Ratios conjecture for primitive quadratic Hecke $L$ -functions

Peng Gao School of Mathematical Sciences, Beihang University, Beijing 100191, China [email protected] and Liangyi Zhao School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia [email protected]

Abstract.

We develop the ratios conjecture with one shift in the numerator and denominator in certain ranges for families of primitive quadratic Hecke $L$ -functions of imaginary quadratic number fields with class number one using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain unconditional asymptotic formulas for the first moments of central values of these families of $L$ -functions with error terms of size that is the square root of that of the primary main terms.

Mathematics Subject Classification (2010): 11M06, 11M41

Keywords: ratios conjecture, mean values, primitive quadratic Hecke $L$ -functions

1. Introduction

The $L$ -functions ratios conjecture is important in number theory, having many important applications. These include it use in developing the density conjecture of N. Katz and P. Sarnak [KS1, K&S] on the distribution of zeros near the central point of a family of $L$ -functions, the mollified moments of $L$ -functions, etc. This conjecture makes predictions on the asymptotic behaviors of the sum of ratios of products of shifted $L$ -functions and has its origin in the work of D. W. Farmer [Farmer93] on the shifted moments of the Riemann zeta function. For general $L$ -functions, the conjecture is formulated by J. B. Conrey, D. W. Farmer and M. R. Zirnbauer [CFZ, Section 5].

Since the work of H. M. Bui, A. Florea and J. P. Keating [BFK21] on quadratic $L$ -functions over function fields, there are now several results available in the literature concerning the ratios conjecture, all valid for certain ranges of the relevant parameters. The first such result over number fields was given by M. Čech [Cech1] on families of both general and primitive quadratic Dirichlet $L$ -functions under the assumption of the generalized Riemann hypothesis (GRH).

The work of Čech makes uses of the powerful tool of multiple Dirichlet series. Due to an extra functional equation for the underlying multiple Dirichlet series, the result for the family of general quadratic Dirichlet $L$ -functions has a better range of the parameters involved. Following this approach, the authors studied the ratios conjecture for general quadratic Hecke $L$ -functions over the Gaussian field $\mathbb{Q}(i)$ in [G&Zhao14] as well as for quadratic twists of modular $L$ -functions in [G&Zhao15].

It is noted in [G&Zhao14] that investigations on ratios conjecture naturally lead to results concerning the first moment of central values of the corresponding family of $L$ -functions, another important subject in number theory. Among the many families of $L$ -functions, the primitive ones received much attention. For families of primitive Dirichlet $L$ -functions, M. Jutila evaluated asymptotically the first moment in [Jutila]. In [DoHo], D. Goldfeld and J. Hoffstein initiated the study on the first moment of the same family using method of double Dirichlet series. It is implicit in their work that an asymptotic formula with an error term of size of the square root of the main term holds for the smoothed first moment and this was later established by M. P. Young [Young1] using a recursive argument. In [Gao20], the first-named author obtained a similar result for the first moment of the family of primitive quadratic Hecke $L$ -functions over the Gaussian field using Young’s recursive method.

Motivated by studies of the first moment of families of primitive $L$ -functions, we are interested in this paper to develop the ratios conjecture for primitive quadratic Hecke $L$ -functions over imaginary quadratic number fields with class number one. We shall show that, as a consequence of the ratios conjecture, asymptotic formulas hold for the first moments of central values of the corresponding families of $L$ -functions with error terms of size of the square root of that of the main terms.

To state our result, let $K$ be an imaginary quadratic number field of class number one. We shall assume throughout the paper that $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathcal{S}$ , where

\displaystyle\mathcal{S}=\{-1,-2,-3,-7,-11,-19,-43,-67,-163\}.

In fact, it is well-known (see [iwakow, (22.77)]) that we obtain all imaginary quadratic number fields with class number one in the above way.

We write $\chi^{(m)}=\left(\frac{m}{\cdot}\right)$ and $\chi_{m}=\left(\frac{\cdot}{m}\right)$ for the quadratic residue symbols to be defined Section 2.3. We define $c_{K}=(1+i)^{5}$ when $d=-1$ , $c_{K}=4\sqrt{-2}$ when $d=-2$ and $c_{K}=8$ for the other $d$ ’s in $\mathcal{S}$ . Let $\mathcal{O}_{K},U_{K}$ and $D_{K}$ denote the ring of integers, the group of units and the discriminant of $K$ , respectively. We also write $|U_{K}|$ for the cardinality of $U_{K}$ . An element $c\in\mathcal{O}_{K}$ is said to be square-free if the ideal $(c)$ is not divisible by the square of any prime ideal. It is shown in [G&Zhao4, Section 2.1] and [G&Zhao2022-4, Section 2.2] that $\chi^{(c_{K}c)}$ is a primitive quadratic character of trivial infinite type for any square-free $c$ .

Let $L(s,\chi)$ stand for the $L$ -function attached to any Hecke character $\chi$ and $\zeta_{K}(s)$ for the Dedekind zeta function of $K$ . We also use the notation $L^{(c)}(s,\chi_{m})$ for the Euler product defining $L(s,\chi_{m})$ but omitting those primes dividing $c$ . Moreover, let $r_{K}$ denote the residue of $\zeta_{K}(s)$ at $s=1$ . We reserve the letter $\varpi$ for a prime element in $\mathcal{O}_{K}$ , by which we mean that the ideal $(\varpi)$ generated by $\varpi$ is a prime ideal. Let also $N(n)$ stand for the norm of any $n\in\mathcal{O}_{K}$ . In the sequel, $\varepsilon$ always, as is standard, denotes a small positive real number which may not be the same at each occurrence.

Our main result in this paper investigates the ratios conjecture with one shift in the numerator and denominator for the family of primitive quadratic Hecke $L$ -functions of $K$ .

Theorem 1.1.

With the notation as above and assuming the truth of GRH, let $K=\mathbb{Q}(\sqrt{-d})$ with $d\in\mathcal{S}$ . Suppose that $w(t)$ is a non-negative Schwartz function and $\widehat{w}(s)$ its Mellin transform. We set

(1.1)

E(\alpha,\beta)=\max\left\{\frac{1}{2},1-\Re(\alpha)-\Re(\beta),1-\frac{\Re(\alpha)}{2}-\frac{\Re(\beta)}{2},\frac{1}{2}-\frac{\Re(\alpha)}{2},\frac{1}{2}-\Re(\alpha)\right\}.

and, for any $n\in\mathcal{O}_{K}$ ,

(1.2)

\displaystyle a(n)=\prod_{\varpi|n}\Big{(}1+\frac{1}{N(\varpi)}\Big{)}^{-1}.

Then we have for $0<|\Re(\alpha)|<1/2$ and $\Re(\beta)>0$ ,

(1.3)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{(c,2)=1}&\frac{L(\frac{1}{2}+\alpha,\chi^{(c_{K}c)})}{L(\frac{1}{2}+\beta,\chi^{(c_{K}c)})}w\left(\frac{N(c)}{X}\right)\\ &=X\widehat{w}(1)\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\frac{\zeta_{K}^{(2)}(1+2\alpha)}{\zeta_{K}^{(2)}(1+\alpha+\beta)}P(\tfrac{1}{2}+\alpha,\tfrac{1}{2}+\beta)\\ &\hskip 28.45274pt+X^{1-\alpha}\widehat{w}(1-\alpha)\frac{(2\pi)^{2\alpha}}{(|D_{K}|N(c_{K}))^{\alpha}}\frac{\Gamma(\tfrac{1}{2}-\alpha)}{\Gamma(\tfrac{1}{2}+\alpha)}\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\frac{\zeta_{K}^{(2)}(1-2\alpha)}{\zeta_{K}^{(2)}(1-\alpha+\beta)}P(\tfrac{1}{2}-\alpha,\tfrac{1}{2}+\beta)\\ &\hskip 28.45274pt+O\left((1+|\alpha|)^{5/2+\varepsilon}(1+|\beta|)^{\varepsilon}X^{E(\alpha,\beta)+\varepsilon}\right),\end{split}

where $\sum^{*}$ means that the sum is restricted to square-free elements $c$ of $\mathcal{O}_{K}$ and

(1.4)

P(w,z)=\prod_{(\varpi,2)=1}\left(1+\frac{1-N(\varpi)^{z-w}}{(N(\varpi)+1)(N(\varpi)^{z+w}-1)}\right).

An evaluation from the ratios conjecture on the left-hand side of (1.3) can be derived following the treatments given in [G&Zhao2022-2, Section 5]. One sees that our result above is consistent with the result therefrom, except that this approach predicts that (1.3) holds uniformly for $|\Re(\alpha)|<1/4$ , $(\log X)^{-1}\ll\Re(\beta)<1/4$ and $\Im(\alpha)$ , $\Im(\beta)\ll X^{1-\varepsilon}$ with an error term $O(X^{1/2+\varepsilon})$ . The advantage of (1.3) here chiefly lies in the absence of any constraint on imaginary parts of $\alpha$ and $\beta$ . Moreover, we do obtain an error term of size $O(X^{1/2+\varepsilon})$ unconditionally when $\beta$ is large enough. In fact, one checks by going through our proof of Theorem 1.1 in Section 3 that the only place we need to assume GRH is (3.10), to estimate the size of $1/L(z,\chi^{(c_{K}c)})$ . However, when $\Re(z)$ is larger than one, then $1/L(z,\chi^{(c_{K}c)})$ is $O(1)$ unconditionally. Replacing this estimation everywhere in the rest of the proof of Theorem 1.1, and then consider the case $\beta\rightarrow\infty$ , we immediately deduce from (1.3) the following unconditional result on the smoothed first moment of values of quadratic Hecke $L$ -functions.

Theorem 1.2.

Using the notation as above and assuming the truth of GRH, let $K=\mathbb{Q}(\sqrt{-d})$ with $d\in\mathcal{S}$ . We have for $0<|\Re(\alpha)|<1/2$ ,

(1.5)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{(c,2)=1}&L(\tfrac{1}{2}+\alpha,\chi^{(c_{K}c)})w\left(\frac{N(c)}{X}\right)\\ =&X\widehat{w}(1)\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\zeta_{K}^{(2)}(1+2\alpha)P(\tfrac{1}{2}+\alpha)\\ &\hskip 28.45274pt+X^{1-\alpha}\widehat{w}(1-\alpha)(|D_{K}|N(c_{K}))^{-\alpha}(2\pi)^{2\alpha}\frac{\Gamma(\tfrac{1}{2}-\alpha)}{\Gamma(\tfrac{1}{2}+\alpha)}\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\zeta_{K}^{(2)}(1-2\alpha)P(\tfrac{1}{2}-\alpha)\\ &\hskip 28.45274pt+O\left((1+|\alpha|)^{5/2+\varepsilon}X^{E(\alpha)+\varepsilon}\right),\end{split}

where

\displaystyle E(\alpha)=\lim_{\beta\rightarrow\infty}E(\alpha,\beta)=\max\left\{\frac{1}{2},\frac{1}{2}-\frac{\Re(\alpha)}{2},\frac{1}{2}-\Re(\alpha)\right\},\;P(w)=\lim_{z\rightarrow\infty}P(w,z)=\prod_{(\varpi,2)=1}\left(1-\frac{1}{(N(\varpi)+1)N(\varpi)^{2w}}\right).

Note that the $O$ -term in (1.5) is uniform in $\alpha$ , we can further take the limit $\alpha\rightarrow 0^{+}$ and deduce the following asymptotic formula for the smoothed first moment of central values of quadratic Hecke $L$ -functions.

Corollary 1.3.

With the notation as above, let $K=\mathbb{Q}(\sqrt{-d})$ for $d\in\mathcal{S}$ . We have,

\displaystyle\begin{split}&\sideset{}{{}^{*}}{\sum}_{(c,2)=1}L(\tfrac{1}{2}+\alpha,\chi^{(c_{K}c)})w\left(\frac{N(c)}{X}\right)=XQ_{K}(\log X)+O\left(X^{1/2+\varepsilon}\right).\end{split}

where $Q_{K}$ is a linear polynomial whose coefficients depend only $K$ , $\widehat{w}(1)$ and $\widehat{w}^{\prime}(1)$ .

As another application of Theorem 1.1, we differentiate with respect to $\alpha$ in (1.3) and then set $\alpha=\beta=r$ to obtain an asymptotic formula concerning the smoothed first moment of $L^{\prime}(\frac{1}{2}+r,\chi^{(c_{K}c)})/L(\frac{1}{2}+r,\chi^{(c_{K}c)})$ averaged over odd, square-free $c$ .

Theorem 1.4.

With the notation as above and assuming the truth of GRH. Let $K=\mathbb{Q}(\sqrt{-d})$ with $d\in\mathcal{S}$ . We have for $0<\varepsilon<\Re(r)<1/2$ ,

(1.6)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{(c,2)=1}&\frac{L^{\prime}(\frac{1}{2}+r,\chi^{(c_{K}c)})}{L(\frac{1}{2}+r,\chi^{(c_{K}c)})}w\left(\frac{N(c)}{X}\right)\\ &=X\widehat{w}(1)\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\left(\frac{(\zeta^{(2)}_{K}(1+2r))^{\prime}}{\zeta^{(2)}_{K}(1+2r)}+\sum_{(\varpi,2)=1}\frac{\log N(\varpi)}{N(\varpi)(N(\varpi)^{1+2r}-1)}\right)\\ &\hskip 28.45274pt-X^{1-r}\widehat{w}(1-r)(|D_{K}|N(c_{K}))^{-r}(2\pi)^{2r}\frac{\Gamma(1/2-r)}{\Gamma(1/2+r)}\frac{|U_{K}|^{2}a(2)}{\zeta^{(2)}_{K}(2-2r)}+O((1+|r|)^{5/2+\varepsilon}X^{1-2r+\varepsilon}).\end{split}

Theorem 1.1 will be established using the approach in the proof of [Cech1, Theorem 1.1] with a refinement on the arguments there. More precisely, we note that the reciprocal of $\zeta_{K}(2s)$ appears in the expression for $A_{K}(s,w,z)$ in (3.5). A similar expression involving $(\zeta(2s))^{-1}$ also emerges in the proof of [Cech1, Theorem 1.1], where $\zeta(s)$ is the Riemann zeta function. In the proof of [Cech1, Theorem 1.1], the factor $(\zeta(2s))^{-1}$ is treated directly so that one needs to restrict $s$ to the region $\Re(s)>1/2$ . Our idea here is to estimate the product of $\zeta_{K}(2s)$ and $A_{K}(s,w,z)$ instead (see (3.6) below), this allows us to obtain the analytic property of $A_{K}(s,w,z)$ in a larger region of $s$ , which leads to improvements on the error term in (1.3) over [Cech1, Theorem 1.1].

We may also apply Theorem 1.4 to compute the one-level density of low-lying zeros of the corresponding families of quadratic Hecke $L$ -functions, as in [Cech1, Corollary 1.5] for the family of quadratic Dirichlet $L$ -functions using [Cech1, Theorem 1.4]. However, one checks that in this case we may only do so for test functions whose Fourier transform being supported in $(-1,1)$ , which is inferior to what may be obtained using other methods. Thus we shall not go in this direction here.

2. Preliminaries

2.1. Imaginary quadratic number fields

Recall that for an arbitrary number field $K$ , $\mathcal{O}_{K},U_{K}$ and $D_{K}$ stand for the ring of integers, the group of units and the discriminant of $K$ , respectively. We say an element $c\in\mathcal{O}_{K}$ is odd if $(c,2)=1$ .

In the rest of this section, let $K$ be an imaginary quadratic number field. The following facts concerning an imaginary quadratic number field $K$ can be found in [iwakow, Section 3.8].

The ring of integers $\mathcal{O}_{K}$ is a free $\mathbb{Z}$ module (see [iwakow, Section 3.8]) such that $\mathcal{O}_{K}=\mathbb{Z}+\omega_{K}\mathbb{Z}$ , where

\displaystyle\omega_{K}

\displaystyle=\begin{cases}\displaystyle\frac{1+\sqrt{d}}{2}\qquad&d\equiv 1\pmod{4},\\ \\ \sqrt{d}\qquad&d\equiv 2,3\pmod{4}.\end{cases}

Note that when $K=\mathbb{Q}(\sqrt{-3})$ , we also have $\mathcal{O}_{K}=\mathbb{Z}+\omega^{2}_{K}\mathbb{Z}$ . The group of units are given by

(2.1)

\displaystyle U_{K}

\displaystyle=\begin{cases}\{\pm 1,\pm i\}\qquad&d=-1,\\ \{\pm 1,\pm\omega_{K},\pm\omega_{K}^{2}\}\qquad&d=-3,\\ \{\pm 1\}\qquad&\text{other $d$}.\end{cases}

Moreover, the discriminants are given by

\displaystyle D_{K}

\displaystyle=\begin{cases}d\qquad&\text{if $d\equiv 1\pmod{4}$},\\ 4d\qquad&\text{if $d\equiv 2,3\pmod{4}$}.\end{cases}

If we write $n=a+b\omega_{K}$ with $a,b\in\mathbb{Z}$ , then

(2.2)

\displaystyle N(n)=\begin{cases}\displaystyle a^{2}+ab+b^{2}\frac{1-d}{4}\qquad&d\equiv 1\pmod{4},\\ \displaystyle a^{2}-db^{2}\qquad&d\equiv 2,3\pmod{4}.\end{cases}

Denote the Kronecker symbol in $\mathbb{Z}$ by $\left(\frac{\cdot}{\cdot}\right)_{\mathbb{Z}}$ . A rational prime ideal $(p)\in\mathbb{Z}$ in $\mathcal{O}_{K}$ factors in the following way:

	$\displaystyle\left(\frac{D_{K}}{p}\right)_{\mathbb{Z}}=0,\text{then $p$ ramifies},\ (p)=\mathfrak{p}^{2}\quad\text{with}\quad N(\mathfrak{p})=p,$
	$\displaystyle\left(\frac{D_{K}}{p}\right)_{\mathbb{Z}}=1,\text{then $p$ splits},\ (p)=\mathfrak{p}\overline{\mathfrak{p}}\quad\text{with}\quad\mathfrak{p}\neq\overline{\mathfrak{p}}\quad\text{and}\quad N(\mathfrak{p})=p,$
	$\displaystyle\left(\frac{D_{K}}{p}\right)_{\mathbb{Z}}=-1,\text{then $p$ is inert},\ (p)=\mathfrak{p}\quad\text{with}\quad N(\mathfrak{p})=p^{2}.$

In particular, the rational prime ideal $(2)\in\mathbb{Q}$ splits if $d\equiv 1\pmod{8}$ , is inert if $d\equiv 5\pmod{8}$ and ramifies for $d=-1$ , $-2$ .

2.2. Primary Elements

It is well-known that when a number field $K$ is of class number one, every ideal of $\mathcal{O}_{K}$ is principal, so that one may fix a unique generator for each non-zero ideal. In this paper, we are interested in ideals that are co-prime to $(2)$ and in this section, we determine for each such ideal a unique generator which we call primary elements. Our choice for these primary elements is based on a result of F. Lemmermeyer [Lemmermeyer05, Theorem 12.17], in order to ensure the validity of a proper form of the quadratic reciprocity law, Lemma 2.4, for these elements.

For $K=\mathbb{Q}(i)$ , $(1+i)$ is the only ideal above the rational ideal $(2)\in\mathbb{Z}$ . We define for any $n\in\mathcal{O}_{K}$ with $(n,2)=1$ to be primary if and only if $n\equiv 1\pmod{(1+i)^{3}}$ by observing that the group $\left(\mathcal{O}_{K}/((1+i)^{3})\right)^{\times}$ is isomorphic to $U_{K}$ . Here and in what follows, for any $q\in\mathcal{O}_{K}$ , let $G_{q}:=\left(\mathcal{O}_{K}/(q)\right)^{\times}$ be the group of reduced residue classes modulo $q$ , i.e. the multiplicative group of invertible elements in $\mathcal{O}_{K}/(q)$ . For any group $G$ , let $G^{2}$ be the subgroup of $G$ consisting of square elements of $G$ .

In what follows we consider the case for $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathcal{S}$ and $d\neq-1$ . Note that we have either $d=4k+1$ for some $k\in\mathbb{Z}$ or $d=-2$ . Recall that our choice for the primary elements is based on [Lemmermeyer05, Theorem 12.17], which requires the determination of the quotient group $G_{4}/G^{2}_{4}$ . We therefore begin by examining the structure of $G_{4}$ .

We first consider the case $d=4k+1$ . Note that by our discussions in Section 2.1, the rational ideal $(2)$ splits in $\mathcal{O}_{K}$ when $2|k$ and is inert otherwise. It follows that

(2.3)

\displaystyle\begin{split}G_{2}=\begin{cases}\{1,-\omega_{K},1-\omega_{K}\},&2\nmid k,\\ \{1\},&2|k.\end{cases}\end{split}

As $|G_{2}|$ is odd in either case above, the natural homomorphism

\displaystyle\begin{split}G_{2}\rightarrow G^{2}_{2}:g\mapsto g^{2},\end{split}

is injective and hence an isomorphism. Now observe that

\displaystyle\begin{split}G_{4}=G_{2}+2l,\quad l\in\{0,1,\omega_{K},1+\omega_{K}\}.\end{split}

As $(a+2l)^{2}\equiv a^{2}\pmod{4}$ , we see that $G^{2}_{4}=G^{2}_{2}\cong G_{2}$ .

Direct computation shows that

(2.4)

\displaystyle\begin{split}G^{2}_{4}=G^{2}_{2}=\begin{cases}\{1,\omega^{2}_{K}=k+\omega_{K},(1+\omega_{K})^{2}=k+1-\omega_{K}\},&2\nmid k,\\ \{1\},&2|k.\end{cases}\end{split}

Here we remark that the reason we choose $-\omega_{K}$ instead of $\omega_{K}$ in the set (2.3) defining $G_{2}$ is to ensure that when $d=-3$ , we have $G^{2}_{2}=U^{2}_{K}$ .

Consider the exact sequence

(2.5)

\displaystyle\begin{split}0\rightarrow\ker{\sigma}\rightarrow G_{4}\xrightarrow{\sigma}G_{2}\rightarrow 0,\end{split}

where $\sigma(g)=g\pmod{2}$ for any $g\in G_{4}$ . It follows that $G_{4}\cong\ker{\sigma}\times G_{2}\cong\ker{\sigma}\times G^{2}_{4}$ . As $\ker{\sigma}$ contains four elements that form a representation of all the residue classes modulo $4$ that are congruent to $1$ modulo $2$ , we see that

(2.6)

\displaystyle\begin{split}\ker{\sigma}=\{\pm 1,\pm(1+2\omega_{K})\}\pmod{4}\cong<-1>\times<1+2\omega_{K}>\pmod{4},\end{split}

where $<a>$ denotes the cyclic group generated by $a$ .

We then conclude that when $d=4k+1$ , we have

(2.7)

\displaystyle\begin{split}G_{4}\cong G^{2}_{4}\times<-1>\times<1+2\omega_{K}>\pmod{4}.\end{split}

When $d=-2\pmod{4}$ , the rational ideal $(2)$ ramifies in $\mathcal{O}_{K}$ and we have

\displaystyle\begin{split}G_{2}=\{1,1+\omega_{K}\}.\end{split}

Note that in this case $G^{2}_{2}=\{1\}$ , but our discussions above imply that

\displaystyle\begin{split}G^{2}_{4}=\{1,(1+\omega_{K})^{2}\}=\{1,-1+2\omega_{K}\}\pmod{4}.\end{split}

One verifies directly that when $d=-2$ ,

(2.8)

\displaystyle\begin{split}G_{4}/G^{2}_{4}=\{\pm 1,\pm(1+\omega_{K})\}=\{\pm 1\}\times\{1,-(1+\omega_{K})\}\pmod{4}.\end{split}

We further note that if $d\neq-3$ , we have $U_{K}\cong<-1>$ by (2.1) and we have $G^{2}_{4}\cong U^{2}_{K}$ for $d=-3$ . Moreover, note that we have $U_{K}\cong U^{2}_{K}\times<-1>$ . It follows from these observations that we can apply the isomorphism in (2.7) and the quotient group given in (2.8) to write $G_{4}$ as a set by

\displaystyle\begin{split}G_{4}=\begin{cases}U_{K}\times G^{2}_{4}\times<1+2\omega_{K}>\pmod{4},&d\neq-2,-3,\\ U_{K}\times G^{2}_{4}\times\{1,-(1+\omega_{K})\}\pmod{4},&d=-2,\\ U_{K}\times<1+2\omega_{K}>\pmod{4},&d=-3.\end{cases}\end{split}

We then define the primary elements to be the ones that are congruent to the group $G_{4}/U_{K}$ modulo $4$ using the above representation. More precisely, the primary elements are the ones

(2.9)

\displaystyle\begin{split}\equiv\begin{cases}G^{2}_{4}\times<1+2\omega_{K}>\pmod{4},&d\neq-2,-3,\\ G^{2}_{4}\times\{1,-(1+\omega_{K})\}\pmod{4},&d=-2,\\ <1+2\omega_{K}>\pmod{4},&d=-3.\end{cases}\end{split}

It is then easy to see that each ideal co-prime to $2$ has a unique primary generator. Moreover, as $G_{4}/U_{K}$ is a group, we see that primary elements are closed under multiplication.

We close this section by pointing out a relationship between the notion of primary elements for $d=-3$ with that of $E$ -primary introduced in [Lemmermeyer, Section 7.3]. Recall that any $n=a+b\omega^{2}_{K}\in\mathcal{O}_{K}$ with $a,b\in\mathbb{Z}$ and $(n,6)=1$ is defined to be $E$ -primary if $n\equiv\pm 1\pmod{3}$ and satisfies

(2.10)

\displaystyle\begin{split}&a+b\equiv 1\pmod{4},\quad\text{if}\quad 2|b,\\ &b\equiv 1\pmod{4},\quad\text{if}\quad 2|a,\\ &a\equiv 3\pmod{4},\quad\text{if}\quad 2\nmid ab.\end{split}

Now, for any primary $n=a+b\omega_{K}\in\mathcal{O}_{K}$ with $a,b\in\mathbb{Z}$ such that $(n,6)=1$ , there is a unique $u\in U^{2}_{K}=\{1,\omega_{K},\omega^{2}_{K}\}$ such that $un\equiv\pm 1\pmod{3}$ . Writing $un$ in the form $c+d\omega^{2}_{K}$ using the observation that $\omega_{K}=-1-\omega^{2}_{K}$ , one checks using (2.10) directly that $un$ is $E$ -primary.

2.3. Quadratic Hecke characters and quadratic Gauss sums

Let $K$ be any imaginary quadratic number field. We say an element $n\in\mathcal{O}_{K}$ is odd if $(n,2)=1$ . For an odd prime $\varpi\in\mathcal{O}_{K}$ , the quadratic symbol $\left(\frac{\cdot}{\varpi}\right)$ is defined for $a\in\mathcal{O}_{K}$ , $(a,\varpi)=1$ by $\left(\frac{a}{\varpi}\right)\equiv a^{(N(\varpi)-1)/2}\pmod{\varpi}$ , with $\left(\frac{a}{\varpi}\right)\in\{\pm 1\}$ . If $\varpi|a$ , we define $\left(\frac{a}{\varpi}\right)=0$ . We then extend the quadratic symbol to $\left(\frac{\cdot}{n}\right)$ for any odd $n$ multiplicatively. We further define $\left(\frac{\cdot}{c}\right)=1$ for $c\in U_{K}$ .

We note the following quadratic reciprocity law concerning primary elements.

Lemma 2.4.

Let $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathcal{S}$ . For any co-prime primary elements $n,m$ with $(nm,2)=1$ , we have

(2.11)

\displaystyle\left(\frac{n}{m}\right)\left(\frac{m}{n}\right)=(-1)^{(N(n)-1)/2\cdot(N(m)-1)/2}.

Proof.

The assertion of the lemma is in [G&Zhao16, Lemma 2.4] for $d\in S\setminus\{-1,-3\}$ and one checks the arguments there also gives that (2.11) holds for $d=-3$ . If $d=-1$ , upon noting that $N(n)\equiv 1\pmod{4}$ for any odd $n$ , we see that (2.11) follows from [G&Zhao4, (2.1)]. This completes the proof. ∎

We also note that it follows from the definition that for any odd $n\in\mathcal{O}_{K}$ ,

(2.12)

\displaystyle\left(\frac{-1}{n}\right)=(-1)^{(N(n)-1)/2}.

Moreover, the following supplementary laws hold for primary odd $n=a+bi\in\mathcal{O}_{K}$ with $a,b\in\mathbb{Z}$ ,

(2.13)

\displaystyle\begin{split}&\left(\frac{i}{n}\right)=(-1)^{(1-a)/2},\quad n=a+bi\in\mathcal{O}_{K},a,b\in\mathbb{Z},K=\mathbb{Q}(i),\\ &\left(\frac{1-\omega_{K}}{n}\right)=1,\quad K=\mathbb{Q}(\sqrt{-3}).\end{split}

The first identity above follows from [BEW, Lemma 8.2.1] and the second follows by noting that $(1-\omega_{K})^{3}=1$ .

We follow the nomenclature of [iwakow, Section 3.8] to define a Dirichlet character $\chi$ modulo $(q)\neq(0)$ to be a homomorphism

\displaystyle\chi:\left(\mathcal{O}_{K}/(q)\right)^{\times}\rightarrow S^{1}:=\{z\in\mathbb{C}:|z|=1\}.

We say that $\chi$ is primitive modulo $(q)$ if it does not factor through $\left(\mathcal{O}_{K}/(q^{\prime})\right)^{\times}$ for any divisor $q^{\prime}$ of $q$ with $N(q^{\prime})<N(q)$ .

Suppose that $\chi(u)=1$ for any $u\in U_{K}$ . Then $\chi$ may be regarded as defined on ideals of $\mathcal{O}_{K}$ since every ideal is principal. In this case, we say that such a character $\chi$ is a Hecke character modulo $(q)$ of trivial infinite type. We also say such a Hecke character is primitive if it is primitive as a Dirichlet character. We say that $\chi$ is a Hecke character modulo $q$ instead of modulo $(q)$ if there is ambiguity.

For any Hecke character $\chi$ modulo $q$ of trivial infinite type and any $k\in\mathcal{O}_{K}$ , we define the associated Gauss sum $g_{K}(k,\chi)$ by

\displaystyle g_{K}(k,\chi)=\sum_{x\negthickspace\negthickspace\negthickspace\pmod{q}}\chi(x)\widetilde{e}_{K}\left(\frac{kx}{q}\right),\quad\mbox{where}\quad\widetilde{e}_{K}(z)=\exp\left(2\pi i\left(\frac{z}{\sqrt{D_{K}}}-\frac{\overline{z}}{\sqrt{D_{K}}}\right)\right).

We write $g_{K}(\chi)$ for $g_{K}(1,\chi)$ and note that our definition above for $g_{K}(k,\chi)$ is independent of the choice of a generator for $(q)$ . We define similarly $g_{K}(k,\chi_{n})$ , $g_{K}(\chi_{n})$ . The following result evaluates $g_{K}(\chi_{n})$ for any primary $n$ .

Lemma 2.5.

Let $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathcal{S}$ and $n$ a primary element in $\mathcal{O}_{K}$ . Then

(2.14)

\displaystyle g_{K}(\chi_{n})=\begin{cases}\sqrt{N(n)}\qquad&\text{for all}\;d,\;N(n)\equiv 1\pmod{4},\\ -i\sqrt{N(n)}\qquad&\text{for}\;d\neq-2,-7,\;N(n)\equiv-1\pmod{4},\\ i\sqrt{N(n)}\qquad&\text{for}\;d=-2,-7,\;N(n)\equiv-1\pmod{4}.\end{cases}

Proof.

We consider (2.14) for the case of $n$ being a primary prime first. Note that this has been established for the case $d=-1$ in [G&Zhao4, Lemma 2.2] for primary primes. In the rest of the proof, we assume that $d\neq-1$ . We note that it is shown in [G&Zhao16, Lemma 2.6] that for any $P$ -primary prime $\varpi$ that is co-prime to $2D_{K}$ and also to $(1-d)/4$ when $d\equiv 1\pmod{4}$ ,

(2.15)

\displaystyle g_{K}(\chi_{\varpi})=\begin{cases}\sqrt{N(\varpi)}\qquad&\text{for all}\;d,\;N(\varpi)\equiv 1\pmod{4},\\ -i\sqrt{N(\varpi)}\qquad&\text{for}\;d\neq-2,-7,\;N(\varpi)\equiv-1\pmod{4}.\end{cases}

Here we recall that when $d\neq-2$ , an odd element $n=a+b\omega_{K}$ with $a,b\in\mathbb{Z}$ is called $P$ -primary if $b\equiv 1\pmod{4}$ or $a+b(1-d)/4\equiv-1\pmod{4}$ when $(b,2)\neq 1$ . When $d=-2$ , an odd element $n=a+b\omega_{K}$ with $a,b\in\mathbb{Z},b\neq 0$ is called $P$ -primary when $b=2^{k}b^{\prime}$ with $k,b^{\prime}\in\mathbb{Z},k\geq 0,b^{\prime}\equiv 1\pmod{4}$ . The arguments in the proof of [G&Zhao16, Lemma 2.6] in fact imply that this is indeed true for any $P$ -primary prime. To see this, first note that when a prime $\varpi$ satisfies $N(\varpi)=p$ with $p$ a rational prime. Then we write $\varpi=a+b\omega_{K}$ to see via (2.2) that

\displaystyle p

\displaystyle=\begin{cases}\displaystyle a^{2}+ab+b^{2}\frac{1-d}{4},\qquad&\text{if}\;d\equiv 1\pmod{4},\\ \displaystyle a^{2}-db^{2},\qquad&\text{if}\;d\equiv 2,3\pmod{4}.\end{cases}

The above then implies that $(ab,p)=1$ . This is clear for the case $d\equiv 2,3\pmod{4}$ since the above expression implies that $a^{2}-db^{2}>p$ when $(p,ab)>1$ . On the other hand, when $d\equiv 1\pmod{4}$ , we have $(1-d)/4\geq 1$ so that $a^{2}+ab+b^{2}\frac{1-d}{4}\geq a^{2}+ab+b^{2}>p$ when $(p,ab)>1$ and $a,b$ are both non-negative or non-positive. When $ab<0$ , we note that $a^{2}+ab+b^{2}=(a+b)^{2}-ab>p$ when $(p,ab)>1$ . So in either case this we must have $(ab,p)=1$ .

For any fixed prime $\varpi$ , we define

\displaystyle E_{K}:=\sum_{x\bmod\varpi}\tilde{e}_{K}\left(\frac{x}{\varpi}\right).

Let $c\pmod{\varpi}$ be such that $\tilde{e}_{K}\left(\frac{c}{\varpi}\right)\neq 1$ . We note that such $c$ must exist, for otherwise $g_{K}(\chi_{\varpi})=\sum_{x\pmod{q}}\chi(x)=0$ , contradicting the well-known fact that $|g_{K}(\chi_{\varpi})|=N(\varpi)^{1/2}$ . Then

\displaystyle\tilde{e}_{K}\left(\frac{c}{\varpi}\right)E_{K}=\sum_{x\bmod\varpi}\tilde{e}_{K}\left(\frac{x+c}{\varpi}\right)=E_{K}.

The above then implies that $E_{K}=0$ . Now the arguments given in the proof of [G&Zhao16, Lemma 2.6] carry through with the above observations to show that the assertion of the lemma is valid for any $P$ -primary prime $\varpi$ in [G&Zhao16, Lemma 2.6].

Now, note that we have the easily verified relation that for any odd $n\in\mathcal{O}_{K}$ and any $u\in U_{K}$ ,

\displaystyle g_{K}(\chi_{un})=\left(\frac{u}{n}\right)g_{K}(\chi_{n}).

As $\left(\frac{u}{n}\right)=1$ for any $u\in U_{K}$ and any odd $n$ with $N(n)\equiv 1\pmod{4}$ by (2.12) and (2.13), we see that in order to establish (2.14) for the case $N(n)\equiv 1\pmod{4}$ , it suffices to show that this is so for any generator of the ideal $(n)$ . In particular, the validity of (2.15) implies that (2.14) is true for any primary prime $\omega$ when $N(\varpi)\equiv 1\pmod{4}$ .

Next, one checks via (2.9) that when a primary prime $\varpi$ has norm congrudent to $-1$ modulo $4$ , then it is also $P$ -primary when $d\neq-2,-7$ , while $-\varpi$ is $P$ -primary when $d=-2,-7$ . In fact, as $N(c)\equiv 1\pmod{4}$ for any $c\in G^{2}_{4}$ , it suffices to check this for elements $c\in<1+2\omega_{K}>$ for $d\neq-2$ and for elements $c\in\{1,-(1+\omega_{K})\}$ when $d=-2$ , which in turn can be easily verified. It follows from this and (2.15) that (2.14) holds for all primary primes.

Lastly, to establish the general case, we note that it follows from the definition of $g_{K}$ that for primary $n_{1},n_{2}$ with $(n_{1},n_{2})=1$ , we have

\displaystyle g_{K}(\chi_{n_{1}n_{2}})=

\displaystyle\left(\frac{n_{2}}{n_{1}}\right)\left(\frac{n_{1}}{n_{2}}\right)g_{K}(\chi_{n_{1}})g_{K}(\chi_{n_{1}}).

The general case of the assertion of the lemma now follows from the above, the case when $n$ is primary prime and Lemma 2.4 by induction on the number of primary primes dividing $n$ . ∎

Recall that $\chi^{(c_{K}c)}$ is a primitive quadratic character of trivial infinite type for any square-free $c$ . The associated Hecke $L$ -function satisfies then functional equation given in (2.25) below. It is expected that the root number $W(\chi)$ there is $1$ for primitive quadratic Hecke characters. Our next lemma confirms this.

Lemma 2.6.

Let $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathcal{S}$ . For any odd, square-free $c\in\mathcal{O}_{K}$ , we have

(2.16)

\displaystyle g_{K}(\chi^{(c_{K}c)})=\sqrt{N(c_{K}c)}.

Proof.

Note first that (2.16) is established in [Gao2, Lemma 2.2] for $d=-1$ . So we may assume that $d\neq-1$ in the rest of the proof. It follows from the Chinese remainder theorem that $x=c_{K}y+cz$ varies over the residue class modulo $c_{K}c$ as $y$ and $z$ vary over the residue class modulo $c$ and $c_{K}$ , respectively. Thus

\displaystyle g_{K}(\chi^{(c_{K}c)})=

\displaystyle\sum_{z\bmod{c_{K}}}\sum_{y\bmod{c}}\left(\frac{c_{K}}{c_{K}y+cz}\right)\left(\frac{c}{c_{K}y+cz}\right)\widetilde{e}_{K}\left(\frac{y}{c}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right).

As $\chi^{(c_{K})}$ is a Hecke character of trivial infinite type modulo $c_{K}$ , we get that

(2.17)

\displaystyle\left(\frac{c_{K}}{c_{K}y+cz}\right)=\chi^{(c_{K})}(c_{K}y+cz)=\chi^{(c_{K})}(cz).

On the other hand, let $s(z)$ denote the unique element in $U_{K}$ such that $s(z)z$ is primary for any $(z,2)=1$ . It follows from the quadratic reciprocity law (2.11) and the observation that $N(cz)\equiv N(c_{K}y+cz)\pmod{4}$ that

(2.18)

\displaystyle\left(\frac{c}{c_{K}y+cz}\right)=\left(\frac{s(z)c_{k}y}{c}\right)(-1)^{((N(c)-1)/2)((N(s(z)cz)-1)/2)}.

We then conclude from (2.17) and (2.18) that

\displaystyle\begin{split}g_{K}(\chi^{(c_{K}c)})=&\sum_{z\bmod{c_{K}}}\sum_{y\bmod{c}}\left(\frac{c_{K}}{cz}\right)\left(\frac{s(z)c_{K}y}{c}\right)(-1)^{((N(c)-1)/2)((N(s(z)cz)-1)/2)}\widetilde{e}_{K}\left(\frac{y}{c}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\\ =&\sum_{z\bmod{c_{K}}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{((N(c)-1)/2)((N(s(z)cz)-1)/2)}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\sum_{y\bmod{c}}\left(\frac{y}{c}\right)\widetilde{e}_{K}\left(\frac{y}{c}\right).\end{split}

Observe that when $N(c)\equiv 1\pmod{4}$ , we have $\left(\frac{s(z)c_{K}y}{c}\right)=1$ and that $((N(c)-1)/2)((N(s(z)cz)-1)/2)\equiv 0\pmod{2}$ . While when $N(c)\equiv-1\pmod{4}$ ,

\displaystyle\frac{N(c)-1}{2}\frac{N(s(z)cz)-1}{2}\equiv\frac{N(s(z)cz)-1}{2}\equiv\frac{N(c)-1}{2}+\frac{N(s(z))-1}{2}+\frac{N(z)-1}{2}\equiv 1+\frac{N(z)-1}{2}\pmod{2}.

We conclude from the above and Lemma 2.5 that

(2.19)

\displaystyle\begin{split}g_{K}(\chi^{(c_{K}c)})=&\begin{cases}\sqrt{N(c)}\displaystyle\sum_{z\bmod{c_{K}}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\qquad&\text{for all}\;d,\;N(c)\equiv 1\pmod{4},\\ i\sqrt{N(c)}\displaystyle\sum_{z\bmod{c_{K}}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\qquad&\text{for}\;d\neq-2,-7,\;N(c)\equiv-1\pmod{4},\\ -i\sqrt{N(c)}\displaystyle\sum_{z\bmod{c_{K}}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\qquad&\text{for}\;d=-2,-7,\;N(c)\equiv-1\pmod{4}.\end{cases}\end{split}

Recall that $c_{K}=8$ if $d\neq-2$ . Similar to (2.5), we see that as sets,

\displaystyle\begin{split}G_{8}=G_{4}\times\{1,5,1+4\omega_{K},5+4\omega_{K}\}\pmod{8},\end{split}

We further note that for $a,b\in\mathbb{Z}$ ,

\displaystyle\widetilde{e}_{K}\left(\frac{a+b\omega_{K}}{c_{K}}\right)=e\left(\frac{b}{c_{K}}\right).

Note also the following supplementary law (see [Lemmermeyer, Propostion 4.2(iii)]) that asserts for $(m,2)=1,m\in\mathcal{O}_{K}$ ,

(2.20)

\displaystyle\left(\frac{2}{m}\right)=\left(\frac{2}{N(m)}\right)_{\mathbb{Z}}.

As we have $N(m)\equiv N(5m)\pmod{8}$ for every $m\in G_{4}$ , we deduce from (2.20) that $\left(\frac{c_{K}}{m}\right)=\left(\frac{c_{K}}{5m}\right)$ for these $m$ . On the other hand, we have $e(b/8)=-e(5b/8)$ for any odd $b\in\mathbb{Z}$ . It follows that

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=m,5m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=\sum_{\begin{subarray}{c}z=m,5m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right).\end{split}

We observe from (2.4) and (2.7) that there are only four elements in $G_{4}$ that can be written as $a+b\omega_{K}$ with $a,b\in\mathbb{Z}$ , $2|b$ . These elements are precisely those in the set given in (2.6). Note that the norms of these elements are all $\equiv\pm 1\pmod{8}$ . Using again $\left(\frac{c_{K}}{m}\right)=\left(\frac{c_{K}}{5m}\right)$ for any $m\in G_{4}$ , we see that

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=m,5m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=2\sum_{\begin{subarray}{c}z=m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right).\end{split}

Using (2.20), a direct computation leads to

(2.21)

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=m,5m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=4.\end{split}

Similarly, we have $N((1+4\omega_{K})m)\equiv N((5+4\omega_{K})m)\pmod{8}$ for every $m\in G_{4}$ , so that $\left(\frac{c_{K}}{(1+4\omega_{K})m}\right)=\left(\frac{c_{K}}{(5+4\omega_{K})m}\right)$ for these $c$ . Moreover, if we write $m=a+b\omega_{K},(1+4\omega_{K})m=a^{\prime}+b^{\prime}\omega_{K}$ with $a,b,a^{\prime},b^{\prime}\in\mathbb{Z}$ , then it is easy to see that $b^{\prime}\equiv b\pmod{4}$ . It follows that

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=(1+4\omega_{K})m,(5+4\omega_{K})m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=&\sum_{\begin{subarray}{c}z=(1+4\omega_{K})m,(5+4\omega_{K})m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\\ =&2\sum_{\begin{subarray}{c}z=(1+4\omega_{K})m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right).\end{split}

A direct computation renders

(2.22)

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=(1+4\omega_{K})m,(5+4\omega_{K})m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\end{subarray}}\left(\frac{c_{K}}{z}\right)\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=4.\end{split}

Our discussions above apply similarly to the sum

\displaystyle\begin{split}\sum_{z\bmod{c_{K}}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right).\end{split}

As $s(z)=s(5z)$ and that $(N(5z)-1)/2\equiv(N(5)-1)/2+(N(z)-1)/2\equiv(N(z)-1)/2\pmod{2}$ , we see that

(2.23)

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=m,5m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\end{subarray}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=&2\sum_{\begin{subarray}{c}z=m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\\ =&\begin{cases}-4i,&d\neq-7,\\ 4i,&d=-7.\end{cases}\end{split}

Here we have $N(1+2\omega_{K})\equiv-1\pmod{8}$ when $d\neq-7$ and $N(1+2\omega_{K})\equiv 3\pmod{8}$ when $d=-7$ .

Similarly, we have $s((1+4\omega_{K})z)=s((5+4\omega_{K})z)$ and that $(N((1+4\omega_{K})z)-1)/2\equiv(N(1+4\omega_{K})-1)/2+(N(z)-1)/2\equiv(N(z)-1)/2\equiv(N((5+4\omega_{K})z)-1)/2\pmod{2}$ , so that we have

(2.24)

\displaystyle\begin{split}\sum_{\begin{subarray}{c}z=(1+4\omega_{K})m,(5+4\omega_{K})m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\end{subarray}}&\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)\\ =&2\sum_{\begin{subarray}{c}z=(1+4\omega_{K})m,m\in G_{4}\\ m=a+b\omega_{K},a,b\in\mathbb{Z}\\ 2|b\end{subarray}}\left(\frac{c_{K}}{z}\right)\left(\frac{s(z)}{c}\right)(-1)^{(N(z)-1)/2}\widetilde{e}_{K}\left(\frac{z}{c_{K}}\right)=\begin{cases}-4i,&d\neq-7,\\ 4i,&d=-7.\end{cases}\end{split}

Here $N(1+4\omega_{K})\equiv 5\pmod{8}$ . We conclude from (2.19), (2.21), (2.22), (2.23) and (2.24) that the assertion of the lemma holds for $d\neq-2$ .

Lastly, for the case $d=-2$ , we have $c_{K}=4\omega_{K}$ in this case. Observe that the residue classes modulo $(\omega_{K})$ consists of $\{0,1\}$ and that $1+4\{0,1\}$ are two distinct elements modulo $4\omega_{K}$ but both congruent to $1$ modulo $4$ . Thus, we have similar to (2.5) that as sets,

\displaystyle\begin{split}G_{c_{K}}=G_{4}\times\{1,5\}\pmod{c_{K}}.\end{split}

Note that this time we have $\widetilde{e}_{K}(m/c_{K})=e(-a/8)$ for any $m=a+b\omega_{K}$ with $a$ , $b\in\mathbb{Z}$ . Moreover, we have by [Lemmermeyer, Propostion 4.2(iii)],

\displaystyle\begin{split}&\left(\frac{\omega_{K}}{1+\omega_{K}}\right)=\left(\frac{\omega_{k}-(1+\omega_{K})}{1+\omega_{K}}\right)=\left(\frac{-1}{1+\omega_{K}}\right)=(-1)^{(N(1+\omega_{K})-1)/2}=-1,\\ &\left(\frac{\omega_{K}}{5}\right)=\left(\frac{N(\omega_{K})}{5}\right)_{\mathbb{Z}}=\left(\frac{2}{5}\right)_{\mathbb{Z}}=-1.\end{split}

One checks directly that the sums given in (2.21) and (2.23) this time equal $4\sqrt{2}$ , $4\sqrt{2}i$ , respectively. We conclude from this and (2.19) that the assertion of the lemma holds for $d=-2$ as well. This completes the proof of the lemma. ∎

2.7. Functional equations for Hecke $L$ -functions

Let $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathcal{S}$ and let $\chi$ be a primitive quadratic Hecke character of trivial infinite type modulo $q$ . A well-known result of E. Hecke asserts that $L(s,\chi)$ has an analytic continuation to the whole complex plane and satisfies the functional equation (see [iwakow, Theorem 3.8])

(2.25)

\displaystyle\Lambda(s,\chi)=W(\chi)\Lambda(1-s,\chi),\;\mbox{where}\;W(\chi)=g_{K}(\chi)(N(q))^{-1/2}

and

(2.26)

\displaystyle\Lambda(s,\chi)=(|D_{K}|N(q))^{s/2}(2\pi)^{-s}\Gamma(s)L(s,\chi).

We apply the above to the case when $\chi=\chi^{(c_{K}c)}$ for odd, square-free $c\in\mathcal{O}_{K}$ and deduce from Lemma 2.6 that $W(\chi^{(c_{K}c)})=1$ in our situation. It then follows from (2.25) and (2.26) that

(2.27)

\displaystyle L(s,\chi^{(c_{K}c)})=(|D_{K}|N(c_{K}c))^{1/2-s}(2\pi)^{2s-1}\frac{\Gamma(1-s)}{\Gamma(s)}L(1-s,\chi^{(c_{K}c)}).

2.8. A mean value estimate for quadratic Hecke $L$ -functions

In the proof of Theorem 1.1, we need the following lemma, which gives an upper bound for the second moment of quadratic Hecke $L$ -functions.

Lemma 2.9.

With the notation as above, let $S(X)$ denote the set of Hecke characters $\chi^{(m)}$ with $N(m)$ not exceeding $X$ . Then we have, for any complex number $s$ and any $\varepsilon>0$ with $\Re(s)\geq 1/2$ , $|s-1|>\varepsilon$ ,

(2.28)

\displaystyle\sum_{\begin{subarray}{c}\chi\in S(X)\end{subarray}}|L(s,\chi)|\ll

\displaystyle X^{1+\varepsilon}|s|^{1/2+\varepsilon}.

Proof.

Let $S^{\prime}(X)$ stand for the set of $\chi^{(m)}$ with $m$ being square-free and $N(m)\leq X$ . It follows from the arguments in the proof of [BGL, Corollary 1.4], upon using the quadratic large sieve for number fields [G&L, Theorem 1.1] that we have

\displaystyle\sum_{\begin{subarray}{c}\chi\in S^{\prime}(X)\end{subarray}}|L(s,\chi)|^{2}\ll

\displaystyle X^{1+\varepsilon}|s|^{1+\varepsilon}.

It follows from the above and the Cauchy-Schwarz inequality that the estimation given in (2.28) is valid with $S(X)$ replaced by $S^{\prime}(X)$ . Now, we write $m=m_{1}m^{2}_{2}$ with $m_{1}$ being square-free to see that

\displaystyle\sum_{\begin{subarray}{c}\chi\in S(X)\end{subarray}}|L(s,\chi)|\ll\sum_{N(m_{2})\geq 1}\sum_{\begin{subarray}{c}\chi\in S^{\prime}(X/N(m^{2}_{2}))\end{subarray}}|L(s,\chi)|\ll

\displaystyle X^{1+\varepsilon}|s|^{1/2+\varepsilon}.

This completes the proof of the lemma. ∎

2.10. Some results on multivariable complex functions

We include in this section some results from multivariable complex analysis. First we need the concept of a tube domain.

Definition 2.11.

An open set $T\subset\mathbb{C}^{n}$ is a tube if there is an open set $U\subset\mathbb{R}^{n}$ such that $T=\{z\in\mathbb{C}^{n}:\ \Re(z)\in U\}.$

For a set $U\subset\mathbb{R}^{n}$ , we define $T(U)=U+i\mathbb{R}^{n}\subset\mathbb{C}^{n}$ . We have the following Bochner’s Tube Theorem [Boc].

Theorem 2.12.

Let $U\subset\mathbb{R}^{n}$ be a connected open set and $f(z)$ be a function holomorphic on $T(U)$ . Then $f(z)$ has a holomorphic continuation to the convex hull of $T(U)$ .

The convex hull of an open set $T\subset\mathbb{C}^{n}$ is denoted by $\widehat{T}$ . Then we quote the result from [Cech1, Proposition C.5] on the modulus of holomorphic continuations of functions in multiple variables.

Proposition 2.13.

Assume that $T\subset\mathbb{C}^{n}$ is a tube domain, $g,h:T\rightarrow\mathbb{C}$ are holomorphic functions, and let $\tilde{g},\tilde{h}$ be their holomorphic continuations to $\widehat{T}$ . If $|g(z)|\leq|h(z)|$ for all $z\in T$ , and $h(z)$ is nonzero in $T$ , then also $|\tilde{g}(z)|\leq|\tilde{h}(z)|$ for all $z\in\widehat{T}$ .

3. Proof of Theorem 1.1

The Mellin inversion renders

(3.1)

\displaystyle\sideset{}{{}^{*}}{\sum}_{(c,2)=1}\frac{L(\frac{1}{2}+\alpha,\chi^{(c_{K}c)})}{L(\frac{1}{2}+\beta,\chi^{(c_{K}c)})}w\left(\frac{N(c)}{X}\right)=\frac{1}{2\pi i}\int\limits_{(2)}A_{K}\left(s,\tfrac{1}{2}+\alpha,\tfrac{1}{2}+\beta\right)X^{s}\widehat{w}(s)\mathrm{d}s,

where

\displaystyle\begin{split}A_{K}(s,w,z):=\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}c\mathrm{\ primary}\end{subarray}}\frac{L(w,\chi^{(c_{k}c)})}{L(z,\chi^{(c_{K}c)})N(c)^{s}},\end{split}

and $\widehat{w}$ is the Mellin transform of $w$ defined by

\displaystyle\widehat{w}(s)=\int\limits^{\infty}_{0}w(t)t^{s}\frac{\mathrm{d}t}{t}.

It follows from (3.1) that in order to prove Theorem 1.1, it suffices to understand the analytical properties of $A_{K}(s,w,z)$ . To that end, we write $\mu_{K}$ for the Möbius function on $\mathcal{O}_{K}$ . For $\Re(s),\Re(w),\Re(z)$ large enough,

(3.2)

\displaystyle\begin{split}A_{K}(s,w,z)=\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}c\mathrm{\ primary}\end{subarray}}\sum_{\begin{subarray}{c}m,k\end{subarray}}\frac{\mu_{K}(k)\chi^{(c_{K}c)}(k)\chi^{(c_{K}c)}(m)}{N(k)^{z}N(m)^{w}N(c)^{s}}=\sum_{\begin{subarray}{c}m,k\end{subarray}}\frac{\mu_{K}(k)\chi^{(c_{K})}(mk)}{N(m)^{w}N(k)^{z}}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}c\mathrm{\ primary}\end{subarray}}\frac{\chi_{mk}(c)}{N(c)^{s}}.\end{split}

Moreover, we apply the functional equation (2.27) to $L(w,\chi^{(c_{K}c)})$ and derive from (3.2) that

(3.3)

\displaystyle\begin{split}A_{K}(s,w,z)=&(|D_{K}|N(c_{K}))^{1/2-w}(2\pi)^{2w-1}\frac{\Gamma(1-w)}{\Gamma(w)}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}c\mathrm{\ primary}\end{subarray}}\frac{L(1-w,\chi^{(c_{K}c)})}{L(z,\chi^{(c_{K}c)})N(c)^{s+w-1/2}}\\ =&(|D_{K}|N(c_{K}))^{1/2-w}(2\pi)^{2w-1}\frac{\Gamma(1-w)}{\Gamma(w)}A_{K}(s+w-\tfrac{1}{2},1-w,z).\end{split}

The above can be regarded as a functional equation for $A_{K}(s,w,z)$ .

3.1. Regions of absolute convergence of $A_{K}(s,w,z)$

Note that when $c$ is primary, we have by Lemma 2.4 that for odd $m,k$ ,

\displaystyle\begin{split}\chi_{mk}(c)=\chi^{s(mk)(mk)}(c)(-1)^{(N(mk)-1)/2\cdot(N(c)-1)/2}=\left(\frac{16s(mk)(mk)(-1)^{(N(mk)-1)/2}}{c}\right):=\chi^{*}_{mk}(c),\end{split}

where we recall that $s(z)$ denotes the unique element in $U_{K}$ such that $s(z)z$ is primary for any $(z,2)=1$ and the second equality above follows from (2.12). Here $\chi^{*}_{mk}$ is a real Hecke character of trivial infinite type whose conductor divides $16mk$ .

The presence of $\chi^{(c_{K})}(mk)$ in (3.2) restricts the sums there to over odd $m$ , $k$ . This allows us to write the inner sum in the last expression of (3.2) as a Euler product, getting

(3.4)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}c\mathrm{\ primary}\end{subarray}}\frac{\chi_{mk}(c)}{N(c)^{s}}=\prod_{(\varpi,2)=1}(1+\chi^{*}_{mk}(\varpi)N(\varpi)^{-s})=\frac{L(s,\chi^{*}_{mk})}{\zeta^{(2mk)}_{K}(2s)}=\frac{L(s,\chi^{*}_{mk})}{\zeta_{K}(2s)}\prod_{\varpi|2mk}(1-N(\varpi)^{-2s})^{-1}.\end{split}

Thus (3.2) and (3.4) yield

(3.5)

\displaystyle\begin{split}A_{K}(s,w,z)=\sum_{\begin{subarray}{c}m,k\end{subarray}}\frac{\mu_{K}(k)\chi^{(c_{K})}(mk)}{N(m)^{w}N(k)^{z}}\frac{L(s,\chi^{*}_{mk})}{\zeta_{K}(2s)}\prod_{\varpi|2mk}(1-N(\varpi)^{-2s})^{-1}.\end{split}

Observe that when $\Re(s)>0$ ,

\displaystyle\prod_{\varpi|2mk}(1-N(\varpi)^{-2s})^{-1}\leq\prod_{\varpi|2mk}\Big{|}(1-2^{-2\Re(s)})^{-1}\Big{|}\leq((1-2^{-2\Re(s)})^{-1})^{\mathcal{W}_{K}(2mk)}\ll N(mk)^{\varepsilon},

where $\mathcal{W}_{K}$ denotes the number of distinct prime factors of $n$ and the last estimation above follows from the well-known bound (which can be derived in a manner similar to the proof of the classical case over $\mathbb{Q}$ given in[MVa1, Theorem 2.10])

\displaystyle\mathcal{W}_{K}(h)\ll\frac{\log N(h)}{\log\log N(h)},\quad\mbox{for}\quad N(h)\geq 3.

From (3.5) and the above, we infer that for $\Re(s)>0$ ,

(3.6)

\displaystyle\begin{split}\zeta_{K}(2s)A_{K}(s,w,z)\ll&\sum_{\begin{subarray}{c}m,k\end{subarray}}\frac{|L(s,\chi^{*}_{mk})|}{N(m)^{\Re(w)-\varepsilon}N(k)^{\Re(z)-\varepsilon}}.\end{split}

Now by Lemma 2.9,

(3.7)

\displaystyle\begin{split}\sum_{N(mk)\leq X}|L(s,\chi^{*}_{mk})|\ll\sum_{N(c)\leq X}d_{K}(c)|L(s,\chi^{*}_{c})|\ll X^{\varepsilon}\sum_{N(c)\leq X}|L(s,\chi^{*}_{c})|\ll X^{1+\varepsilon}|s|^{1/2+\varepsilon},\end{split}

where $d_{K}(n)$ is the divisor function on $\mathcal{O}_{K}$ and we have, similar to the classical bound for the divisor function on $\mathbb{Z}$ given in [MVa1, Theorem 2.11], that

\displaystyle\begin{split}d_{K}(n)\ll N(n)^{\varepsilon}.\end{split}

We thus conclude from (3.6), (3.7) and partial summation that the function $A_{K}(s,w,z)$ converges absolutely in the region

S_{1,1}=\{(s,w,z):\Re(s)>0,\ \Re(w)>1,\ \Re(z)>1,\ \Re(s+w)>3/2,\ \Re(s+z)>3/2\}.

In what follows, we shall define similar regions $S_{i,j}$ and $S_{j}$ and adopt the convention that for any real number $\delta$ ,

\displaystyle\begin{split}S_{i,j,\delta}:=\{(s,w)+\delta(1,1):(s,w)\in S_{i,j}\}\quad\mbox{and}\quad S_{j,\delta}:=\{(s,w)+\delta(1,1):(s,w)\in S_{j}\}.\end{split}

Using this notation, we see that in the region $S_{1,1,\varepsilon}$ ,

(3.8)

\displaystyle\begin{split}|(s-1)\zeta_{K}(2s)A_{K}(s,w,z)|\ll&(1+|s|)^{3/2+\varepsilon}.\end{split}

We note also from (3.2) that

(3.9)

\displaystyle\begin{split}A_{K}(s,w,z)\ll&\sideset{}{{}^{*}}{\sum}_{c\mathrm{\ primary}}\frac{|L(w,\chi^{(c_{K}c)})|}{|L(z,\chi^{(c_{K}c)})|N(c)^{\Re(s)}}.\end{split}

When $\Re(w)\geq 1/2$ , we apply from (2.28) and partial summation to infer that the sum above is convergent in the region

S_{1,2}=\{(s,w):\ \Re(s)>1,\ \Re(w)\geq 1/2,\ \Re(z)>1/2\}.

Recall from [iwakow, Theorem 5.19] that, on GRH,

(3.10)

\displaystyle\begin{split}\frac{1}{|L(z,\chi^{(c_{K}c)})|}\ll&(|z|N(c_{K}c))^{\varepsilon},\quad\Re(z)\geq 1/2+\varepsilon.\end{split}

Thus in the region $S_{1,2,\varepsilon}$ ,

(3.11)

\displaystyle\begin{split}|A_{K}(s,w,z)|\ll&|z|^{\varepsilon}(1+|w|)^{1/2+\varepsilon}.\end{split}

If $\Re(w)<1/2$ , we apply first the functional equation (2.27) and then deduce from (2.28) via partial summation that the sum in (3.9) is also convergent in the region

S_{1,3}=\{(s,w):\ \Re(s)>1,\ \Re(w)<1/2,\ \Re(s+w)>3/2,\ \Re(z)>1/2\}.

now Stirling’s formula (see [iwakow, (5.113)]) implies that

(3.12)

\displaystyle\frac{\Gamma(1-s)}{\Gamma(s)}\ll(1+|s|)^{1-2\Re(s)}.

This estimate, together with our discussions above, implies that in the region $S_{1,3,\varepsilon}$ ,

(3.13)

\displaystyle\begin{split}|A_{K}(s,w,z)|\ll&|z|^{\varepsilon}(1+|w|)^{1/2+1-2\Re(w)+\varepsilon}.\end{split}

We denote $S_{1,4}$ for the union of $S_{1,2}$ and $S_{1,3}$ so that

S_{1,4}=\{(s,w):\ \Re(s)>1,\ \Re(s+w)>3/2,\ \Re(z)>1/2\}.

We then deduce from (3.11) and (3.13) that in the region $S_{1,4,\varepsilon}$ , under GRH,

(3.14)

\displaystyle\begin{split}|A_{K}(s,w,z)|\ll&|z|^{\varepsilon}(1+|w|)^{1/2+\max\{1-2\Re(w),0\}+\varepsilon}.\end{split}

The point $(1/2,1,1)$ is in $S_{1,1}$ while $(1,1/2,1/2)$ , $(1,1/2,1)$ are in $S_{1,2}$ and the three points determine a plane whose equation is given by $\Re(s+w)=3/2$ . Similarly, the point $(1/2,1,1)$ is in $S_{1,1}$ and $(1,1/2,1/2)$ , $(1,1,1/2)$ are in $S_{1,2})$ and the three points determine a plane whose equation is $\Re(s+z)=3/2$ . It follows that the convex hull of $S_{1,1}$ and $S_{1,4}$ equals

S_{1}=\{(s,w,z):\ \Re(s)>0,\ \Re(z)>1/2,\ \Re(s+w)>3/2,\ \Re(s+z)>3/2\}.

Moreover, Proposition 2.13, (3.8), (3.14) and the bound $\zeta_{K}(2s)\ll 1$ for $\Re(s)>1$ yeild that in the region $S_{1,\varepsilon}$ , on GRH,

(3.15)

\displaystyle\begin{split}|(s-1)\zeta_{K}(2s)A_{K}(s,w,z)|\ll&(1+|s|)^{3/2+\varepsilon}|z|^{\varepsilon}(1+|w|)^{1/2+\max\{1-2\Re(w),0\}+\varepsilon}.\end{split}

We now apply the functional equation for $A_{K}(s,w,z)$ given in (3.3) and follow our discussions above to see that $A_{K}(s,w,z)$ is also holomorphic in the region

S_{2}=\{(s,w,z):\ \Re(s+w)>1/2,\ \Re(z)>1/2,\ \Re(s)>1,\ \Re(s+w+z)>2\}.

We also deduce from (3.3), (3.12) and (3.15) that in the region $S_{2,\varepsilon}$ for any $\varepsilon>0$ , we have on GRH,

(3.16)

\displaystyle\begin{split}|(s+w-3/2)&\zeta_{K}(2s+2w-1)A_{K}(s,w,z)|\\ \ll&(1+|s+w-1/2|)^{3/2+\varepsilon}|z|^{\varepsilon}(1+|1-w|)^{1/2+\max\{1-2\Re(w),0\}+\varepsilon}(1+|w|)^{1-2\Re(w)}\\ \ll&(1+|s|)^{3/2+\varepsilon}|z|^{\varepsilon}(1+|w|)^{3-2\Re(w)+\max\{1-2\Re(w),0\}+\varepsilon}.\end{split}

The union of $S_{1}$ and $S_{2}$ is connected and the convex hull of $S_{1},S_{2}$ equals

(3.17)

\displaystyle\begin{split}&S_{3}=\{(s,w,z):\ \Re(s)>0,\ \Re(z)>1/2,\ \Re(s+w+z)>2,\\ &\hskip 36.135pt\ \Re(2s+w+z)>3,\ \Re(2s+w)>3/2,\ \Re(s+w)>1/2,\ \Re(s+z)>3/2\}.\end{split}

To see this, note that the two planes $\Re(s+w)=3/2$ and $\Re(s+z)=3/2$ intersect when $z=1/2$ at the line given by the intersection of $\Re(s+w)=3/2$ and $\Re(s)=1$ . Note further that the plane $\Re(s+w+z)=2$ can be written as $\Re(s+w)=2-z$ and $2-z=3/2$ for $z=1/2$ . Note also that $2-z=1/2$ when $z=3/2$ . It follows that the convex hull of $S_{1}$ and $S_{2}$ contains the plane determined by the two lines: $\Re(s+w)=3/2$ , $\Re(s+z)=3/2$ on $S_{1}$ and $\ \Re(s+w+z)=2,\ \Re(s)=1$ on $S_{2}$ , intersecting with the two planes: $z=1/2$ and $z=3/2$ . Note that the two lines both intersect the plane $z=1/2$ at the point $(1,1/2,1/2)$ . The first line intersects the plane $z=3/2$ at $(0,3/2,3/2)$ , the second at $(1,-1/2,3/2)$ . These three points then determine a plane with the equation $\Re(2s+w+z)=3$ . Moreover, when $z>3/2$ , the convex hull continues with the plane that is determined by the lines: $\Re(s)=0,\ \Re(s+w)=3/2$ in $S_{1}$ and $\Re(s)=1,\ \Re(s+w)=1/2$ in $S_{2}$ . As the point $(0,3/2,3/2)$ is on the first line and the point $(1,-1/2,3/2)$ is on the second line, we see easily that the plane has the equation: $\Re(2s+w)=3/2$ .

Consequently, Theorem 2.12 gives that $(s-1)(s+w-1)\zeta_{K}(2s)\zeta_{K}(2w+2s-1)A_{K}(s,w,z)$ converges absolutely in the region $S_{3}$ . Moreover, Proposition 2.13 supplies us with the bound, inherited from (3.15) and (3.16), that in the region

S_{\varepsilon}:=S_{3,\varepsilon}\cap\{(s,w,z):\ \Re(s)\geq 1/2+\varepsilon,\ \Re(s+w)\geq 1+\varepsilon,\ \Re(z)\geq 1/2+\varepsilon\},

for any $0<\varepsilon<1/100$ ,

\displaystyle\begin{split}|(s-1)(s+w-1)\zeta_{K}(2s)\zeta_{K}(2w+2s-1)A(s,w)|\ll&|\zeta_{K}(2s)\zeta_{K}(2w+2s-1)|(1+|s|)^{5/2+\varepsilon}|z|^{\varepsilon}(1+|w|)^{5/2+\varepsilon}.\end{split}

Furthermore, when $\Re(s)\geq 1/2+\varepsilon$ and $\Re(s+w)\geq 1+\varepsilon$ , we have $1\ll\zeta(2s)$ and $\zeta(2w+2s-1)\ll 1$ . So from (3.13), in the region $S_{\varepsilon}$ ,

(3.18)

\displaystyle\begin{split}|(s-1)(s+w-1)A_{K}(s,w,z)|\ll&(1+|s|)^{5/2+\varepsilon}|z|^{\varepsilon}(1+|w|)^{5/2+\varepsilon}.\end{split}

3.2. Residues of $A_{K}(s,w,z)$ at $s=1$ and $s+w=3/2$

It follows from (3.4) that $A_{K}(s,w,z)$ has a pole at $s=1$ arising from the terms with $mk=u\square$ with $u\in U_{K}$ , where $\square$ denotes a perfect square. In this case,

L\left(s,\chi^{*}_{mk}\right)=\zeta_{K}(s)\prod_{\varpi|2mk}\left(1-\frac{1}{N(\varpi)^{s}}\right).

Recall that we denote the residue of $\zeta_{K}(s)$ at $s=1$ by $r_{K}$ and for any $n\in\mathcal{O}_{K}$ . We then deduce from (3.5) that

(3.19)

\displaystyle\mathrm{Res}_{s=1}A_{K}(s,w,z)=\frac{r_{K}|U_{K}|^{2}}{\zeta_{K}(2)}\sum_{\begin{subarray}{c}mk=\square\\ m,k\mathrm{\ primary}\end{subarray}}\frac{\mu_{K}(k)a(2mk)}{N(m)^{w}N(k)^{z}}=\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\sum_{\begin{subarray}{c}mk=\square\\ m,k\mathrm{\ primary}\end{subarray}}\frac{\mu_{K}(k)a(mk)}{N(m)^{w}N(k)^{z}},

where $a(n)$ is defined in (1.2).

We now express the last sum in (3.19) as an Euler product via a direct computation similar to that done in [Cech1, (4.11)] to obtain that

(3.20)

\displaystyle\mathrm{Res}_{s=1}A_{K}(s,w,z)=\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\frac{\zeta_{K}^{(2)}(2w)}{\zeta_{K}^{(2)}(w+z)}P(w,z),

where $P(w,z)$ is given in (1.4).

Similarly, we deduce from (3.3) that

(3.21)

\displaystyle\begin{split}\mathrm{Res}_{s=3/2-w}A_{K}(s,w,z)=&(|D_{K}|N(c_{K}))^{1/2-w}(2\pi)^{2w-1}\frac{\Gamma(1-w)}{\Gamma(w)}\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\frac{\zeta_{K}^{(2)}(2-2w)}{\zeta_{K}^{(2)}(1-w+z)}P(1-w,z).\end{split}

3.3. Completion of proof

Now repeated integration by parts gives that for any integer $A\geq 0$ ,

(3.22)

\displaystyle\widehat{w}(s)\ll\frac{1}{(1+|s|)^{A}}.

We evaluate the integral in (3.1) by shifting the line of integration to $\Re(s)=E(\alpha,\beta)+\varepsilon$ , where $E(\alpha,\beta)$ is given in (1.1). Applying (3.18) and (3.22) gives that the integral on the new line can be absorbed into the $O$ -term in (1.3). We encounter simple poles at $s=1$ and $s=1-\alpha$ in the process whose residues are given in (3.20) and (3.21), respectively. This yields the main terms in (1.3) and completes the proof of Theorem 1.1.

4. Proof of Theorem 1.4

We first recast the expression in (1.3) as

(4.1)

\displaystyle\begin{split}&\sideset{}{{}^{*}}{\sum}_{(c,2)=1}\frac{L(\frac{1}{2}+\alpha,\chi^{(c_{K}c)})}{L(\frac{1}{2}+\beta,\chi^{(c_{K}c)})}w\left(\frac{N(c)}{X}\right)=XM_{1}(\alpha,\beta)+X^{1-\alpha}M_{2}(\alpha,\beta)+R(X,\alpha,\beta),\end{split}

where $R(X,\alpha,\beta)$ is the error term in (1.3) and we define

\displaystyle\begin{split}M_{1}(\alpha,\beta)=&X\widehat{w}(1)\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\frac{\zeta_{K}^{(2)}(1+2\alpha)}{\zeta_{K}^{(2)}(1+\alpha+\beta)}P(\tfrac{1}{2}+\alpha,\tfrac{1}{2}+\beta),\\ M_{2}(\alpha,\beta)=&X^{1-\alpha}\widehat{w}(1-\alpha)(|D_{K}|N(c_{K}))^{-\alpha}(2\pi)^{2\alpha}\frac{\Gamma(\tfrac{1}{2}-\alpha)}{\Gamma(\tfrac{1}{2}+\alpha)}\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\frac{\zeta_{K}^{(2)}(1-2\alpha)}{\zeta_{K}^{(2)}(1-\alpha+\beta)}P(\tfrac{1}{2}-\alpha,\tfrac{1}{2}+\beta).\end{split}

Note that the expression on the left-hand side of (4.1) and both $M_{1}(\alpha,\beta)$ and $M_{2}(\alpha,\beta)$ are analytic functions of $\alpha,\beta$ , so is $E(X,\alpha,\beta)$ .

We now differentiate the above terms with respect to $\alpha$ for a fixed $\beta=r$ with $\Re(\beta)>\varepsilon$ , and then set $\alpha=\beta=r$ to see that

(4.2)

\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}\alpha}XM_{1}(\alpha,\beta)\Big{|}_{\alpha=\beta=r}&=X\widehat{w}(1)\frac{r_{K}|U_{K}|^{2}a(2)}{\zeta_{K}(2)}\left(\frac{(\zeta^{(2)}_{K}(1+2r))^{\prime}}{\zeta^{(2)}_{K}(1+2r)}+\sum_{(\varpi,2)=1}\frac{\log N(\varpi)}{N(\varpi)(N(\varpi)^{1+2r}-1)}\right).\end{split}

For the second term, we notice that due to the factor $1/\zeta_{K}(1-\alpha+\beta)$ , only one term remains. Moreover, $1/\zeta_{K}(s)=r^{-1}_{K}(s-1)+O((s-1)^{2})$ and $P(1-\alpha,1+\beta)=\zeta^{(2)}_{K}(2)/\zeta^{(2)}_{K}(2-2r)$ . It follows that

(4.3)

\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}\alpha}&X^{1-\alpha}M_{2}(\alpha,\beta)\Big{|}_{\alpha=\beta=r}=-X^{1-r}\widehat{w}(1-r)(|D_{K}|N(c_{K}))^{-r}(2\pi)^{2r}\frac{\Gamma(\tfrac{1}{2}-r)}{\Gamma(\tfrac{1}{2}+r)}\frac{|U_{K}|^{2}a(2)}{\zeta^{(2)}_{K}(2-2r)}.\end{split}

Next, as $R(X,\alpha,\beta)$ is analytic in $\alpha$ , Cauchy’s integral formula yields

\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}\alpha}R(X,\alpha,\beta)=\frac{1}{2\pi i}\int\limits_{C_{\alpha}}\frac{R(X,z,\beta)}{(z-\alpha)^{2}}\mathrm{d}z,\end{split}

where $C_{\alpha}$ is a circle centered at $\alpha$ of radius $\rho$ with $\varepsilon/2<\rho<\varepsilon$ . It follows that

\displaystyle\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}\alpha}R(X,\alpha,\beta)\right|\ll\frac{1}{\rho}\cdot\max_{z\in C_{\alpha}}|R(X,z,\beta)|\ll(1+|\alpha|)^{5/2+\varepsilon}(1+|\beta|)^{\varepsilon}X^{E(\alpha,\beta)+\varepsilon}.\end{split}

We now set $\alpha=\beta=r$ to deduce from (1.1) and the above that

(4.4)

\displaystyle\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}\alpha}R(X,\alpha,\beta)\right|\ll(1+|r|)^{5/2+\varepsilon}X^{1-2r+\varepsilon}.\end{split}

We conclude from (4.1)–(4.4) that (1.6) holds, completing the proof of Theorem 1.4.

Acknowledgments. P. G. is supported in part by NSFC grant 11871082 and L. Z. by the FRG Grant PS43707 at the University of New South Wales.

Ratios conjecture for primitive quadratic Hecke LL-functions

Abstract.

1. Introduction

Theorem 1.1.

Theorem 1.2.

Corollary 1.3.

Theorem 1.4.

2. Preliminaries

2.1. Imaginary quadratic number fields

2.2. Primary Elements

2.3. Quadratic Hecke characters and quadratic Gauss sums

Lemma 2.4.

Proof.

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

2.7. Functional equations for Hecke LL-functions

2.8. A mean value estimate for quadratic Hecke LL-functions

Lemma 2.9.

Proof.

2.10. Some results on multivariable complex functions

Definition 2.11.

Theorem 2.12.

Proposition 2.13.

3. Proof of Theorem 1.1

3.1. Regions of absolute convergence of AK​(s,w,z)A_{K}(s,w,z)

3.2. Residues of AK​(s,w,z)A_{K}(s,w,z) at s=1s=1 and s+w=3/2s+w=3/2

3.3. Completion of proof

4. Proof of Theorem 1.4

References

Ratios conjecture for primitive quadratic Hecke $L$ -functions

2.7. Functional equations for Hecke $L$ -functions

2.8. A mean value estimate for quadratic Hecke $L$ -functions

3.1. Regions of absolute convergence of $A_{K}(s,w,z)$

3.2. Residues of $A_{K}(s,w,z)$ at $s=1$ and $s+w=3/2$