Non-vanishing of central values of angular Hecke $L$-functions

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

Introduction

Let $K = \mathbb{Q}(\sqrt{d})$ be an imaginary quadratic number field of class number one. Here $d$ is a negative, square-free integer and it is well-known that (see [3, (22.77)]) the class number of $K$ is one if and only if

We denote $\mathcal{O}_K$, $U_K$, $D_K$ the ring of integers, the group of units and the discriminant of $K$, respectively. Note that (see [2, Section 2.1]) the order $\lvert U_K\rvert$ of $U_K$ equals 4 when $d=-1$, equals 6 when $d=-3$ and equals 2 for other values of $d$ listed in (1.1). Throughout the paper, we denote $M_K$ the integer such that $2M_K=\lvert U_K\rvert$.

Denote $\mathfrak{a}$ an integral ideal of $\mathcal{O}_K$. As $\mathcal{O}_K$ is a principal ideal domain, the generators of $\mathfrak{a}=(\alpha)\subseteq\mathcal{O}_K$ differ by multiplication of an element in $U_K$. In what follows we shall therefore fix a generator $\alpha$ of $(\alpha)$ such that $\theta_{\mathfrak{a}}:=\theta_\alpha$ lies in $[-\pi/\lvert U_K\rvert,\pi/\lvert U_K\rvert)$, where $\theta_\alpha$ denotes the argument of $\alpha$.

For any $l\in\mathbb{Z}$, we define the Hecke character

Note that as $\lambda_{M_Kl}$ is trivial on $U_K$, it defines a Hecke character of trivial infinite type. We write the associated Hecke $L$-function as

where $N(\mathfrak{a})$ denotes the norm of the integral ideal $\mathfrak{a}$, and we reserve the letter $\varpi$ for a prime in $\mathcal{O}_K$ throughout the paper, by which we mean that the ideal $(\varpi)$ generated by it is a prime ideal.

Let $X>1$ be a large real number, interesting properties concerning the family of Hecke $L$-functions $\{L(s,\lambda_{M_Kl}):1\le l\le X\}$ have been investigated in [8], [2] for its one-level density, in [9] for its extreme values.

Under the generalized Riemann hypothesis, the one-level density results established in [8], [2] (see [8, Corollary 1.3]) imply that at least 75% of the members of the family of angular Hecke $L$-functions do not vanish at the central point.

It is the aim of this paper to establish a non-vanishing result unconditionally for these families of $L$-functions by evaluating their first and second moments at the central point.

Our first result evaluates asymptotically the first moment as follows.

Our next result evaluates asymptotically the second moment.

It follows from Theorems 1.1–1.2 and the standard arguments using Cauchy-Schwarz inequality that we have the following non-vanishing result concerning the family of angular Hecke $L$-functions.

Preliminaries

Angular Hecke characters

It is shown in [8 Section 2] that for $l\neq0$, the complete Hecke $L$-function

is entire and have an analytic continuation to the whole complex plane that satisfies the functional equation

The above allows us to establish the following approximate functional equation for $L(1/2,\lambda_{M_Kl})$.

where

and where

Proof. Consider the integral

Then by Cauchy’s theorem, we have

Note that by (4) and (9),

On the other hand, we may expand $\Lambda_{M_Kl}^2\left(\frac{1}{2}+s\right)$ into Dirichlet series and grouping the ideals according to $\mathfrak{n}=N(\mathfrak{a})N(\mathfrak{b})$ to evaluate $I$. We deduce from this and (12)–(13) that the expression given in (10) is valid. This completes the proof of the lemma. □

Proof of Theorem 1.1

We apply Lemma 2.2 and (7) to see that

where

Note that when $\mathfrak{a}=(n)$ with $n\in\mathbb{N}$, we have $\lambda_{M_Kl}(\mathfrak{a})=1$ for all $X<l\le2X$. It follows that we have

where $\zeta(s)$ is the Riemann zeta function. We now shift the line of integration above to $\Re(s)=-1$. As $\zeta(s)$ has a simple pole at $s=1$ with the corresponding residue being $1$, we see that we encounter a double pole at $s=0$ in the process, whose residue can be easily computed to be

where the implied constant depends on $K$ only. It follows that

where

Note that by Stirling’s formula (see [3 (5.112)]), we have

It follows from this that for $|\sigma|\leq10$ and $X<u\leq2X$, we have

The above implies that if $|\sigma|\leq10$ and $X<u\leq2X$, we have

Note also that we have for $x\geq1$, $y\in\mathbb{R}$,

We apply (16) and (17) to see that

It follows that

where the last estimation above follows from the well-known functional equation (see [6 Corollary 10.3]) for $\zeta(s)$ that

Stirling’s formula (see [3 (5.113)]) yields

We deduce from this and (18) that

We apply the above into (15) to see that

We further note from [6 (C.17)] that we have

It follows from this and (19) that we have by partial summation,

Next, we estimate $R_1$ and $R_2$ by keeping the line of integration in $V_{M_Kl}$ to be at $\Re(s)=3$ for $R_2$ while shifting the line of integration to $\Re(s)=-1$ for $R_1$. In this process we encounter a simple pole with residue $1$ at $s=0$, whose contribution to $R_1$ is denoted by $R_{1,1}$. We denote $R_{1,2}$ the contribution to $R_1$ on the new line of integration, so that we have

where

We note that when $\mathfrak{a}\ne(n)$ with $n\in\mathbb{N}$, we have $\theta_{\mathfrak{a}}\notin\pi\mathbb{Z}$, so that similar to [8 (6.3)], we have for any real $u\geq1$,

Now, our choice of $\theta_K\in[-\pi/|U_K|,\pi/|U_K|)$ and the value of $M_K=|U_K|/2$ ensures that we have $M_K\theta_{\mathfrak{a}}\in[-\pi/2,\pi/2)$. It then follows from the well-known estimation $2x/\pi\leq\sin x\leq x$ for $0<x\leq\pi/2$ that we have

Recall (see [4 Corollary 2, Chap. 2]) that the ring of integers $\mathcal{O}_K$ is a free $\mathbb{Z}$ module such that $\mathcal{O}_K=\mathbb{Z}+\omega_K\mathbb{Z}$, where

We apply this to see that we may write any $\mathfrak{a}\ne(n)$ with $n\in\mathbb{N}$ as $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$. It follows from this that we have

so that

We deduce from (23)-(25) that for any $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$ and any $X\leq u\leq2X$,

We note moreover that for any $u\geq5$,

In particular, this together with (26) implies that

We further deduce from (26) and partial summation that when $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$, we have

where we set

We apply [9 Lemma 3] to see that when $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$,

where the last estimation above follows from the well-known bound that $1+x\le e^x$ for any $x\in\mathbb{R}$.

We substitute (30) into (29) to deduce that for $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$,

Using the relation given in (17), we infer from the above that for $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$,

where we set $s=\sigma+it$ with $\sigma,t\in\mathbb{R}$.

We apply (16) into (31) to see that for $|\sigma|\le10$, we have for $\mathfrak{a}=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$,

It follows from the above and (22) that we have

where the last estimation above follows from (27) and partial summation.

Similarly, we have

The asymptotical expression given in (2) now follows from (14), (20), (21), (28), (33) and (34). This completes the proof of the Theorem 1.1.

Proof of Theorem 1.2

We apply Lemma 2.3 and (11) to see that

where

Note again that when $n=(n)$ with $n\in\mathbb{N}$, we have $\lambda_{M_Kl}(n)=1$ for all $X<l\le2X$. It follows that we have

Note that (see [3 Section 3.8]) a rational prime $p$ ramifies in $O_K$ if and only if $\chi_d(p)=0$, where $\chi_d=\left(\frac{d}{\cdot}\right)$ is the Kronecker symbol. Also, $p$ splits (resp. is inert) in $O_K$ if and only if $\chi_d(p)=1$ (resp. $\chi_d(p)=-1$). We apply this to rewrite the Dirichlet series $\sum_{n\geq1}\frac{\tau_K(n)}{n^{1+2s}}$ into Euler products to see that we have

where $L(s,\chi_d)$ is the Dirichlet $L$-function associated to $\chi_d$ and $F_K(s)$ is analytical in the region $\Re(s)\geq-1/4+\varepsilon$ for any $\varepsilon>0$.

We now shift the line of integration above to $\Re(s)=-1/4+\varepsilon$ to encounter a pole of order 4 at $s=0$, whose residue can be easily computed to be

where the implied constant depends on $K$ only.

Note that (see [10 §12.31]) we have for $\Re(z)\geq0$,

The above implies that

Differentiating both sides above with respect to $z$ now implies that for $\Re(z)\geq0$,

We apply the above to see that the expression in (4.2) equals

It follows that

where

We recall the convexity bound for $L(s,\chi_d)$ (see [3 Exercise 3, p. 100]) asserts that

Note that the above estimation also holds by replacing $L(s,\chi_d)$ with $\zeta(s)$ and $d$ with 1.

We apply the above and (3.4) to see that

where the last estimation above follows from (3.3).

We apply the above into (4.3) to see that

where $C_K$ is a constant depending on $K$ only.

Next, we estimate $T_1$ and $T_2$ by keeping the line of integration in $W_{M_Kl}$ to be at $\Re(s)=3$ for $T_2$ while shifting the line of integration to $\Re(s)=-1$ for $T_1$. In this process we encounter a simple pole with residue 1 at $s=0$, whose contribution to $T_1$ is denoted by $T_{1,1}$. We denote $T_{1,2}$ the contribution to $T_1$ on the new line of integration, so that we have

where

We now apply (3.13) to see that

Note that for any $a\mid n$, we have $N(a)\mid N(n)$. Moreover, if $a=(\alpha)$ and $b=(\beta)$ satisfy that $N(a)=N(b)$, then we have $\alpha=\beta u$ with $u\in U_K$. It follows that we have

where $\tau$ is the usual divisor function on $\mathbb{Z}$. Thus we have

Note that for $n=(a+b\omega_K)$ we have $N(n)=f_K(a;b)$, where

Observe that for any fixed $b\in\mathbb{Z}$, $f_K(a;b)$ is an irreducible quadratic polynomial of $a$ with integer coefficients. We now recall a result of J. McKee [5 (12)] that for any $x\geq2$,

where for any positive integer $m$, $\rho(m)$ denotes the number of solutions to the congruence equation

Let $f(x,y)=Ax^2+Bxy+Cy^2$ with $A,B,C\in\mathbb{Z}$ and be a positive definite binary quadratic form, such that its discriminant $\Delta=B^2-4AC<0$, and $A>0$. Given

we define $f^\beta(x,y)=f(\beta_1x+\beta_3y,\beta_2x+\beta_4y)$. Then $f^\beta$ has the same discriminant as $f$. Forms related in this way are called equivalent. The form $f$ is called reduced if $|B|\leq A\leq C$, with $|B|=0$ if either $A=|B|$ or $A=C$. We recall from [5 Section 2] that each positive definite binary quadratic form is equivalent to precisely one reduced form. The number of reduced forms with discriminant $\Delta$ is finite, and is denoted $H(\Delta)$. The subgroup of $\mathrm{SL}_2(\mathbb{Z})$ consisting of those $\beta$ such that $f^\beta=f$ is called the automorphism group of $f$.

Write $(A,B,C)$ as a shorthand for $Ax^2+Bxy+Cy^2$. Let $w(A,B,C)$ denote the size of the automorphism group of $(A,B,C)$. Then $w(A,A,A)=6$, $w(A,0,A)=4$, and otherwise, if $(A,B,C)$ is reduced, we have $w(A,B,C)=2$. Define the Hurwitz class number $H^*(\Delta)$ as

Note that the discriminant $\Delta_K$ of our function $f_K(n;b)$ equals

It is then shown in [5 (16)] that for any $x \ge2$,

where $\mu$ is the Möbius function.

Observe that the inner sum is the number of non-trivial integer points within the ellipse $Ax^2+Bxy+Cy^2=x/q^2$, we diagonalize it to see that its equation can be recasted as

where $\nu_1\nu_2=-\Delta_K/4$. The area of the ellipse given in (46) then equals

It follows that the number of non-trivial integer points within the ellipse $Ax^2+Bxy+Cy^2=x/q^2$ is

We apply the above and proceed as in the paragraph below [5 (16)] to see that for any $x \ge2$,

It follows from the above and partial summation that we have

We deduce from (43)–(45), (47) and (48) that for $u \ge2$,

Note that we have $H^*(\Delta_K)\ll H(\Delta_K)$. We then note from [1 §6, (15)] that

where the last estimation above follows by noting that $\chi_{\Delta_K}$ is induced by the character $\chi_{-d}$ so that $L(1,\chi_{\Delta_K})\ll1$.

We substitute (50) into (49) to see that

We apply (51) into (42) to see that

We now estimate $T_{1,2}$ by noting that similar to (32), we have for $n=(a+b\omega_K)$ with $a,b\in\mathbb{Z}$, $b\ne0$,

We apply the above to see from (4.6) that

where the last estimation above follows from (51) and partial summation.

Note that the above procedure also implies that

The asymptotical expression given in (3) now follows from (35), (38), (39), (52)–(54). This completes the proof of Theorem 1.2.

Acknowledgments. P. G. is supported in part by NSFC grant 12471003.

School of Mathematical Sciences, Beihang University, Beijing 100191, China

Email address: [email protected]