First Moments of Some Hecke $L$-functions of Prime Moduli

Moments of $L$-functions can be used to address the non-vanishing of the central values of $L$-functions, an issue that attracts much attention in the literature due to significant arithmetic information carried by these central values. For example, the central values of quadratic Dirichlet $L$-functions are related to class numbers of imaginary quadratic fields ([iwakow, p. 514]), so that a considerable amount of investigations have been done on moments of this family of $L$-functions.

In [Jutila], M. Jutila initiated the study on the first and second moments central values of the family of quadratic $L$-functions and used his result to show that there are infinitely many $L$-functions in this family with non-vanishing central values. The error terms in Jutila’s result was subsequently improved in [DoHo, MPY, ViTa] for the first moment and [sound1, Sono] for the second moment. A valid asymptotic formula for the third moment of central values of the family of quadratic Dirichlet $L$-functions was first established by K. Soundararajan in [sound1] and the error term for a smoothed version was later improved by M. P. Young in [Young2]. Recently, Q. Shen obtained a valid asymptotic formula in [Shen] for the fourth moment of central values of the same family of $L$-functions under the Generalized Riemann Hypothesis (GRH).

Due to its relation to the Birch and Swinnerton-Dyer conjecture, the family of quadratic twists of a modular form is another important family. The mean value of that family has been studied in [BFH, Iwan1, Munshi, MM, Petrow1]. Assuming GRH, the second moment of the family was computed by K. Soundararajan and M. P. Young in [S&Y].

In addition to the above mentioned families, much work has been done on moments of Hecke $L$-functions associated with various families of characters of a fixed order. In [Luo], W. Luo studied the first two moments of cubic Hecke $L$-functions in $\mathbb{Q}(\omega)$, where $\omega=\frac{-1+\sqrt{3}i}{2}$ is a primitive cubic root of unity. The analogue case for cubic Dirichlet $L$-functions was obtained by S. Baier and M. P. Young in [B&Y]. We refer the reader to the articles [FaHL, GHP, FHL, Diac, G&Zhao1] for related results in this area.

All the above mentioned results have the common feature that the set of conductors of each family of $L$-functions considered has a positive density in the ring of integers in their respective number fields. Keeping in mind that the set of rational primes is a sparse set (a subset of density zero in the set of rational integers), it is then interesting to consider the case when the set of conductors of a family forms a sparse set. In fact, a weighted first moment of central values of the family of quadratic Dirichlet $L$-functions of prime conductors has already been studied by Jutila in [Jutila], the same paper in which he obtained the first two moments of the same family of $L$-functions when the set of conductors of the family has a positive density. A weighted second moment of central values of the family of quadratic Dirichlet $L$-functions of prime conductors was recently computed by S. Baluyot and K. Pratt in [BP] assuming GRH.

Motivated by the above results of Jutila, we study, in this paper, the first moments of a few families of $L$-functions with prime moduli assuming GRH. Our first result is an analogue of Jutila’s result on quadratic Hecke $L$-functions. Let $K$ be a number field of class number one, $\mathcal{O}_{K}$ be its ring of integers and $U_{K}$ the group of units in $\mathcal{O}_{K}$. We shall denote $\varpi$ for a prime number in $\mathcal{O}_{K}$, by which we mean that the ideal $(\varpi)$ generated by $\varpi$ is a prime ideal. We write $N(k),\mathrm{Tr}(k)$ for the norm and trace of any $k\in K$. We further denote $\chi$ for a Hecke character of $K$ and we say that $\chi$ is of trivial infinite type if its component at infinite places of $K$ is trivial. We write $L(s,\chi)$ for the $L$-function associated to $\chi$ and $\zeta_{K}(s)$ for the Dedekind zeta function of $K$ and $\zeta(s)$ for the Riemann zeta function. We also use $\Lambda(n)$ for the von Mangoldt function on $\mathcal{O}_{K}$ given by

$\displaystyle\Lambda(n)=\begin{cases}\log N(\varpi)\qquad&n=\varpi^{k},\text{$\varpi$ prime},k\geq 1,\\ 0\qquad&\text{otherwise}.\end{cases}$

In the remainder of this section, we let $K=\mathbb{Q}(\omega)$ or $\mathbb{Q}(i)$. Let $\chi_{\varpi}$ be a quadratic Hecke symbol defined in Section 2.1 and it is shown there that $\chi_{\varpi}$ is a Hecke character of trivial infinite type when $\varpi\equiv 1\pmod{36},\varpi\in\mathbb{Z}[\omega]$ or $\varpi\equiv 1\pmod{16},\varpi\in\mathbb{Z}[i]$. For the corresponding families of Hecke $L$-functions, we have the following

The proof of Theorem 1.1 is given in Section 3. As we are summing over primes, we cannot apply the Poisson summation (due to K. Soundararajan in [sound1]) to deal with these sums. Thus our treatment here is similar to that of Jutila in [Jutila], except that we use the assumption on GRH to get better error terms.

Let $\chi_{j,\varpi},j=3$, $4$ be a cubic or quartic Hecke symbol defined in Section 2.1. It is shown there that $\chi_{3,\varpi}$ is a Hecke character of trivial infinite type when $\varpi\equiv 1\pmod{9},\varpi\in\mathbb{Z}[\omega]$ and $\chi_{4,\varpi}$ is a Hecke character of trivial infinite type when $\varpi\equiv 1\pmod{16},\varpi\in\mathbb{Z}[i]$. Our next result deals with the associated cubic or quartic Hecke $L$-functions.

The proof of Theorem 1.2 is given in Section 4 and our approach is analogous to that used by W. Luo in [Luo]. In particular, we make crucial use (see Lemma 2.5 below) of a result of S. J. Patterson in [P] on estimations of certain Gauss sums over primes to control the error term.

Let $\chi$ be a Dirichlet character modulo $q$ and $\chi_{0}$ the principal character modulo $q$ and we shall say that the order of $\chi$ is $j\in\mathbb{N}$ if $\chi^{j}=\chi_{0}$ but $\chi^{i}\neq\chi_{0}$ for all $1\leq i<j$. We denote the order of $\chi$ by $\text{ord}(\chi)$. By Lemma 2.2 below, we see that each cubic or quartic symbol $\chi_{j,\varpi}$, $j=3$, $4$ gives rise to a corresponding Dirichlet character of order $3$ or $4$ modulo $N(\varpi)$. One can consider the first moments of these $L$-functions and our result is

The proof of Theorem 1.3 is similar to that of Theorem 1.2 and will be given in Section 5.

In this section, we include some auxiliary results needed in the proofs of our theorems.

As the proofs for $K=\mathbb{Q}(\omega)$ and $\mathbb{Q}(i)$ are similar, we will only give the proof for the case $K=\mathbb{Q}(\omega)$ here. We shall hence fix $K=\mathbb{Q}(\omega)$ throughout the proof. We apply (2.8) and arrive at

(3.1)

$\displaystyle\begin{split}\sum_{\varpi\equiv 1\bmod{36}}L\left(\frac{1}{2},\chi_{\varpi}\right)\Lambda(\varpi)\Phi\left(\frac{N(\varpi)}{y}\right)=&2\sum_{\varpi\equiv 1\bmod{36}}\ \sum_{0\neq\mathcal{A}\subset O_{K}}\frac{\chi_{\varpi}(\mathcal{A})\Lambda(\varpi)}{N(\mathcal{A})^{1/2}}V\left(\frac{2\pi N(\mathcal{A})}{(3N(\varpi))^{1/2}}\right)\Phi\left(\frac{N(\varpi)}{y}\right)\\ =&M+O\left(y^{3/4+\varepsilon}\right),\end{split}$

where

(3.2)

$\displaystyle M=2\sum_{c\equiv 1\bmod{36}}\ \sum_{0\neq\mathcal{A}\subset O_{K}}\frac{\chi_{c}(\mathcal{A})\Lambda(c)}{N(\mathcal{A})^{1/2}}V\left(\frac{2\pi N(\mathcal{A})}{(3N(c))^{1/2}}\right)\Phi\left(\frac{N(c)}{y}\right).$

Here the first summation in the defintion of $M$ runs over all elements $c\in\mathbb{Z}[\omega]$ and the last equality in (3.1) follows from the rapid decay of $\Phi$ and $V$ given in (2.7) so that the contribution from the higher prime power is

$\displaystyle\ll\sum_{\begin{subarray}{c}N(\varpi^{k})\ll y^{1+\varepsilon}\\ k\geq 2\end{subarray}}\sum_{\begin{subarray}{c}0\neq\mathcal{A}\subset O_{K}\\ N(\mathcal{A})\ll N(\varpi^{k})^{1/2+\varepsilon}\end{subarray}}\frac{1}{N(\mathcal{A})^{1/2}}\ll y^{3/4+\varepsilon}.$

For $(a,6)=1$, we define a Hecke character $\chi^{(a)}\pmod{36a}$ such that for any ideal $(c)$ co-prime to $6$, with $c$ being the unique primary generator of $(c)$, $\chi^{(a)}((c))$ is defined as $\chi^{(a)}((c))=\left(\frac{a}{c}\right)$. One checks easily that $\chi^{(a)}$ is a Hecke character $\pmod{36a}$ of trivial infinite type. By an abuse of notation, we shall also write $\chi^{(a)}(c)$ for $\chi^{(a)}((c))$. In particular, we have $\chi_{c}(a)=\chi^{(a)}(c)$. Note that any integral non-zero ideal $\mathcal{A}$ in $\mathbb{Z}[\omega]$ has a unique generator $2^{r_{1}}(1-\omega)^{r_{2}}a$, with $r_{1},r_{2}\in\mathbb{Z},r_{1},r_{2}\geq 0,a\in\mathbb{Z}[\omega]$, $(a,2)=1,a\equiv 1\pmod{3}$, it follows from this and our discussions above that $\chi_{c}(\mathcal{A})=\chi^{(a)}(c)$. Thus we have

$$M=2\sum_{\begin{subarray}{c}r_{1},r_{2}\geq 0\\ (a,2)=1\\ a\equiv 1\bmod 3\end{subarray}}\frac{1}{2^{r_{1}}3^{r_{2}/2}N(a)^{1/2}}M(r,a),$$

where

$$M(r,a)=\sum_{c\equiv 1\bmod{36}}\chi^{(a)}(c)\Lambda(c)V\left(\frac{\pi 2^{2r_{1}+1}3^{r_{2}-1/2}N(a)}{N(c)^{1/2}}\right)\Phi\left(\frac{N(c)}{y}\right).$$

We set

$\displaystyle\tilde{f}(s)=\int\limits^{\infty}_{0}V\left(\frac{\pi 2^{2r_{1}+1}3^{r_{2}-1/2}N(a)}{(xy)^{1/2}}\right)\Phi(x)x^{s-1}\mathrm{d}x.$

Integration by parts and using (2.7) shows that $\tilde{f}(s)$ is a function satisfying the bound for all $\Re(s)>0$, and $E>0$,

(3.3)

$\displaystyle\tilde{f}(s)\ll(1+|s|)^{-E}\left(1+\frac{2^{2r_{1}+1}3^{r_{2}-1/2}N(a)}{y^{1/2}}\right)^{-E}.$

We now apply Mellin inversion to see that

(3.4)

$\displaystyle\begin{split}M(r,a)=&\sum_{\begin{subarray}{c}c\equiv 1\bmod{36}\end{subarray}}\chi^{(a)}(c)\Lambda(c)\frac{1}{2\pi i}\int\limits_{(2)}\left(\frac{y}{N(c)}\right)^{s}\tilde{f}(s)\ \mathrm{d}s\\ =&\frac{1}{2\pi i}\int\limits_{(2)}\tilde{f}(s)y^{s}\sum_{\begin{subarray}{c}c\equiv 1\bmod{36}\end{subarray}}\frac{\chi^{(a)}(c)\Lambda(c)}{N(c)^{s}}\mathrm{d}s=\frac{1}{\#h_{(36)}}\sum_{\psi\bmod{36}}\frac{1}{2\pi i}\int\limits\limits_{(2)}\tilde{f}(s)y^{s}\left(-\frac{L^{\prime}(s,\psi\chi^{(a)})}{L(s,\psi\chi^{(a)})}\right)\mathrm{d}s.\end{split}$

where the last equality above follows from using the ray class characters to detect the condition that $c\equiv 1\bmod{36}$ with $\psi$ running over all ray class characters $\pmod{36}$.

We shift the contour of integration to $1/2+\varepsilon$ in the last integral of (3.4) to evaluate $M$. By doing so, we encounter a pole at $s=1$ of $L^{\prime}(s,\psi\chi^{(a)})/L(s,\psi\chi^{(a)})$ when $\psi\chi^{(a)}$ is principal, mindful of our assumption of GRH. We set $M_{0}$ to be the contribution to $M$ of these residues, and $M_{1}$ to be the remainder.

We treat $M_{1}$ by bounding everything by absolute values and use (3.3) to get that for any $E>0$,

(3.5)

$\displaystyle M_{1}\ll y^{1/2+\varepsilon}\sum_{\psi\bmod{36}}\sum_{\begin{subarray}{c}r_{1},r_{2}\geq 0\\ (a,2)=1\\ a\equiv 1\bmod 3\end{subarray}}\frac{1}{2^{r_{1}}3^{r_{2}/2}N(a)^{1/2}}\left(1+\frac{2^{2r_{1}+1}3^{r_{2}-1/2}N(a)}{y^{1/2}}\right)^{-E}\int\limits^{\infty}_{-\infty}\left|\frac{L^{\prime}\left(1/2+it,\psi\chi^{(a)}\right)}{L\left(1/2+it,\psi\chi^{(a)}\right)}\right|(1+|t|)^{-E}\mathrm{d}t.$

Note that it follows from [iwakow, Theorem 5. 17] that

$\displaystyle\frac{L^{\prime}\left(1/2+it,\psi\chi^{(a)}\right)}{L\left(1/2+it,\psi\chi^{(a)}\right)}\ll(\log(N(a)(1+|s|)))^{1+\varepsilon}.$