First Moment of Hecke $L$-functions with quartic characters at the central point

TL;DR

This paper investigates the average behavior of Hecke L-functions at the central point for quartic characters, providing insights into their distribution and value distribution.

Contribution

It introduces a new analysis of the first moment of Hecke L-functions with quartic characters at the central point, advancing understanding of their value distribution.

Findings

Derived explicit formulas for the first moment

Identified non-vanishing results at the central point

Provided bounds on the average size of these L-values

Abstract

In this paper, we study the first moment of central values of Hecke $L$-functions associated with quartic characters.

Equations

$$L(s,{\chi }_c)=\sum _{0\ne \mathcal{A}\subset {\mathcal{O}}_K}{\chi }_c(\mathcal{A})(N(\mathcal{A}){)}^{-s},$$

$$\Lambda (s,{\chi }_c)=W({\chi }_c)(N(c){)}^{-1/2}\Lambda (1-s,{\overline{\chi }}_c),$$

$$W(\chi_{c})=\sum_{a\in\mathcal{O}_{K}/(c)}\chi_{c}(a)e\Big{(}\text{Tr}\Big{(}\frac{a}{\delta c}\Big{)}\Big{)},$$

$$\Lambda (s,{\chi }_c)=(|D_K|N(c){)}^{s/2}(2\pi {)}^{-s}\Gamma (s)L(s,{\chi }_c),$$

$$\text{sideset}{}^{\ast }{\sum }_{c\equiv 1(\mathrm{mod}16)}L\left(\frac{1}{2},{\chi }_c\right)\exp \left(-\frac{N(c)}{y}\right)=Ay+O_{\epsilon }(y^{13/14+\epsilon }).$$

$${\left(\frac{m}{n}\right)}_4={\left(\frac{n}{m}\right)}_4(-1{)}^{((N(n)-1)/4)((N(m)-1)/4)}.$$

$$g(n)=\sum _{x\mathrm{mod}n}{\left(\frac{x}{n}\right)}_4\tilde{e}\left(\frac{x}{n}\right)\text{mbox}and{g}_2(n)=\sum _{x\mathrm{mod}n}{\left(\frac{x}{n}\right)}_4^2\tilde{e}\left(\frac{x}{n}\right),$$

$$|g(n)|$$

$$g(r,n)=\sum _{x\mathrm{mod}n}{\left(\frac{x}{n}\right)}_4\tilde{e}\left(\frac{rx}{n}\right).$$

$$g(rs,n)$$

$$g(r,n_1{n}_2)$$

$$g({\pi }^k,{\pi }^l)$$

$${\left(\frac{i}{n}\right)}_4=i^{(1-a)/2}\text{mbox}and{\left(\frac{1+i}{n}\right)}_4=i^{(a-b-1-b^2)/4}.$$

$${\chi }_c(1+i)=1.$$

$$\frac{1}{2\pi i}\underset{(2)}{\int }(2\pi {)}^{-(s+1/2)}\Gamma \left(s+\frac{1}{2}\right)L\left(s+\frac{1}{2},{\chi }_c\right)x^s\frac{\mathrm{d}s}{s}$$

$$\begin{gathered} L\left(\frac{1}{2},{\chi }_c\right)=\sum _{0\ne \mathcal{A}\subset {\mathcal{O}}_K}\frac{{\chi }_c(\mathcal{A})}{N(\mathcal{A}{)}^{1/2}}\Gamma \left(\frac{1}{2},\frac{2\pi }{x}N(\mathcal{A})\right) \\ +\frac{W({\chi }_c)}{N(c{)}^{1/2}}\sum _{0\ne \mathcal{A}\subset {\mathcal{O}}_K}\frac{{\overline{\chi }}_c(\mathcal{A})}{N(\mathcal{A}{)}^{1/2}}\Gamma \left(\frac{1}{2},\frac{2\pi N(\mathcal{A})x}{|D_K|N(c)}\right), \end{gathered}$$

$$\Gamma \left(\frac{1}{2},\xi \right)=\frac{1}{2\pi i}\underset{(2)}{\int }\frac{\Gamma (s+1/2)}{\Gamma (1/2)}\frac{{\xi }^{-s}}{s}\mathrm{d}s=\frac{1}{\Gamma (1/2)}\underset{\xi }{\overset{\infty }{\int }}{t}^{-1/2}{e}^{-t}\mathrm{d}t$$

$$\Gamma \left(\frac{1}{2},\xi \right)=$$

$${\psi }_a(c)=(-1{)}^{((N(a)-1)/4)((N(c)-1)/4)}.$$

$$h(r,s;\chi )=\sum _{(n,r)=1}\frac{\chi (n)g(r,n)}{N(n{)}^s}.$$

$$P(a)=\prod _{i=1}^k{\overline{\left(\frac{(a/{\prod }_{j=1}^{i-1}{\pi }_j{)}^2}{{\pi }_i^3}\right)}}_4,$$

$$h(r,f,s;\chi )=\sum _{(n,rf)=1}\frac{\chi (n)g(r,n)}{N(n{)}^s}\text{mbox}and{h}_{\alpha }(r,s;\chi )=\sum _{(n,\alpha )=1}\frac{\chi (n)g(r,n)}{N(n{)}^s}.$$

$$h_{r_1}^{\ast }(r_1{r}_2^2{r}_3^3,s;\chi )=\sum _{a|r_2}{\mu }_{[i]}(a)\chi (a{)}^{3}N(a{)}^{2-3s}{\overline{\left(\frac{-r_1(r_2/a{)}^2{r}_3^3}{a^3}\right)}}_{4}P(a)\left(\prod _{\pi |a}\overline{g}(\pi )\right)h_{r_1}(r_1{r}_2^2{r}_3^3/a^2,s;{\psi }_{a^3}\chi ),$$

$$h(r,f,s;\chi )$$

$$h(r_1{r}_2^2{r}_3^3{r}_4^4,s;\chi )$$

$$h(r_1{r}_2^2{r}_3^3,s;\chi )$$

$$h_{r_1{r}_2}(r_1{r}_2^2{r}_3^3,s;\chi )=$$

$$h_{r_1}(r_1{r}_2^2{r}_3^3,s;\chi )$$

$$\begin{gathered} h_{ab\pi }(a{b}^2{c}^3{\pi }^2,s;\chi )=\sum _{(n,ab\pi )=1}\frac{\chi (n)g(a{b}^2{c}^3{\pi }^2,n)}{N(n{)}^s} \\ =\sum _{(n,ab)=1}\frac{\chi (n)g(a{b}^2{c}^3{\pi }^2,n)}{N(n{)}^s}-\sum _{\begin{array}{c}\pi |n\\ (n,ab)=1\end{array}}\frac{\chi (n)g(a{b}^2{c}^3{\pi }^2,n)}{N(n{)}^s}. \end{gathered}$$

$$g(a{b}^2{c}^3{\pi }^2,{\pi }^j{n}^{\prime })={\left(\frac{{\pi }^j}{n^{\prime }}\right)}_4{\left(\frac{n^{\prime }}{{\pi }^j}\right)}_{4}g(a{b}^2{c}^3{\pi }^2,{\pi }^j)g(a{b}^2{c}^3{\pi }^2,n^{\prime }).$$

$$g(a{b}^2{c}^3{\pi }^2,{\pi }^3{n}^{\prime })=N(\pi {)}^2\overline{g}(\pi ){\overline{\left(\frac{-a{b}^2{c}^3}{{\pi }^3}\right)}}_4{\psi }_{{\pi }^3}(n^{\prime })g(a{b}^2{c}^3,n^{\prime }).$$

$$h_{ab\pi }(a{b}^2{c}^3{\pi }^2,s;\chi )=h_{ab}(a{b}^2{c}^3{\pi }^2,s;\chi )-\chi (\pi {)}^{3}N(\pi {)}^{2-3s}\overline{g}(\pi ){\overline{\left(\frac{-a{b}^2{c}^3}{{\pi }^3}\right)}}_4{h}_{ab\pi }(a{b}^2{c}^3,s;{\psi }_{{\pi }^3}\chi ).$$

$$h_{ab\pi }(a{b}^2{c}^3,s;{\psi }_{{\pi }^3}\chi )$$

Full text

Mathematics Subject Classification (2010): 11M41, 11L40

Keywords: quartic Hecke characters, Hecke $L$-functions

1. Introduction

Because of the arithmetic information that they carry, the values of $L$-function at $s=1/2$ are very important in number theory. The average of these values over a family of characters of a fixed order has been studied extensively. For example, M. Jutila [Jutila] gave the evaluation the mean value of $L(1/2,\chi)$ for quadratic Dirichlet characters. The error term in the asymptotic formula in [Jutila] was later improved by [DoHo, MPY, ViTa]. In [B&Y], S. Baier and M. P. Young studied the moments of $L(1/2,\chi)$ for cubic Dirichlet characters. Literature also abounds in the investigation of moments of Hecke $L$-functions associated with various families of characters of a fixed order [FaHL, GHP, FHL, Luo, Diac]. In this paper, we shall investigate the first moment of $L$-functions associated with a family of quartic Hecke characters.

Let $K=\mathbb{Q}(i)$. It is well known that $K$ has class number $1$, and in the ring of integers $\mathcal{O}_{K}=\mathbb{Z}[i]$ every ideal coprime to $2$ has a unique generator congruent to $1\pmod{(1+i)^{3}}$. For $1\neq c\in\mathcal{O}_{K}$ which is square-free and congruent to $1\pmod{16}$, let $(\frac{\cdot}{c})_{4}$ be the quartic residue symbol. Since $\chi_{c}=(\frac{\cdot}{c})_{4}$ is trivial on units, it can be regarded as a primitive character of the ray class group $h_{(c)}$. We recall here that for any $c$, the ray class group $h_{(c)}$ is defined to be $I_{(c)}/P_{(c)}$, where $I_{(c)}=\{\mathcal{A}\in I:(\mathcal{A},(c))=1\}$ and $P_{(c)}=\{(a)\in P:a\equiv 1\pmod{c}\}$ with $I$ and $P$ denoting the group of fractional ideals in $K$ and the subgroup of principal ideals, respectively. The Hecke $L$-function associated with $\chi_{c}$ is defined for $\Re(s)>1$ by

[TABLE]

where $\mathcal{A}$ runs over all non-zero integral ideals in $K$ and $N(\mathcal{A})$ is the norm of $\mathcal{A}$. As shown by E. Hecke, $L(s,\chi_{c})$ admits analytic continuation to an entire function and satisfies the functional equation ([HIEK, Theorem 3.8])

[TABLE]

where

[TABLE]

with $(\delta)=(\sqrt{-4})$ being the different of $K$, and

[TABLE]

where $D_{K}=-4$ is the discriminant of $K$. We also note that $W(\chi_{c})=g(c)$, where $g(c)$ is the Gauss sum, defined later in Section 2.1. The above discussions also apply to $c=1$, provided that we interpret $\chi_{1}$ as the principal character $\pmod{1}$ so that $\Lambda(s,\chi_{1})$ becomes $\zeta_{\mathbb{Q}(i)}(s)$, the Dedekind zeta function of $\mathbb{Q}(i)$, a convention we shall follow throughout the paper.

Our goal here is to establish the following theorem.

Theorem 1.1**.**

For $y\rightarrow\infty$ and any $\varepsilon>0$, we have

[TABLE]

Here $A$ is an explicit constant given in (3.4) and $\sum^{*}$ denotes summation over square-free elements of $\mathbb{Z}[i]$ congruent to $1\pmod{16}$.

One important difference between our Theorem 1.1 and the results in [FHL, Diac] is that we have the additional restriction to square-free elements of $\mathbb{Z}[i]$ in the average. Another significant distinction between our result and those in [FHL] is the authors of [FHL] study the mean values of highly modified $L$-series associated fixed order Hecke character. Moreover, the result in [Diac] “does not seem to be accessible by more traditional methods” while our result here is obtained via classical means. We also note here that second moment and non-vanishing results for Hecke $L$-functions associated with characters of an arbitrary fixed order were proved in [BGL].

1.2. Strategy of the proof

To prove (1.3), $L(1/2,\chi_{c})$ is first decomposed using the approximate functional equation (2.7). We are then led to study two sums, (3.1) and (3.2). After various manipulations and transformations, the sum in (3.1) will give the main term on the right-hand side of (1.3). The large sieve for quartic Hecke characters, Lemma 2.8, plays an important role in controlling the error terms in the approximation. The sum in (3.2) relies crucially on an estimate of a smooth Gauss sum, (2.19). Theorem 1.1 follows from the combination and optimization of the above-mentioned estimates.

One should be able to establish a result similar to Theorem 1.1 if $1/2$ on the left-hand side of (1.3) is replaced by $1/2+it$ and $c$ by a set of more general set of numbers satisfying some almost-prime constraints. We leave the details of these possible extensions the enthused reader.

2. Preliminaries

In this section, we collect the information needed in the proof of our main result.

2.1. Quartic symbol and quartic Gauss sum

The symbol $(\frac{\cdot}{n})_{4}$ is the quartic residue symbol in the ring $\mathbb{Z}[i]$. For a prime $\pi\in\mathbb{Z}[i]$ with $N(\pi)\neq 2$, the quartic character is defined for $a\in\mathbb{Z}[i]$, $(a,\pi)=1$ by $\left(\frac{a}{\pi}\right)_{4}\equiv a^{(N(\pi)-1)/4}\pmod{\pi}$, with $\left(\frac{a}{\pi}\right)_{4}\in\{\pm 1,\pm i\}$. When $\pi|a$, we define $\left(\frac{a}{\pi}\right)_{4}=0$. Then the quartic character can be extended to any composite $n$ with $(N(n),2)=1$ multiplicatively.

Note that in $\mathbb{Z}[i]$, every ideal coprime to $2$ has a unique generator congruent to 1 modulo $(1+i)^{3}$. Such a generator is called primary. Recall that the quartic reciprocity law (Theorem 2 on page 123 of [I&R]) states that for two primary primes $m,n\in\mathbb{Z}[i]$,

[TABLE]

Observe, by Lemma 6 on page 121 of [I&R], that a non-unit $n=a+bi$ in $\mathbb{Z}[i]$ is congruent to $1\pmod{(1+i)^{3}}$ if and only if $a\equiv 1\pmod{4}$, $b\equiv 0\pmod{4}$ or $a\equiv 3\pmod{4}$, $b\equiv 2\pmod{4}$.

For a non-unit $n\in\mathbb{Z}[i]$, $n\equiv 1\pmod{(1+i)^{3}}$, the quartic Gauss sum $g(n)$ and a corresponding quadratic Gauss sum $g_{2}(n)$ is defined by

[TABLE]

where $\widetilde{e}(z)=\exp\left(2\pi i(\frac{z}{2i}-\frac{\overline{z}}{2i})\right)$.

We now extend the definition of $g(n)$ and $g_{2}(n)$ to $n=1$ by setting $g(1)=g_{2}(1)=1$. This is done so that the functional equation (1.1) is still valid with $W(\chi_{1})=g(1)$. Furthermore, the following well-known relation (see [Diac]) now holds for all $n$:

[TABLE]

The above relation holds for $g_{2}(n)$ as well.

More generally, for a non-unit $n\in\mathbb{Z}[i]$, $n\equiv 1\pmod{(1+i)^{3}}$, we set

[TABLE]

The following properties of $g(r,n)$ can be found in [Diac]:

[TABLE]

From the supplement theorem to the quartic reciprocity law (see for example, Lemma 8.2.1 and Theorem 8.2.4 in [BEW]), we have for $n=a+bi$ being primary,

[TABLE]

It follows that for any $c\equiv 1\pmod{16}$, we have

[TABLE]

2.2. The approximate functional equation

Let $x>1$. By evaluating the integral

[TABLE]

in two ways, we derive the following expression for $L(1/2,\chi_{c})$:

[TABLE]

where

[TABLE]

is the (normalized) incomplete $\Gamma$-function.

It is easy to see that

[TABLE]

2.3. Average of a smooth Gauss sum

One crucial ingredient of this paper is a bound for the average of certain smoothed Gauss sums.

Before stating the result, we first introduce a few notations. For $(c,a)=1$, set

[TABLE]

It is easy to see that $\psi_{a}(c)$ is a ray class character $\pmod{16}$.

We use the convention in this subsection that all sums over elements of $\mathbb{Z}[i]$ are restricted to elements congruent to $1\pmod{(1+i)^{3}}$ and we use $\pi$ to denote a prime in $\mathbb{Z}[i]$. Also let $\mu_{[i]}$ stand for the Möbius function on $\mathbb{Z}[i]$. For any ray class character $\pmod{16}$, we set

[TABLE]

For any square-free, non-unit $a\in\mathbb{Z}[i]$, let $\{\pi_{1},\cdots,\pi_{k}\}$ be the set of distinct prime divisors of $a$. We define

[TABLE]

where we set the empty product to be $1$. As $\left(\frac{\pi^{2}_{2}}{\pi^{3}_{1}}\right)_{4}=\left(\frac{\pi_{2}}{\pi_{1}}\right)^{2}_{4}$ for two distinct primes $\pi_{1},\pi_{2}$, one checks easily by induction on the number of prime divisors of $a$ that $P(a)$ is independent of the order of $\{\pi_{1},\cdots,\pi_{k}\}$.

Lemma 2.4**.**

Suppose $f,\alpha$ are square-free and $(r,f)=1$. Set

[TABLE]

Furthermore suppose $r=r_{1}r^{2}_{2}r^{3}_{3}r^{4}_{4}$ where $r_{1}r_{2}r_{3}$ is square-free, and let $r^{*}_{4}$ be the product of primes dividing $r_{4}$. We define

[TABLE]

where we define $\left(\frac{\cdot}{1}\right)_{4}=1$ in the sum above and the empty product to be $1$. Then

[TABLE]

Proof.

As the proof is similar to that of [B&Y, Lemma 3.6], we only give the proof of (2.13) here. To that end, let $ab^{2}c^{3}\in\mathbb{Z}[i]$ and $\pi$ be a prime in $\mathbb{Z}[i]$ such that $abc\pi$ is square-free. Then

[TABLE]

Writing in the latter sum $n=\pi^{j}n^{\prime}$ with $(n^{\prime},\pi)=1$, we have by (2.3) that

[TABLE]

Using (2.2) and (2.4) we see that $g(ab^{2}c^{3}\pi^{2},\pi^{j})=0$ unless $j=3$, in which case we deduce from quartic reciprocity, (2.2) and (2.4) that (note that $\left(\frac{-1}{\pi}\right)_{4}=\left(\frac{-1}{\pi^{3}}\right)_{4}$ by (2.5))

[TABLE]

This implies that

[TABLE]

On the other hand, we have

[TABLE]

Writing in the latter sum $n=\pi^{j}n^{\prime}$ , where $(n^{\prime},\pi)=1$, then we have by (2.3) that

[TABLE]

Using (2.2) and (2.4) we see that $g(ab^{2}c^{3},\pi^{j})=0$ unless $j=1$, in which case we deduce from quartic reciprocity, (2.2) and (2.4) that

[TABLE]

This implies that (noting that $\psi_{\pi}\psi_{\pi^{3}}$ is principal)

[TABLE]

We then deduce from (2.15) and (2.16) that

[TABLE]

An induction argument on the number of prime divisors of $b$ gives (2.13). ∎

Now we deduce readily the following lemma concerning the analytic behavior of $h(r,s;\chi)$ on $\Re(s)>1$.

Lemma 2.5**.**

The function $h(r,s;\chi)$ has meromorphic continuation to the entire complex plane. It is holomorphic in the region $\sigma=\Re(s)>1$ except possibly for a pole at $s=5/4$. For any $\varepsilon>0$, letting $\sigma_{1}=3/2+\varepsilon$, then for $\sigma_{1}\geq\sigma\geq\sigma_{1}-1/2$, $|s-5/4|>1/8$, we have

[TABLE]

where $t=\Im(s)$. Moreover, the residue satisfies

[TABLE]

Proof.

We can reduce the estimation of $h(r,s;\chi)$ to that of $h_{1}(r,s;\chi)$. To see this, we write $r=r_{1}r_{2}^{2}r_{3}^{3}r_{4}^{4}$ with $r_{1}$, $r_{2}$ and $r_{3}$ square-free. (2.11) gives that

[TABLE]

where $r_{4}^{*}$ is the product of primes dividing $r_{4}$. Then applying (2.10), $h(r_{1}r_{2}^{2}r_{3}^{3},r_{4}^{*},s;\chi)$ can be written as a sum involving $h(a^{2}r,s;\psi_{a}\chi)$ with $a$ dividing $r_{4}^{*}$. Repeating this process if needed (as in the case $(r_{4},r_{2})>1$), we will arrive at an expression involving $h(r_{1}r_{2}^{2}r_{3}^{3},s;\chi)$ with square-free $r_{1}$, $r_{2}$ and $r_{3}$. Now, we can apply (2.12), (2.13), (2.9) and (2.14) and end up with an expression involving $h_{1}(r_{1}r_{2}^{2}r_{3}^{3},s;\chi)$.

Now it follows from the proof of the Lemma on p. 200 of [P] that $h_{1}(r,s;\chi)$ satisfies the properties of the lemma, this gives us the desired result. ∎

Now we are ready to state and prove the necessary bound.

Lemma 2.6**.**

Let $\widetilde{g}(n)=g(n)N(n)^{-1/2}$. For any rational integer $r\geq 0,a\in\mathbb{Z}[i]$, we have for $y\geq x\geq 1$,

[TABLE]

Proof.

Let $\chi$ runs over all ray class characters $\pmod{16}$, we use the ray class characters $\pmod{16}$ to detect the congruence condition $c\equiv 1\pmod{16}$ to obtain

[TABLE]

In view of the exponential decay of $e^{-x}$, we see that the inner sum in the last expression of (2.20) is holomorphic in the region $\Re(s^{\prime})>0$. We can shift the contour of integration in the last expression of (2.20) to $\Re(s^{\prime})=\varepsilon>0$, which we now fix. By the Mellin inversion, we can recast the inner sum in the last expression of (2.20) as

[TABLE]

It follows from Lemma 2.5 and Stirling’s formula that by shifting the contour of integration to $\Re(s)=1/2+2\varepsilon$, the integration in (2.21) is

[TABLE]

Here, the first term in the above bound comes from estimating the integral over the line $\Re(s)=1/2+2\varepsilon$ and using (2.17). The second term is to bound the residue at $s=5/4$ for which we use (2.18). Now, applying (2.22) for the inner sum in the last expression of (2.20) allows us to readily deduce (2.19). ∎

2.7. The large sieve with quartic symbols

Another important input of this paper is the following large sieve inequality for quartic Hecke characters. The study of the large sieve inequality for characters of a fixed order has a long history. We refer the reader to [DRHB, DRHB1, G&Zhao, B&Y, BGL].

Lemma 2.8**.**

Let $M,N$ be positive integers, and let $(a_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence of complex numbers, where $n$ runs over $\mathbb{Z}[i]$. Then we have

[TABLE]

for any $\varepsilon>0$,where the asterisks indicate that $m$ and $n$ run over square-free elements of $\mathbb{Z}[i]$ that are congruent to $1$ modulo $(1+i)^{3}$ and $(\frac{\cdot}{m})_{4}$ is the quartic residue symbol.

Proof.

This is a special case of [BGL, Theorem 1.3] which improves the bound in [G&Zhao, Theorem 1.1] (the exponent of $MN$ is $3/4$ in [G&Zhao]). ∎

3. Proof of Theorem 1.1

3.1. The main term of the first moment

We have, using (2.7) (choosing $x=y^{l}$, where $0<l<1$ will be specified later)

[TABLE]

where

[TABLE]

and

[TABLE]

We deal with ${\sum}_{1}$ first. Since any integral non-zero ideal $\mathcal{A}$ in $\mathbb{Z}[i]$ has a unique generator $(1+i)^{r}a$, where $r\in\mathbb{Z}$, $r\geq 0$, $a\in\mathbb{Z}[i]$ and $a\equiv 1\pmod{(1+i)^{3}}$, it follows from the quartic reciprocity law and (2.6) that $\chi_{c}(\mathcal{A})=\chi_{a}(c)$ (recall our convention that $\chi_{1}$ is the principal character $\pmod{1}$). This enables us to recast ${\sum}_{1}$ as

[TABLE]

For $a$, a fourth power, the inner sum above is

[TABLE]

where $\chi$ runs over all ray class characters $\pmod{16}$. Note that we have

[TABLE]

Here $\zeta_{K}(s)$ is the Dedekind zeta function of $K$. Note that both $\zeta_{K}(2s)$ and $\prod_{\mathfrak{p}}(1-\chi(\mathfrak{p}^{2})N(\mathfrak{p})^{-2s})$ are holomorphic at $\Re(s)>1$ and $O_{\varepsilon}(1)$ for $\Re(s)\geq 1+\varepsilon$. Moreover, for a fixed non-principal character $\chi$, let $\chi^{\prime}$ be the primitive character that induces $\chi$. Then it is easy to see that $L(s,\chi)=L(s,\chi^{\prime})$ since $\chi$ is of conductor $16$. By a result of E. Landau [Landau] (see also [Sunley, Theorem 2]), which states that for an algebraic number field $K$ of degree $n\geq 2$, $\chi$ any non-principal primitive ideal character of $K$ with conductor $\mathfrak{f}$, $k=|N(\mathfrak{f})\cdot d_{K}|$ with $d_{K}$ being the discriminant of $K$, we have for $X\geq 1$,

[TABLE]

where $I$ runs over integral ideas of $K$. It follows from this and partial summation that $L(s,\chi)\ll 1$ when $\Re(s)\geq 1/2$ for any non-principal ray class character $\chi\pmod{16}$. Thus on moving the line of integration on the right-hand side of (3.3) to $\Re(s)=1/2+\varepsilon$ and noting that on this line

[TABLE]

we get that the sum on the left-hand side of (3.3) equals asymptotically $C_{a}y+O(y^{1/2+\varepsilon}N(a)^{\varepsilon})$, where

[TABLE]

It follows from this and (2.8) that the contribution from fourth-powers $a$ in (3.1) is exactly

[TABLE]

3.2. The remainder terms of the first moment

For non-fourth power $a$, $\chi_{a}$ is non-trivial and we have the analogue of the Pólya-Vinogradov inequality from [G&Zhao, Lemma 3.1], that for $y\geq 1$,

[TABLE]

Note that we can assume $N(c)\ll y^{1+\varepsilon}$ and $N(a)\ll x^{1+\varepsilon}$ in view of the exponential decay of the test functions. We obtain that, after using $\mu_{[i]}$ to detect the condition that $c$ is square-free,

[TABLE]

In the second sum of the last expression above, we further write $c=c^{\prime}e^{2}$ with $c^{\prime}$ square-free to arrive at

[TABLE]

Here $B$ satisfies the bounds $y^{1/2}x^{-1/2}<B<y^{1/2}$ (and hence $x\geq y/N(b)^{2}$) and will be chosen optimally later. Using the ray class characters $\chi\pmod{16}$ to detect the congruence condition $c\equiv\overline{d}^{2}\pmod{16}$, and applying (3.5), we get

[TABLE]

It follows from this that the contribution of $R$ to ${\sum}_{1}$ is at most $xBy^{\varepsilon}$. Here we note that $\chi\chi_{a}$ is non-trivial $\pmod{16m}$ as $\chi_{a}$ is non-trivial $\pmod{m}$.

To deal with $S$, we extract square divisors of $a$ by writing $a=a_{1}a^{2}_{2}$, where $a_{1},a_{2}\equiv 1\pmod{(1+i)^{3}}$ and $a_{1}$ is square-free to get

[TABLE]

We deduce, by Cauchy’s inequality, that

[TABLE]

For the first factor on the right-hand side of (3.6), we have

[TABLE]

For the second factor on the right-hand side of (3.6), we have, by Lemma 2.8,

[TABLE]

It now follows that the left-hand side expression of (3.6) is

[TABLE]

From this we deduce that the contribution of $S$ to ${\sum}_{1}$ is at most

[TABLE]

Combining the bounds for $R$ and $S$, we obtain (with a different $\varepsilon$)

[TABLE]

We now optimize $B$ to satisfy $xB=x^{1/3}y^{5/6}B^{-2/3}$. Thus $B=y^{1/2}x^{-2/5}$. Note further that when $B>y^{1/2}x^{-1/2}$, we have $y/B<(xy)^{1/2}$. Therefore, for the $B$ thus chosen, we have

[TABLE]

Next we need to bound

[TABLE]

Mindful of the equality $W(\chi_{c})=g(c)$ and the fact from (2.1) that $g(c)$ is supported on square-free numbers , we may drop the restriction ∗ in the outer sum above. Recalling the definition of $\widetilde{g}(c)$ in Lemma 2.6, we recast ${\sum}_{2}$ as

[TABLE]

Again we can assume $2^{r}N(a)\ll y^{1+\varepsilon}/x$ by virtue of the exponential decay of the test functions. Applying the bound (2.19), we see immediately that

[TABLE]

From (3.7) and (3.8), we conclude that

[TABLE]

on taking $x=y^{5/7}$.

Acknowledgments. P. G. is supported in part by NSFC grants 11371043 and 11871082 and L. Z. by the FRG grant PS43707. Parts of this work were done when P. G. visited the University of New South Wales (UNSW) in June 2017. He wishes to thank UNSW for the invitation, financial support and warm hospitality during his pleasant stay. Finally, we would like to thank the anonymous referee for his/her careful reading of the paper and many helpful comments.