ , and we let $s\in \text{Gal}(M/\mathbb{Q})$ be the non-trivial automorphism fixing the subfield $K_{1}=\mathbb{Q}(i,\sqrt{1+i})$ . Then $D_{8}\cong \text{Gal}(M/\mathbb{Q})\cong \langle r,s\rangle$ , with $r$ of order $4$ , $s$ of order $2$ , and $sr=r^{3}s$ . Hereinafter, we refer to the following field diagram.

By the Čebotarëv density theorem, for each $\unicode[STIX]{x1D70C}\in ({\mathcal{O}}_{M}/(\unicode[STIX]{x1D6E5}_{M}))^{\times }$ , we can choose an inverse $\unicode[STIX]{x1D70C}^{\prime }\in {\mathcal{O}}_{M}$ such that $\unicode[STIX]{x1D70C}^{\prime }{\mathcal{O}}_{M}$ is a prime of degree one. Fix a set of such $\unicode[STIX]{x1D70C}^{\prime }$ and call it ${\mathcal{R}}$ . Define $F$ to be the rational integer

(3.1) $$\begin{eqnarray}F=\unicode[STIX]{x1D6E5}_{M}\cdot \mathop{\prod }_{\unicode[STIX]{x1D70C}\in ({\mathcal{O}}_{M}/(\unicode[STIX]{x1D6E5}_{M}))^{\times }}\text{N}_{M/\mathbb{Q}}(\unicode[STIX]{x1D70C}^{\prime }).\end{eqnarray}$$

This is not really analogous to $F$ on [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, p. 723], but we denote it by the same letter because it will play an analogous role later on in the estimation of certain congruence sums.

3.2 Quadratic reciprocity

Let $L$ be a number field and let ${\mathcal{O}}_{L}$ be its ring of integers. We say that an ideal $\mathfrak{a}$ in ${\mathcal{O}}_{L}$ is odd if $\text{N}(\mathfrak{a})$ is odd; similarly, an element $\unicode[STIX]{x1D6FC}$ in ${\mathcal{O}}_{L}$ is called odd if the principal ideal generated by $\unicode[STIX]{x1D6FC}$ is odd. If $\mathfrak{p}$ is an odd prime ideal in ${\mathcal{O}}_{L}$ , and $\unicode[STIX]{x1D6FC}$ is an element in ${\mathcal{O}}_{L}$ , then one defines

$$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x1D6FC}}{\mathfrak{p}}\biggr)_{L}=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }\unicode[STIX]{x1D6FC}\in \mathfrak{p},\\ 1\quad & \text{if }\unicode[STIX]{x1D6FC}\notin \mathfrak{p}\text{ and }\unicode[STIX]{x1D6FC}\text{ is a square modulo }\mathfrak{p},\\ -1\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

If $\mathfrak{b}$ is an odd ideal in ${\mathcal{O}}_{L}$ , one defines

$$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x1D6FC}}{\mathfrak{b}}\biggr)_{L}=\mathop{\prod }_{\mathfrak{p}^{k_{\mathfrak{p}}}\Vert \mathfrak{b}}\biggl(\frac{\unicode[STIX]{x1D6FC}}{\mathfrak{p}}\biggr)_{L}^{k_{\mathfrak{p}}}.\end{eqnarray}$$

If $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in {\mathcal{O}}_{L}$ with $\unicode[STIX]{x1D6FD}$ odd, we define

$$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FD}}\biggr)_{L}=\biggl(\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FD}{\mathcal{O}}_{L}}\biggr)_{L}.\end{eqnarray}$$

A weak (but sufficient to us) version of the law of quadratic reciprocity for number fields can be stated as follows (see for instance [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, Lemma 2.1, p. 703]).

Lemma 3.1. Suppose $L$ is a totally complex number field, and let $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in {\mathcal{O}}_{L}$ be odd. Then

$$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FD}}\biggr)_{L}=\unicode[STIX]{x1D700}\cdot \biggl(\frac{\unicode[STIX]{x1D6FD}}{\unicode[STIX]{x1D6FC}}\biggr)_{L},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}\in \{\pm 1\}$ depends only on the congruence classes of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ modulo $8{\mathcal{O}}_{L}$ .◻

When $\unicode[STIX]{x1D6FC}$ is not odd, the following supplement to the law of quadratic reciprocity will suffice for our purposes (see [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, Proposition 2.2, p. 703]).

3.3 Short character sums

Here we state the conjecture that we assume in the proof of Theorem 1. It stipulates power-savings in short character (modulo $q$ ) sums of length $q^{1/8}$ and is essentially the same as the case $n=8$ of Conjecture  $C_{n}$ in [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, p. 738].

Conjecture 1. There exist absolute constants $\unicode[STIX]{x1D6FF}>0$ and $C>0$ such that if $\unicode[STIX]{x1D712}$ is a non-principal real-valued Dirichlet character modulo a squarefree integer $q>2$ and $N<q^{1/8}$ , then

$$\begin{eqnarray}\biggl|\mathop{\sum }_{M\leqslant n\leqslant M+N}\unicode[STIX]{x1D712}(n)\biggr|\leqslant Cq^{1/8-\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

for all integers $M$ .

We feel that Conjecture 1 is of a genuinely different nature than the arithmetic applications that follow. It is the oscillation of spins over the set of prime ideals that yields the various arithmetic applications. In the sieving methods we use, proving oscillation of spins over prime ideals requires us to first prove oscillation over the set of all ideals. There we encounter character sums in the number field $M$ that one wishes to relate to character sums in $\mathbb{Q}$ , where oscillation of character sums is better understood. In passing from $M$ to $\mathbb{Q}$ , one suffers from the fact that, in some fixed integral basis for ${\mathcal{O}}_{M}$ , a nicely chosen element of norm $X$ generally has coordinates of size $X^{1/8}$ . Conductors of characters in question have size similar to the norm, while the length of character sums in question is essentially limited by the size of the coordinates. We also remark that thanks to the work of Burgess [Reference BurgessBur62, Reference BurgessBur63], Conjecture 1 is known to be true when $1/8$ is replaced with any real number $\unicode[STIX]{x1D703}>1/4$ , in which case the exponent $\unicode[STIX]{x1D6FF}$ and the constant $C$ depend on  $\unicode[STIX]{x1D703}$ .

Instead of directly appealing to Conjecture 1, we will instead need a corollary of Conjecture 1 for arithmetic progressions. For $q$ odd and squarefree, let $\unicode[STIX]{x1D712}_{q}$ be the real Dirichlet character $(\frac{\cdot }{q})$ . Following [Reference Friedlander, Iwaniec, Mazur and RubinFIMR15, 7, pp. 924–925] we will prove the following corollary.

Corollary 7. Assume Conjecture 1. Then there exist absolute constants $\unicode[STIX]{x1D6FF}>0$ and $C>0$ such that for all odd squarefree integers $q>1$ , all integers $N<q^{1/8}$ , all integers $M$ , $l$ , and $k$ satisfying $q\nmid k$ we have

$$\begin{eqnarray}\biggl|\mathop{\sum }_{\substack{ M\leqslant n\leqslant M+N \\ n\equiv l\hspace{0.2em}{\rm mod}\hspace{0.2em}k}}\unicode[STIX]{x1D712}_{q}(n)\biggr|\leqslant Cq^{1/8-\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

Proof. Write $n=km+l$ . Then we have

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{q}(n)=\unicode[STIX]{x1D712}_{(q,k)}(l)\unicode[STIX]{x1D712}_{q/(q,k)}(k)\unicode[STIX]{x1D712}_{q/(q,k)}(m+r),\end{eqnarray}$$

where $r$ satisfies $kr\equiv l\hspace{0.2em}{\rm mod}\hspace{0.2em}q/(q,k)$ . It follows that

$$\begin{eqnarray}\biggl|\mathop{\sum }_{\substack{ M\leqslant n\leqslant M+N \\ n\equiv l\hspace{0.2em}{\rm mod}\hspace{0.2em}k}}\unicode[STIX]{x1D712}_{q}(n)\biggr|\leqslant \biggl|\mathop{\sum }_{M^{\prime }\leqslant m\leqslant M^{\prime }+(N/k)}\unicode[STIX]{x1D712}_{q/(q,k)}(m)\biggr|,\end{eqnarray}$$

where $M^{\prime }=(M-l)k^{-1}+r$ . By our assumption $q\nmid k$ , we see that $q/(q,k)$ is an odd squarefree integer greater than one. Hence $\unicode[STIX]{x1D712}_{q/(q,k)}$ is a non-principal real-valued Dirichlet character. Now apply Conjecture 1.◻

3.4 Vinogradov’s method, after Friedlander, Iwaniec, Mazur, and Rubin

Vinogradov’s method [Reference VinogradovVin47, Reference VinogradovVin54] has been substantially simplified by Vaughan [Reference VaughanVau77], and Friedlander et al. [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, §5, pp. 717–722] gave a nice generalization to number fields. Morally speaking, power-saving estimates in sums over primes follow from power-saving estimates in linear congruence sums (sums of type I) and general bilinear sums (sums of type II). Precisely, by [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, Proposition 5.2, p. 722] with $\unicode[STIX]{x1D717}=\unicode[STIX]{x1D6FF}/4$ and $\unicode[STIX]{x1D703}=1/48$ , Theorem 1 is a direct consequence of the following two propositions.

Proposition 3.3. Assume Conjecture 1 holds with $\unicode[STIX]{x1D6FF}>0$ . Then for all $\unicode[STIX]{x1D716}>0$ , we have

$$\begin{eqnarray}\mathop{\sum }_{\text{N}(\mathfrak{a})\leqslant x,~\mathfrak{m}|\mathfrak{a}}{s_{\mathfrak{a}}\ll }_{\unicode[STIX]{x1D716}}x^{1-\unicode[STIX]{x1D6FF}/4+\unicode[STIX]{x1D716}}\end{eqnarray}$$

uniformly for all non-zero ideals $\mathfrak{m}$ of ${\mathcal{O}}_{M}$ and all $x\geqslant 2$ .

Proposition 3.4. For each $\unicode[STIX]{x1D716}>0$ , there exists a constant $c_{\unicode[STIX]{x1D716}}>0$ such that

$$\begin{eqnarray}\mathop{\sum }_{\text{N}(\mathfrak{a})\leqslant M}\mathop{\sum }_{\text{N}(\mathfrak{b})\leqslant N}v_{\mathfrak{a}}w_{\mathfrak{b}}{s_{\mathfrak{a}\mathfrak{b}}\ll }_{\unicode[STIX]{x1D716}}(M+N)^{1/48}(MN)^{47/48+\unicode[STIX]{x1D716}}\end{eqnarray}$$

uniformly for all $M,N\geqslant 2$ and all sequences of complex numbers $\{v_{\mathfrak{a}}\}$ and $\{w_{\mathfrak{b}}\}$ satisfying $|v_{\mathfrak{a}}|,|w_{\mathfrak{a}}|\leqslant c_{\unicode[STIX]{x1D716}}\text{N}(\mathfrak{a})^{\unicode[STIX]{x1D716}}$ .

Note that Proposition 3.4 is unconditional – it is only for the sums of type I featuring in Proposition 3.3 that we have to assume Conjecture 1. The proof of Proposition 3.4 is rather standard at this point; similar results in slightly different settings can be found in [Reference Friedlander and IwaniecFI98, Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, Reference MilovicMil17b, Reference MilovicMil18, Reference Koymans and MilovicKM18], among others. The substantially more difficult proof of Proposition 3.3 requires us to make a genuine improvement to the argument of Friedlander et al. [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, §6].

3.5 A fundamental domain for the action of ${\mathcal{O}}_{M}^{\times }$

In the definition of $s_{\mathfrak{a}}$ in (1.3), we chose a generator $\unicode[STIX]{x1D6FC}$ for the ideal $\mathfrak{a}$ . As we will see in the proofs of Propositions 3.3 and 3.4, when summing over multiple ideals $\mathfrak{a}$ , it will be useful to work with a compatible set of generators. Here we present a suitable set of such generators, given by a standard fundamental domain for the action of ${\mathcal{O}}_{M}^{\times }$ on ${\mathcal{O}}_{M}$ .

Recall that ${\mathcal{O}}_{M}^{\times }=\langle \unicode[STIX]{x1D701}_{8}\rangle \times V$ , where $V$ is free of rank $3$ . The group $V$ acts on ${\mathcal{O}}_{M}$ by multiplication, i.e., there is an action

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:V\times {\mathcal{O}}_{M}\rightarrow {\mathcal{O}}_{M}\end{eqnarray}$$

given by $\unicode[STIX]{x1D6F9}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}$ . Up to units of finite order, the orbits of $\unicode[STIX]{x1D6F9}$ correspond to ideals in ${\mathcal{O}}_{M}$ .

Fix an integral basis for ${\mathcal{O}}_{M}$ , say $\unicode[STIX]{x1D702}=\{\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{8}\}$ . If $\unicode[STIX]{x1D6FC}=a_{1}\unicode[STIX]{x1D702}_{1}+\cdots +a_{8}\unicode[STIX]{x1D702}_{8}\in {\mathcal{O}}_{M}$ with $a_{i}\in \mathbb{Z}$ , we call $a_{i}$ the coordinates of $\unicode[STIX]{x1D6FC}$ in the basis $\unicode[STIX]{x1D702}$ . The ideal in ${\mathcal{O}}_{M}$ generated by $\unicode[STIX]{x1D6FC}$ is also generated by $\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}$ for any unit $\unicode[STIX]{x1D707}\in V$ . As $V$ is infinite, one can choose $\unicode[STIX]{x1D707}$ so that the coordinates of $\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}$ in the integral basis $\unicode[STIX]{x1D702}$ are arbitrarily large. The following classical result ensures that one can choose $\unicode[STIX]{x1D707}$ so that the coordinates of $\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}$ are reasonably small.

For a proof, see [Reference Koymans and MilovicKM18], based on [Reference LangLan86, Lemma 1, p. 131]. We are now ready to prove Propositions 3.3 and 3.4, thereby proving Theorem 1.

4.1 Sums of type I

In this section, we prove Proposition 3.3. Define $F$ as in (3.1). We recall that we fixed a rank $3$ subgroup $V$ of ${\mathcal{O}}_{M}$ and a set of representatives $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{8}$ for $V/V^{2}$ . Let $\mathfrak{m}$ be an ideal of ${\mathcal{O}}_{M}$ coprime with $F$ . Recall the definition of $s_{\mathfrak{a}}$ in (1.3). After using Lemma 3.5 to transform a sum over ideals in ${\mathcal{O}}_{M}$ to a sum over elements in the fundamental domain ${\mathcal{D}}$ , our goal becomes to bound the following sum

$$\begin{eqnarray}A(x)=\frac{1}{64}\mathop{\sum }_{\substack{ \text{N}(\mathfrak{a})\leqslant x \\ (\mathfrak{a},F)=1,\mathfrak{m}\mid \mathfrak{a}}}\mathop{\sum }_{i=1}^{8}\mathop{\sum }_{j=1}^{8}[\unicode[STIX]{x1D707}_{i}\unicode[STIX]{x1D701}_{8}^{j}\unicode[STIX]{x1D6FC}]=\frac{1}{64}\mathop{\sum }_{i=1}^{8}\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}\in {\mathcal{D}};\text{N}(\unicode[STIX]{x1D6FC})\leqslant x \\ (\unicode[STIX]{x1D6FC},F)=1,\mathfrak{m}\mid \unicode[STIX]{x1D6FC}}}[\unicode[STIX]{x1D707}_{i}\unicode[STIX]{x1D6FC}],\end{eqnarray}$$

where, for convenience of notation, we have set $[\unicode[STIX]{x1D6FD}]=\unicode[STIX]{x1D713}(\unicode[STIX]{x1D6FD}\hspace{0.2em}{\rm mod}\hspace{0.2em}F)[\unicode[STIX]{x1D6FD}]_{r}$ for $\unicode[STIX]{x1D6FD}\in {\mathcal{O}}_{M}$ . The rough strategy of our proof will be the same as the strategy in [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, §6], although we will have to make the appropriate adjustments in numerous places. We can simplify several steps thanks to the special properties of the field $M$ as described in § 3.1. At some point, however, the strategy of [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, §6] will no longer suffice, and we will need a new ingredient.

By making changes of variables $\unicode[STIX]{x1D6FC}\mapsto \unicode[STIX]{x1D707}_{i}^{-1}\unicode[STIX]{x1D6FC}$ , we rewrite the sum above as

$$\begin{eqnarray}A(x)=\frac{1}{64}\mathop{\sum }_{i=1}^{8}\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}};\text{N}(\unicode[STIX]{x1D6FC})\leqslant x \\ (\unicode[STIX]{x1D6FC},F)=1,\mathfrak{m}\mid \unicode[STIX]{x1D6FC}}}[\unicode[STIX]{x1D6FC}]\end{eqnarray}$$

and after splitting the sum into congruence classes modulo $F$ , we get

$$\begin{eqnarray}A(x)=\frac{1}{64}\mathop{\sum }_{i=1}^{8}\mathop{\sum }_{\substack{ \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F; \\ (\unicode[STIX]{x1D70C},F)=1}}\unicode[STIX]{x1D713}(\unicode[STIX]{x1D70C})A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}),\end{eqnarray}$$

where

$$\begin{eqnarray}A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}};\text{N}(\unicode[STIX]{x1D6FC})\leqslant x \\ \unicode[STIX]{x1D6FC}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ \unicode[STIX]{x1D6FC}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}}}[\unicode[STIX]{x1D6FC}]_{r}.\end{eqnarray}$$

Our goal is to estimate $A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ for each congruence class $\unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F$ , $(\unicode[STIX]{x1D70C},F)=1$ and unit $\unicode[STIX]{x1D707}_{i}$ . As a $\mathbb{Z}$ -module, the ring ${\mathcal{O}}_{M}$ decomposes as ${\mathcal{O}}_{M}=\mathbb{Z}\oplus \mathbb{M}$ , where $\mathbb{M}$ is a free $\mathbb{Z}$ -module of rank  $7$ , so that we can write

$$\begin{eqnarray}\mathbb{M}=\unicode[STIX]{x1D714}_{2}\mathbb{Z}+\cdots +\unicode[STIX]{x1D714}_{8}\mathbb{Z}\end{eqnarray}$$

for some $\unicode[STIX]{x1D714}_{2},\ldots ,\unicode[STIX]{x1D714}_{8}\in {\mathcal{O}}_{M}$ . This means that $\unicode[STIX]{x1D6FC}$ can be written uniquely as

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=a+\unicode[STIX]{x1D6FD}\quad \text{with }a\in \mathbb{Z},\unicode[STIX]{x1D6FD}\in \mathbb{M},\end{eqnarray}$$

so the four summation conditions above are equivalent to

$$\begin{eqnarray}a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}},\quad \text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x,\quad a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F,\quad a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}.\end{eqnarray}$$

Part 3 of Lemma 3.5 implies that the conjugates of $\unicode[STIX]{x1D6FD}$ , say $\unicode[STIX]{x1D6FD}^{(i)}$ for $1\leqslant i\leqslant 8$ , satisfy $|\unicode[STIX]{x1D6FD}^{(i)}|\ll x^{1/8}$ for any embedding $M{\hookrightarrow}\mathbb{C}$ . Because our field $M$ and the integral basis $\{1,\unicode[STIX]{x1D714}_{2},\ldots ,\unicode[STIX]{x1D714}_{8}\}$ is fixed, the implied constant is absolute.

Perhaps the main step of [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, §6] is a trick on p. 725, which we use to rewrite $[\unicode[STIX]{x1D6FC}]_{r}=(\frac{r(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FC}})$ as

$$\begin{eqnarray}\biggl(\frac{r(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FC}}\biggr)=\biggl(\frac{r(a+\unicode[STIX]{x1D6FD})}{a+\unicode[STIX]{x1D6FD}}\biggr)=\biggl(\frac{r(\unicode[STIX]{x1D6FD})-\unicode[STIX]{x1D6FD}}{a+\unicode[STIX]{x1D6FD}}\biggr).\end{eqnarray}$$

Morally speaking, this allows us to fix $\unicode[STIX]{x1D6FD}$ and vary $a$ , thereby creating a genuine character sum in which the variable of summation does not depend on the conductor of the character. If $\unicode[STIX]{x1D6FD}=r(\unicode[STIX]{x1D6FD})$ , then $\unicode[STIX]{x1D6FD}$ does not contribute to the sum. So we can and will assume $\unicode[STIX]{x1D6FD}\neq r(\unicode[STIX]{x1D6FD})$ . By property (P1) in § 3.1, we can write

$$\begin{eqnarray}r(\unicode[STIX]{x1D6FD})-\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D702}^{2}c_{0}c\end{eqnarray}$$

with $c_{0},c,\unicode[STIX]{x1D702}\in {\mathcal{O}}_{M}$ , $c_{0}\mid F$ squarefree, $\unicode[STIX]{x1D702}\mid F^{\infty }$ , and $(c,F)=1$ . Then

$$\begin{eqnarray}\biggl(\frac{r(\unicode[STIX]{x1D6FD})-\unicode[STIX]{x1D6FD}}{a+\unicode[STIX]{x1D6FD}}\biggr)=\biggl(\frac{\unicode[STIX]{x1D702}^{2}c_{0}c}{a+\unicode[STIX]{x1D6FD}}\biggr)=\biggl(\frac{c_{0}c}{a+\unicode[STIX]{x1D6FD}}\biggr)=\biggl(\frac{c_{0}}{a+\unicode[STIX]{x1D6FD}}\biggr)\biggl(\frac{c}{a+\unicode[STIX]{x1D6FD}}\biggr).\end{eqnarray}$$

By Lemma 3.2, the factor $(\frac{c_{0}}{a+\unicode[STIX]{x1D6FD}})$ depends only on the congruence class of $a+\unicode[STIX]{x1D6FD}$ modulo  $8c_{0}$ , and, as $c_{0}$ is squarefree and divides $F$ , it depends only on $\unicode[STIX]{x1D70C}$ .

Next we claim that

$$\begin{eqnarray}\biggl(\frac{c}{a+\unicode[STIX]{x1D6FD}}\biggr)=\unicode[STIX]{x1D700}_{1}\cdot \biggl(\frac{a+\unicode[STIX]{x1D6FD}}{c}\biggr),\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{1}\in \{\pm 1\}$ depends only on $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D6FD}$ . Indeed, $\unicode[STIX]{x1D70C}$ determines the congruence class of $a+\unicode[STIX]{x1D6FD}$ modulo $8$ and $c$ depends only on $\unicode[STIX]{x1D6FD}$ , so an application of Lemma 3.1 proves the claim. Combining everything gives

$$\begin{eqnarray}\biggl(\frac{r(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FC}}\biggr)=\unicode[STIX]{x1D700}_{2}\cdot \biggl(\frac{a+\unicode[STIX]{x1D6FD}}{c}\biggr),\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{2}=\unicode[STIX]{x1D700}_{2}(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FD})\in \{\pm 1\}$ depends only on $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D6FD}$ . Having rewritten $(\frac{r(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FC}})$ in a desirable form, we can now split $A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ as follows

$$\begin{eqnarray}\displaystyle A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}) & = & \displaystyle \mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}};\text{N}(\unicode[STIX]{x1D6FC})\leqslant x \\ \unicode[STIX]{x1D6FC}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ \unicode[STIX]{x1D6FC}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}}}\biggl(\frac{r(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FC}}\biggr)=\mathop{\sum }_{\substack{ a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}};\text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x \\ a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}}}\biggl(\frac{r(a+\unicode[STIX]{x1D6FD})}{a+\unicode[STIX]{x1D6FD}}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \mathbb{M}}\mathop{\sum }_{\substack{ a\in \mathbb{Z}; \\ a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}},\text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x \\ a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}}}\biggl(\frac{r(a+\unicode[STIX]{x1D6FD})}{a+\unicode[STIX]{x1D6FD}}\biggr)=\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \mathbb{M}}\mathop{\sum }_{\substack{ a\in \mathbb{Z}; \\ a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}};\text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x \\ a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}}}\unicode[STIX]{x1D700}_{2}(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FD})\biggl(\frac{a+\unicode[STIX]{x1D6FD}}{c}\biggr)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \mathbb{M}}|T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})|,\nonumber\end{eqnarray}$$

where $T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ is defined as

$$\begin{eqnarray}T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})=\mathop{\sum }_{\substack{ a\in \mathbb{Z}; \\ a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}},\text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x \\ a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}}}\biggl(\frac{a+\unicode[STIX]{x1D6FD}}{c}\biggr).\end{eqnarray}$$

From now on we treat $\unicode[STIX]{x1D6FD}$ as fixed and estimate $T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ . Recall that $c$ is odd and hence no ramified prime can divide the ideal $(c)=c{\mathcal{O}}_{M}$ by property (P2) in § 3.1. This implies that $(c)$ can be factored as

$$\begin{eqnarray}(c)=\mathfrak{g}\mathfrak{q},\end{eqnarray}$$

where, similarly as in [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13, (6.21), p. 727], $\mathfrak{g}$ consists of all prime ideals dividing $(c)$ that are of degree greater than one or unramified primes of degree one for which some conjugate is also a factor of $(c)$ . By construction $\mathfrak{q}$ consists of all the remaining primes dividing $c{\mathcal{O}}_{M}$ . Then $q:=N\mathfrak{q}$ is a squarefree integer and $g:=N\mathfrak{g}$ is a squarefull number coprime with $q$ . There exists a rational integer $b$ with $b\equiv \unicode[STIX]{x1D6FD}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{q}$ by an application of the Chinese remainder theorem. Again, as $c$ depends on $\unicode[STIX]{x1D6FD}$ and not on $a$ , so also $b$ is a rational integer that depends on $\unicode[STIX]{x1D6FD}$ and not on $a$ . We get

$$\begin{eqnarray}\biggl(\frac{a+\unicode[STIX]{x1D6FD}}{c}\biggr)=\biggl(\frac{a+\unicode[STIX]{x1D6FD}}{\mathfrak{g}}\biggr)\biggl(\frac{a+\unicode[STIX]{x1D6FD}}{\mathfrak{q}}\biggr)=\biggl(\frac{a+\unicode[STIX]{x1D6FD}}{\mathfrak{g}}\biggr)\biggl(\frac{a+b}{\mathfrak{q}}\biggr).\end{eqnarray}$$

Define $g_{0}$ as the radical of $g$ , i.e.,

$$\begin{eqnarray}g_{0}=\mathop{\prod }_{p\mid g}p.\end{eqnarray}$$

Note that the quadratic residue symbol $(\frac{\unicode[STIX]{x1D6FC}}{\mathfrak{g}})$ is periodic in $\unicode[STIX]{x1D6FC}$ modulo $\mathfrak{g}^{\ast }=\prod _{\mathfrak{p}\mid \mathfrak{g}}\mathfrak{p}$ . Since $\mathfrak{g}^{\ast }$ divides $g_{0}$ , we conclude that the symbol $(\frac{a+\unicode[STIX]{x1D6FD}}{\mathfrak{g}})$ is periodic of period $g_{0}$ as a function of $a\in \mathbb{Z}$ . We split $T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ into congruence classes modulo $g_{0}$ , giving

(4.1) $$\begin{eqnarray}|T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})|\leqslant \mathop{\sum }_{a_{0}\hspace{0.2em}{\rm mod}\hspace{0.2em}g_{0}}|T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i},a_{0})|,\end{eqnarray}$$

where

$$\begin{eqnarray}T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i},a_{0})=\mathop{\sum }_{\substack{ a\in \mathbb{Z}; \\ a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}},\text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x \\ a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F \\ a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m} \\ a\equiv a_{0}\hspace{0.2em}{\rm mod}\hspace{0.2em}g_{0}}}\biggl(\frac{a+b}{\mathfrak{q}}\biggr).\end{eqnarray}$$

Note that $a+\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}}$ implies that $a\ll x^{1/8}$ , where the implied constant depends only on one of the eight units $\unicode[STIX]{x1D707}_{i}$ . The condition $\text{N}(a+\unicode[STIX]{x1D6FD})\leqslant x$ for fixed $\unicode[STIX]{x1D6FD}$ and $x$ is a polynomial inequality of degree $8$ in $a$ . So the summation variable $a\in \mathbb{Z}$ runs over a collection of at most $8$ intervals whose endpoints depend on $\unicode[STIX]{x1D6FD}$ and $x$ . But from $a\ll x^{1/8}$ we see that for the length $L$ of each such interval we have $L\ll x^{1/8}$ .

Furthermore, the congruences $a+\unicode[STIX]{x1D6FD}\equiv \unicode[STIX]{x1D70C}\hspace{0.2em}{\rm mod}\hspace{0.2em}F$ , $a+\unicode[STIX]{x1D6FD}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{m}$ and $a\equiv a_{0}\hspace{0.2em}{\rm mod}\hspace{0.2em}g_{0}$ mean that $a$ runs over a certain arithmetic progression of modulus $k$ , which divides $g_{0}mF$ , where $m:=\text{N}\mathfrak{m}$ . Hence, we see that the inner sum in (4.1) can be rewritten as at most $8$ sums, each of which runs over an arithmetic progression of modulus $k$ in a single segment of length $\ll x^{1/8}$ .

As $q=\text{N}(\mathfrak{q})$ is squarefree, $(\frac{\cdot }{\mathfrak{q}})$ is the real primitive Dirichlet character of modulus $q$ , and hence we have at most $8$ incomplete character sums of length $\ll x^{1/8}$ and modulus $q\ll x$ . When the modulus $q$ of the Dirichlet character divides the modulus $k$ of the arithmetic progression, one can not expect to get cancellation. For now we assume that $q\nmid k$ , and we will deal with the case $q\mid k$ later on. Corollary 7 implies that

$$\begin{eqnarray}T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i},a_{0})\ll x^{1/8-\unicode[STIX]{x1D6FF}},\end{eqnarray}$$

and hence that

(4.2) $$\begin{eqnarray}T(x;\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll g_{0}x^{1/8-\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

Just as in [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13], the implied constant above does not depend on $\unicode[STIX]{x1D6FD}$ because Conjecture 1, and so also Corollary 7, encompasses all incomplete character sums of length $\ll x^{1/8}$ , regardless of the endpoints of the interval being summed over.

We still need to deal with the case $q\mid k$ . Certainly, this implies $q\mid m$ . So (4.2) holds if $q\nmid m$ . Hence, by the definition of $(c)$ and the factorization $(c)=\mathfrak{g}\mathfrak{q}$ , we have (4.2) unless

(4.3) $$\begin{eqnarray}p\mid \text{N}(\unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC}))\;\Longrightarrow \;p^{2}\mid mF\text{N}(\unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC})).\end{eqnarray}$$

We write $A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ for the contribution to $A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ with (4.3). We have

$$\begin{eqnarray}A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\leqslant |\{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}}:N\unicode[STIX]{x1D6FC}\leqslant x,~p\mid \text{N}(\unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC}))\;\Longrightarrow \;p^{2}\mid mF\text{N}(\unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC}))\}|.\end{eqnarray}$$

Decompose ${\mathcal{O}}_{M}$ as

$$\begin{eqnarray}{\mathcal{O}}_{M}=\mathbb{Z}[\sqrt{-2}]\oplus \mathbb{M}^{\prime },\end{eqnarray}$$

where $\mathbb{M}^{\prime }$ is a free $\mathbb{Z}$ -module of rank $6$ . Then we get an injective map $\mathbb{M}^{\prime }\rightarrow {\mathcal{O}}_{M}$ given by $\unicode[STIX]{x1D6FC}\mapsto \unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC})$ . Since $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D707}_{i}{\mathcal{D}}$ and $\text{N}(\unicode[STIX]{x1D6FC})\leqslant x$ , we know that all the conjugates $|\unicode[STIX]{x1D6FC}^{(k)}|$ are $\ll x^{1/8}$ . If we write

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=a+b\sqrt{-2}+m^{\prime }\end{eqnarray}$$

with $a,b\in \mathbb{Z}$ , and $m^{\prime }\in \mathbb{M}^{\prime }$ , then it follows that $|a|,|b|\leqslant y$ and furthermore all the conjugates of $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC})$ satisfy $|\unicode[STIX]{x1D6FE}^{(k)}|\leqslant y$ for some $y\asymp x^{1/8}$ . Therefore, we have

$$\begin{eqnarray}A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\leqslant y^{2}|\{\unicode[STIX]{x1D6FE}\in {\mathcal{O}}_{M}:|\unicode[STIX]{x1D6FE}^{(k)}|\leqslant y,~p\mid \text{N}(\unicode[STIX]{x1D6FE})\;\Longrightarrow \;p^{2}\mid mF\text{N}(\unicode[STIX]{x1D6FE})\}|.\end{eqnarray}$$

Since it is easier to count ideals than integers, we replace $\unicode[STIX]{x1D6FE}$ by the principal ideal it generates. We remark that an ideal $\mathfrak{b}$ with $\text{N}\mathfrak{b}\leqslant y^{8}$ has $\ll (\log y)^{8}$ generators satisfying $|\unicode[STIX]{x1D6FE}^{(k)}|\leqslant y$ for all $k$ . Hence

$$\begin{eqnarray}A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll x^{1/4}(\log x)^{8}|\{\mathfrak{b}\subseteq {\mathcal{O}}_{M}:\text{N}\mathfrak{b}\leqslant y^{8},~p\mid \text{N}\mathfrak{b}\;\Longrightarrow \;p^{2}\mid mF\text{N}\mathfrak{b}\}|.\end{eqnarray}$$

Now we can use the multiplicative structure of the ideals in ${\mathcal{O}}_{M}$ , giving the bound

$$\begin{eqnarray}A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll x^{1/4}(\log x)^{8}\mathop{\sum }_{\substack{ b\leqslant y^{8} \\ p\mid b\;\Longrightarrow \;p^{2}\mid mFb}}\unicode[STIX]{x1D70F}(b),\end{eqnarray}$$

where $b$ runs over the positive rational integers and $\unicode[STIX]{x1D70F}(b)$ counts the number of ideals in $M$ with norm $b$ . Then we have $\unicode[STIX]{x1D70F}(b)\ll b^{\unicode[STIX]{x1D716}}$ . Note that we can assume $m\leqslant x$ because otherwise $A(x)$ is the empty sum. Hence, recalling that $y\asymp x^{1/8}$ , we conclude that

$$\begin{eqnarray}A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll x^{3/4+\unicode[STIX]{x1D716}},\end{eqnarray}$$

where the implied constant depends only on $\unicode[STIX]{x1D716}$ .

Define $A_{0}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ to be the contribution of $A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ of the terms $\unicode[STIX]{x1D6FC}=a+\unicode[STIX]{x1D6FD}$ not satisfying (4.3). We have

$$\begin{eqnarray}A(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})=A_{\Box }(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})+A_{0}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}).\end{eqnarray}$$

To estimate $A_{0}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ we can use (4.2) for every relevant $\unicode[STIX]{x1D6FD}$ . Unfortunately, the bound (4.2) is only good when $g_{0}$ is small. So we make the further partition

$$\begin{eqnarray}A_{0}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})=A_{1}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})+A_{2}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}),\end{eqnarray}$$

where the components run over $\unicode[STIX]{x1D6FC}=a+\unicode[STIX]{x1D6FD}$ with $\unicode[STIX]{x1D6FD}$ satisfying

$$\begin{eqnarray}\displaystyle & \displaystyle g_{0}\leqslant Z\quad \text{in the sum }A_{1}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle g_{0}>Z\quad \text{in the sum }A_{2}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}). & \displaystyle \nonumber\end{eqnarray}$$

Here $Z$ is at our disposal and we choose it later. It is here that we must improve on the bounds of [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13]. In their proof they define three sums

$$\begin{eqnarray}\displaystyle \displaystyle g_{0}\leqslant Z\quad & \text{in the sum }A_{1}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}), & \displaystyle \nonumber\\ \displaystyle \displaystyle g_{0}>Z,g\leqslant Y\quad & \text{in the sum }A_{2}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}), & \displaystyle \nonumber\\ \displaystyle \displaystyle g_{0}>Z,g>Y\quad & \text{in the sum }A_{3}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i}), & \displaystyle \nonumber\end{eqnarray}$$

with $Z\leqslant Y$ at their disposal. Following the proof in [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13] would give

$$\begin{eqnarray}A_{0}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll x^{\unicode[STIX]{x1D716}}(Zx^{1-\unicode[STIX]{x1D6FF}}+Y^{-1/2}x^{1+1/4}+Z^{-1}\log Yx+Y^{5/2}x^{1/4}),\end{eqnarray}$$

and it is easily seen that there is no choice of $Z\leqslant Y$ that makes $A_{0}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll x^{1-\unicode[STIX]{x1D703}_{1}}$ for some $\unicode[STIX]{x1D703}_{1}>0$ . Our proof is conceptually simpler and provides sharper bounds.

We estimate $A_{1}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ as in [Reference Friedlander, Iwaniec, Mazur and RubinFIMR13] by using (4.2) and summing over $\unicode[STIX]{x1D6FD}\in \mathbb{M}$ satisfying $|\unicode[STIX]{x1D6FD}^{(1)}|,\ldots ,|\unicode[STIX]{x1D6FD}^{(8)}|\ll x^{1/8}$ to obtain

$$\begin{eqnarray}A_{1}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})\ll Zx^{1-\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

Our next goal is to estimate $A_{2}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})$ . We keep the condition $\unicode[STIX]{x1D6FC}-r(\unicode[STIX]{x1D6FC})\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{g}$ , giving

(4.4) $$\begin{eqnarray}|A_{2}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})|\leqslant y^{2}\mathop{\sum }_{\substack{ \mathfrak{g} \\ g_{0}>Z}}E_{\mathfrak{g}}(y),\end{eqnarray}$$

where $y\asymp x^{1/8}$ and

$$\begin{eqnarray}E_{\mathfrak{g}}(y):=|\{\unicode[STIX]{x1D6FE}\in \mathbb{M}^{\prime \prime }:\unicode[STIX]{x1D6FE}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{g},|\unicode[STIX]{x1D6FE}^{(k)}|\leqslant y\text{ for all }k\}|.\end{eqnarray}$$

Here $\mathbb{M}^{\prime \prime }$ is by definition the image of $\mathbb{M}^{\prime }$ under the map $\unicode[STIX]{x1D6FD}\mapsto \unicode[STIX]{x1D6FD}-r(\unicode[STIX]{x1D6FD})$ . Let $\unicode[STIX]{x1D702}_{3},\ldots ,\unicode[STIX]{x1D702}_{8}$ be a $\mathbb{Z}$ -basis of $\mathbb{M}^{\prime \prime }$ . We view $\mathbb{M}^{\prime \prime }\subseteq \mathbb{R}^{6}$ via $a_{3}\unicode[STIX]{x1D702}_{3}+\cdots +a_{8}\unicode[STIX]{x1D702}_{8}\mapsto (a_{3},\ldots ,a_{8})$ . In this way, we identify $\mathbb{M}^{\prime \prime }$ with $\mathbb{Z}^{6}$ , so $\mathbb{M}^{\prime \prime }$ becomes a lattice in $\mathbb{R}^{6}$ . Furthermore, define $\unicode[STIX]{x1D6EC}_{\mathfrak{g}}$ as

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathfrak{g}}:=\{\unicode[STIX]{x1D6FE}\in \mathbb{M}^{\prime \prime }:\unicode[STIX]{x1D6FE}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{g}\}.\end{eqnarray}$$

Then it is easily seen that $\unicode[STIX]{x1D6EC}_{\mathfrak{g}}$ is a sublattice of $\mathbb{M}^{\prime \prime }$ .

We further define

$$\begin{eqnarray}S_{x}=\{(a_{3},\ldots ,a_{8})\in \mathbb{R}^{6}:|a_{i}|\leqslant c_{1}x^{1/8}\},\end{eqnarray}$$

where the constant $c_{1}>0$ is taken large enough such that

(4.5) $$\begin{eqnarray}E_{\mathfrak{g}}(y)\leqslant |S_{x}\cap \unicode[STIX]{x1D6EC}_{\mathfrak{g}}|.\end{eqnarray}$$

Note that $S_{x}=x^{1/8}S_{1}$ , which implies that $\text{Vol}(S_{x})=x^{3/4}\text{Vol}(S_{1})$ . Because $S_{1}$ is a $6$ -dimensional hypercube, it has $12$ sides. Hence, there exist an absolute constant $L$ and functions $\unicode[STIX]{x1D711}_{1},\ldots ,\unicode[STIX]{x1D711}_{12}:[0,1]^{5}\rightarrow \mathbb{R}^{6}$ satisfying a Lipschitz condition

$$\begin{eqnarray}|\unicode[STIX]{x1D711}_{i}(a)-\unicode[STIX]{x1D711}_{i}(b)|\leqslant L|a-b|\end{eqnarray}$$

for $a,b\in [0,1]^{5}$ , $i=1,\ldots ,12$ such that the boundary of $S_{1}$ , denoted by $\unicode[STIX]{x2202}S_{1}$ , is covered by the images of the $\unicode[STIX]{x1D711}_{i}$ . Then $x^{1/8}\unicode[STIX]{x1D711}_{1},\ldots ,x^{1/8}\unicode[STIX]{x1D711}_{12}$ are Lipschitz functions for $\unicode[STIX]{x2202}S_{x}=\unicode[STIX]{x2202}x^{1/8}S_{1}=x^{1/8}\unicode[STIX]{x2202}S_{1}$ . Hence, we can choose $x^{1/8}L$ as the Lipschitz constant for $S_{x}$ .

We now apply [Reference WidmerWid10, Theorem 5.4], which gives

(4.6) $$\begin{eqnarray}\biggl||S_{x}\cap \unicode[STIX]{x1D6EC}_{\mathfrak{g}}|-\frac{\text{Vol}(S_{x})}{\det \unicode[STIX]{x1D6EC}_{\mathfrak{g}}}\biggr|\ll _{L}\max _{0\leqslant i<6}\frac{x^{i/8}}{\unicode[STIX]{x1D706}_{\mathfrak{g},1}\cdot \ldots \cdot \unicode[STIX]{x1D706}_{\mathfrak{g},i}},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{\mathfrak{g},1},\ldots ,\unicode[STIX]{x1D706}_{\mathfrak{g},6}$ are the successive minima of $\unicode[STIX]{x1D6EC}_{\mathfrak{g}}$ and $\ll _{L}$ means that the implied constant may depend on $L$ . Our next goal is to give a lower bound for $\unicode[STIX]{x1D706}_{\mathfrak{g},1}$ .

So let $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6EC}_{\mathfrak{g}}$ be non-zero. Then $\mathfrak{g}\mid \unicode[STIX]{x1D6FE}$ and hence $g\mid \text{N}(\unicode[STIX]{x1D6FE})$ . Write $\unicode[STIX]{x1D6FE}=(a_{3},\ldots ,a_{8})$ . We fix some small $\unicode[STIX]{x1D716}>0$ . If $a_{3},\ldots ,a_{8}\leqslant c_{2}g^{1/8-\unicode[STIX]{x1D716}}$ for some sufficiently small absolute constant $c_{2}>0$ , we obtain $\text{N}(\unicode[STIX]{x1D6FE})<g$ . Since $g\mid \text{N}(\unicode[STIX]{x1D6FE})$ , we conclude that $\text{N}(\unicode[STIX]{x1D6FE})=0$ , contradiction. Hence there is an $i$  with $a_{i}>c_{2}g^{1/8-\unicode[STIX]{x1D716}}$ . This implies that the length of $\unicode[STIX]{x1D6FE}$ satisfies $\Vert \unicode[STIX]{x1D6FE}\Vert \gg g^{1/8-\unicode[STIX]{x1D716}}$ and therefore

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{\mathfrak{g},1}\gg g^{1/8-\unicode[STIX]{x1D716}}.\end{eqnarray}$$

By Minkowski’s second theorem and (4.7) we find that

(4.8) $$\begin{eqnarray}\det \unicode[STIX]{x1D6EC}_{\mathfrak{g}}\gg g^{3/4-6\unicode[STIX]{x1D716}}.\end{eqnarray}$$

Combining (4.6)–(4.8) gives

(4.9) $$\begin{eqnarray}|S_{x}\cap \unicode[STIX]{x1D6EC}_{\mathfrak{g}}|\ll \frac{x^{3/4}}{g^{3/4-6\unicode[STIX]{x1D716}}}+\frac{x^{5/8}}{g^{5/8-5\unicode[STIX]{x1D716}}}\ll \frac{x^{3/4}}{g^{3/4-6\unicode[STIX]{x1D716}}}.\end{eqnarray}$$

Plugging (4.5) and (4.9) back into (4.4) gives

$$\begin{eqnarray}|A_{2}(x;\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707}_{i})|\leqslant y^{2}\mathop{\sum }_{\substack{ \mathfrak{g} \\ g_{0}>Z}}E_{\mathfrak{g}}(y)\leqslant y^{2}\mathop{\sum }_{\substack{ \mathfrak{g} \\ g_{0}>Z}}|S_{x}\cap \unicode[STIX]{x1D6EC}_{\mathfrak{g}}|\ll \mathop{\sum }_{\substack{ \mathfrak{g} \\ g_{0}>Z}}\frac{x}{g^{3/4-6\unicode[STIX]{x1D716}}}.\end{eqnarray}$$

We rewrite the last sum as

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ \mathfrak{g} \\ g_{0}>Z}}\frac{x}{g^{3/4-6\unicode[STIX]{x1D716}}} & = & \displaystyle x\mathop{\sum }_{\substack{ g\leqslant x \\ g\text{ squarefull} \\ g_{0}>Z}}\frac{\unicode[STIX]{x1D70F}(g)}{g^{3/4-6\unicode[STIX]{x1D716}}}\ll x^{1+\unicode[STIX]{x1D716}^{\prime }}\mathop{\sum }_{\substack{ g\leqslant x \\ g\text{ squarefull} \\ g_{0}>Z}}\frac{1}{g^{3/4-6\unicode[STIX]{x1D716}}}\nonumber\\ \displaystyle & = & \displaystyle x^{1+\unicode[STIX]{x1D716}^{\prime }}\mathop{\sum }_{\substack{ g\leqslant x \\ g\text{ squarefull} \\ g_{0}>Z}}g^{-1/4+6\unicode[STIX]{x1D716}}\frac{1}{g^{1/2}}\leqslant x^{1+\unicode[STIX]{x1D716}^{\prime }}Z^{-1/2+3\unicode[STIX]{x1D716}}\mathop{\sum }_{\substack{ g\leqslant x \\ g\text{ squarefull} \\ g_{0}>Z}}\frac{1}{g^{1/2}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle x^{1+\unicode[STIX]{x1D716}^{\prime }}Z^{-1/2+3\unicode[STIX]{x1D716}}\mathop{\sum }_{\substack{ g\leqslant x \\ g\text{ squarefull}}}\frac{1}{g^{1/2}}\ll x^{1+\unicode[STIX]{x1D716}^{\prime }}Z^{-1/2+3\unicode[STIX]{x1D716}}\log x.\nonumber\end{eqnarray}$$

By picking $Z=X^{\unicode[STIX]{x1D6FF}/2}$ , $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D716}^{\prime }$ sufficiently small, we get the desired result with $\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D6FF}/4$ .