Advertisement

The Ramanujan Journal

, Volume 37, Issue 2, pp 257–268 | Cite as

Some arithmetic problems related to class group \(L\)–functions

  • S. Dittmer
  • M. Proulx
  • S. Seybert
Article

Abstract

We prove that for each fundamental discriminant \(-D<0\), there exists at least one ideal class group character \(\chi \) of \(\mathbb {Q}(\sqrt{-D})\) such that the \(L\)–function \(L(\chi ,s)\) is nonvanishing at \(s=\tfrac{1}{2}\). In addition, assuming that the quadratic Dirichlet \(L\)-function \(L(\chi _D,\tfrac{1}{2}) \ge 0\), we prove that the class number \(h(-D)\) satisfies the effective lower bound
$$\begin{aligned} h(-D) \ge 0.1265 \cdot \varepsilon D^{\frac{1}{4}}\log (D) \end{aligned}$$
for each fundamental discriminant \(-D < 0\) with \(D \ge (8 \pi /e^{\gamma })^{(\frac{1}{2}-\varepsilon )^{-1}}\) where \(0 < \varepsilon < 1/2\) is arbitrary and fixed (here \(\gamma \) is Euler’s constant).

Keywords

\(L\)-functions Ideal class group Class number problem 

Mathematics Subject Classification

11M41 11F67 

1 Introduction and statement of results

It is well known that the \(L\)–function \(L(\chi ,s)\) of an ideal class group character \(\chi \) of an imaginary quadratic field \(K=\mathbb Q(\sqrt{-D})\) can be expressed in terms of values of the real-analytic Eisenstein series for \(SL_2(\mathbb Z)\) at Heegner points. In this paper, we will exploit this relationship to study two related arithmetic problems. First, we show that for each fundamental discriminant \(-D < 0\), there exists at least one \(\chi \) such that the central value \(L(\chi ,\tfrac{1}{2}) \ne 0\). Next, by building on ideas of Iwaniec and Sarnak [7] and Iwaniec and Kowalski [6], we show that if \(L(\chi _D,\tfrac{1}{2}) \ge 0\) where \(L(\chi _D,s)\) is the quadratic Dirichlet \(L\)-function associated to the Kronecker symbol \(\chi _D\), then the class number \(h(-D)\) of \(K\) satisfies the effective lower bound:
$$\begin{aligned} h(-D) \ge 0.1265 \cdot \varepsilon D^{\frac{1}{4}}\log (D) \end{aligned}$$
for each fundamental discriminant \(-D < 0\) with \(D \ge (8 \pi /e^{\gamma })^{(\frac{1}{2}-\varepsilon )^{-1}}\) where \(0 < \varepsilon < 1/2\) is arbitrary and fixed (here \(\gamma \) is Euler’s constant).
In order to describe our results, we review some facts regarding class group \(L\)-functions (see, e.g., [6, Sect. 22.3]). Let \(-D < 0\) be a fundamental discriminant, \(K = \mathbb {Q}(\sqrt{-D})\) be an imaginary quadratic field, \(\mathcal {O}_D\) be the ring of integers, \(\omega \) be the number of units in \(\mathcal {O}_D\), \(Cl(\mathcal {O}_D)\) be the ideal class group of \(K\), \(h(-D)\) be the class number, and \(\widehat{Cl(\mathcal {O}_D)}\) be the group of characters of \(Cl(\mathcal {O}_D)\). Given \(\chi \in \widehat{Cl(\mathcal {O}_D)}\), the class group \(L\)-function is defined by
$$\begin{aligned} L(\chi , s) = \sum _{\mathcal {C} \in Cl(\mathcal {O}_D)} \chi (\mathcal {C}) \zeta _\mathcal {C}(s), \end{aligned}$$
where
$$\begin{aligned} \zeta _\mathcal {C}(s) = \sum _{\begin{array}{c} 0 \ne {\mathfrak a}\in \mathcal {C} \\ {\mathfrak a}\> integral \end{array}} N({\mathfrak a})^{-s}, \quad \mathfrak {Re}(s) > 1, \end{aligned}$$
and \(N({\mathfrak a})\) is the norm of \({\mathfrak a}\). It is known that if \(\chi \) is nontrivial, then \(L(\chi , s)\) extends to an entire function on \(\mathbb {C}\) and satisfies the functional equation:
$$\begin{aligned} \Lambda (s)=\Lambda (1-s), \end{aligned}$$
where
$$\begin{aligned} \Lambda (s):=(2\pi )^{-s}\Gamma (s)D^{s/2}L(\chi ,s). \end{aligned}$$
The nonvanishing of central values of automorphic \(L\)–functions is a problem of great importance in number theory. While it is difficult to determine whether an individual \(L\)–function is nonvanishing, progress can often be made by studying \(L\)-functions in families. The class group \(L\)–functions provide an interesting example of such a family (see [1, 3, 4] ). The nonvanishing of their central values was studied by Blomer [1], who used deep techniques in analytic number theory to prove that, for \(D\) large enough,
$$\begin{aligned} \frac{ \# \{ \chi \in \widehat{Cl(\mathcal {O}_D)}: L \left( \chi , \tfrac{1}{2} \right) \ne 0 \}}{h(-D)} \ge c \prod _{p \mid D} \left( 1 - \frac{1}{p} \right) \end{aligned}$$
(1)
for some explicit \(c > 0\). This result is ineffective in the sense that one does not know how large \(D\) must be for (1) to hold due to an application of Siegel’s theorem in the proof.

We will show that there is always at least one \(\chi \in \widehat{Cl(\mathcal {O}_D)}\) such that \(L(\chi , \tfrac{1}{2}) \ne 0\).

Theorem 1.1

For each fundamental discriminant \(-D < 0\), there exists at least one \(\chi \in \widehat{Cl(\mathcal {O}_D)}\) such that \(L \left( \chi , \tfrac{1}{2} \right) \ne 0\).

It is expected that for each \(-D < 0\) one has \(L \left( \chi , \tfrac{1}{2} \right) \ne 0\) for all \(\chi \in \widehat{CL(\mathcal {O}_D)}\), due to theoretical reasons (see, e.g., [1, Sect. 1]) and numerical evidence. We have calculated the following table of \(L\)-values for small \(D\) with prime class number \(h(-D)\) using the identity (2) (Table 1).
Table 1

\(L\)–function values for small \(D\)

\(-D\)

\(h(-D)\)

\(L\)-function values at \(s=\tfrac{1}{2}\)

\(-3\)

\(1\)

\(-0.702237\)

\(-4\)

\(1\)

\(-0.975066\)

\(-7\)

\(1\)

\(-1.67442\)

\(-8\)

\(1\)

\(-1.60701\)

\(-11\)

\(1\)

\(-1.44805\)

\(-15\)

\(2\)

\(-2.69732\), \(0.111442\)

\(-19\)

\(1\)

\(-1.17474\)

\(-20\)

\(2\)

\(-2.45292\), \(0.154738\)

\(-23\)

\(3\)

\(-3.5857\), \(0.174036\), \(0.174036\)

\(-24\)

\(2\)

\(-2.29555\), \(0.179696\)

\(\vdots \)

\(\vdots \)

\(\vdots \)

\(-47\)

\(5\)

\(-4.82435\), \(0.359728\), \(0.247743\), \(0.247743\), \(0.359728\)

\(\vdots \)

\(\vdots \)

\(\vdots \)

\(-71\)

\(7\)

\(-5.99259\), \(0.521411\), \(0.417899\), \(0.252331\), \(0.252331\), \(0.417899\), \(0.521411\)

\(\vdots \)

\(\vdots \)

\(\vdots \)

Our proof of Theorem \(1.1\) is inspired by the discussion of Michel and Venkatesh in [8, Sect. 1.1]. To study the nonvanishing of \(L(\chi ,\tfrac{1}{2})\), we use the identity:
$$\begin{aligned} L(\chi , s) = \frac{2}{w} \left( \frac{\sqrt{D}}{2} \right) ^{-s} \sum \limits _{[{\mathfrak a}] \in Cl(\mathcal O_D)} \chi ({\mathfrak a}) f(z_{{\mathfrak a}}, s), \end{aligned}$$
(2)
and the resulting exact formula for the first moment:
$$\begin{aligned} \frac{1}{h(-D)} \sum _{\chi \in \widehat{Cl(\mathcal {O}_D)}} L(\chi , \tfrac{1}{2}) = \frac{2}{w} \left( \frac{\sqrt{D}}{2} \right) ^{-\frac{1}{2}} f(z_{\mathcal {O}_D}, \tfrac{1}{2}), \end{aligned}$$
where \(f(z, \tfrac{1}{2})\) is essentially the central derivative of the real-analytic Eisenstein series for \(SL_2(\mathbb {Z})\) and \(z_{\mathcal {O}_D}\) is the Heegner point corresponding to the trivial ideal class (see Propositions 2.1 and 2.2). The Heegner point \(z_{\mathcal {O}_D}\) lives high in the cusp of \(SL_2 (\mathbb Z)\backslash \mathbb H\). Since the Fourier expansion of \(f(z, \tfrac{1}{2})\) is dominated by its constant term, which depends only on the imaginary part of \(z\), we are able to prove that \(f(z_{\mathcal {O}_D}, \tfrac{1}{2}) \ne 0\) for all \(D\).
Another problem of great importance in number theory is that of finding effective lower bounds for the class number \(h(-D)\). Very strong effective lower bounds which are conditional on the location of the zeros of the quadratic Dirichlet \(L\)–function \(L(\chi _D,s)\) have been known for many years. For example, in 1918, Hecke and Landau proved that if \(L(\chi _D,s)\) does not vanish in the region \(s > 1-a/\log (D)\), then
$$\begin{aligned} h(-D) > b \frac{D^{\frac{1}{2}}}{\log (D)} \end{aligned}$$
where \(a\) and \(b\) are effective, positive constants (see [6, Proposition 22.2]). However, it is natural to ask if strong effective lower bounds can be obtained without assuming anything about the location of the zeros. This question was discussed by Iwaniec and Sarnak [7, Sect. 5] in the context of nonnegativity of central values of automorphic \(L\)–functions. Namely, Iwaniec and Sarnak remarked that if one knew that \(L(\chi _D,\tfrac{1}{2}) \ge 0\), then one could “eliminate in part the Landau-Siegel lacuna” discussed in [7, Sect. 2]. This question was revisited by Iwaniec and Kowalski [6, Sect. 22.3], where they briefly explained how to use the condition \(L(\chi _D,\tfrac{1}{2}) \ge 0\) to establish an effective lower bound of the form:
$$\begin{aligned} h(-D) \gg D^{\frac{1}{4}}\log (D). \end{aligned}$$
Using methods similar to those in the proof of Theorem 1.1, we will refine the argument of Iwaniec and Kowalski and make this lower bound completely explicit.

Theorem 1.2

Let \(-D < 0\) be a fundamental discriminant with \(D \ge (8 \pi /e^{\gamma })^{(\frac{1}{2}-\varepsilon )^{-1}}\) where \(0 < \varepsilon < 1/2\) is arbitrary and fixed (here \(\gamma \) is Euler’s constant). Assume that \(L(\chi _D, \tfrac{1}{2}) \ge 0\). Then,
$$\begin{aligned} h(-D) \ge 0.1265 \cdot \varepsilon D^{\frac{1}{4}} \log (D). \end{aligned}$$

It is not difficult to show that GRH implies \(L(\chi _D,\tfrac{1}{2}) \ge 0\). As remarked by Iwaniec and Kowalski [6, Sect. 22.3], this result “may conceivably be established sometime without recourse to the GRH.” In fact, it should be emphasized that there are many examples of automorphic \(L\)–functions which are known unconditionally to have nonnegative central values (see [7, Sect. 5]). This is striking considering the difficulty of proving such a result in the “simplest” case of a quadratic Dirichlet \(L\)–function.

Remark

The condition \(L(\chi _0,\tfrac{1}{2}) \le 0\) is equivalent to \(L(\chi _D,\tfrac{1}{2}) \ge 0\) since \(L(\chi _0,s)=\zeta (s)L(\chi _D,s)\). We use as our assumption \(L(\chi _D,\tfrac{1}{2}) \ge 0\), following the literature, though our proof will, in fact, require the equivalent statement \(L(\chi _0,\tfrac{1}{2}) \le 0\).

2 Averaging \(L\)–functions

In this section, we review the well-known relationship between class group \(L\)–functions and Eisenstein series. A detailed discussion of this can be found in [6, Sect. 22.3]. Recall that each ideal class \(\mathcal {C} \in Cl(\mathcal {O}_D)\) contains a reduced, primitive integral ideal:
$$\begin{aligned} {\mathfrak a}= \mathbb {Z} a + \mathbb {Z} \left( \frac{b + \sqrt{-D}}{2}\right) \end{aligned}$$
with \(a = N({\mathfrak a})\), where an integral ideal is reduced and primitive precisely when the associated binary quadratic form is (see, e.g., [6, Sect. 22.2]). Moreover, the point
$$\begin{aligned} z_{\mathfrak a}= \frac{b + \sqrt{-D}}{2a} \end{aligned}$$
lies in the standard fundamental domain for \(\Gamma = SL_2(\mathbb {Z})\). Following convention, we call these points Heegner points. Next, define the Eisenstein series
$$\begin{aligned} E(z, s) = \sum _{\gamma \in \Gamma _{\infty } \backslash \Gamma } \mathfrak {Im}(\gamma z)^s, \quad \mathfrak {Re}(s) > 1 \end{aligned}$$
where
$$\begin{aligned} \Gamma _\infty = \left\{ \left. \left( \begin{array}{cc} 1 &{} n \\ 0 &{} 1 \end{array} \right) \right| n \in \mathbb Z \right\} . \end{aligned}$$
Finally, to make certain equations simpler, we introduce the convention
$$\begin{aligned} f(z,s):=\zeta (2s) E(z, s). \end{aligned}$$

Proposition 2.1

We have
$$\begin{aligned} \frac{1}{h(-D)} \sum _{\chi \in \widehat{Cl(\mathcal {O}_D)}} L(\chi , s) = \frac{2}{w} \left( \frac{\sqrt{D}}{2} \right) ^{-s} f(z_{\mathcal O_D}, s). \end{aligned}$$

Proof

Recall the following classical formula due to Hecke:
$$\begin{aligned} \zeta _{[{\mathfrak a}]}(s) = \frac{2}{w} \left( \frac{\sqrt{D}}{2} \right) ^{-s} f(z_{{\mathfrak a}}, s). \end{aligned}$$
Then,
$$\begin{aligned} L(\chi , s) = \frac{2}{w} \left( \frac{\sqrt{D}}{2} \right) ^{-s} \sum \limits _{[{\mathfrak a}] \in Cl(\mathcal O_D)} \chi ({\mathfrak a}) f(z_{{\mathfrak a}}, s), \end{aligned}$$
so we have
$$\begin{aligned} \sum \limits _{\chi \in \widehat{Cl(\mathcal {O}_D)}} L(\chi , s) = \frac{2}{w} \left( \frac{\sqrt{D}}{2} \right) ^{-s} \sum \limits _{[{\mathfrak a}] \in Cl(\mathcal O_D)} f(z_{{\mathfrak a}}, s)\sum \limits _{\chi \in \widehat{Cl(\mathcal {O}_D)}} \chi ({\mathfrak a}). \end{aligned}$$
Applying the orthogonality relations,
$$\begin{aligned} \sum _{\chi \in \widehat{Cl(\mathcal O_D)}} \chi ({\mathfrak a}) = {\left\{ \begin{array}{ll} h(-D), &{} [{\mathfrak a}] = [\mathcal O_D] \\ 0, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
gives the desired result immediately. \(\square \)

Proposition 2.2

For \(z = x + iy \), we have
$$\begin{aligned} f(z,s)&= \sqrt{y} \>( \log (y) - \log (4\pi e^{-\gamma }))\\&\quad + 4 \sqrt{y} \sum _{n=1}^\infty \tau _{s- \frac{1}{2}}(n) K_{s - \frac{1}{2}} (2 \pi n y) \cos (2 \pi n x)+ O(s - \tfrac{1}{2}), \end{aligned}$$
where \(\gamma \) is Euler’s constant,
$$\begin{aligned} \tau _{s}(n) = \sum _{ab = n} \left( \frac{a}{b} \right) ^{s} \end{aligned}$$
and \(K_{s}(t)\) is the \(K\)-Bessel function.

Proof

Recall the Fourier expansion (see [6, Eq. (22.46)]):
$$\begin{aligned} f(z,s)&= y^s\zeta (2s) + \sqrt{\pi }\frac{\Gamma (s-\tfrac{1}{2}) \zeta (2s - 1)}{\Gamma (s)} y^{1-s} \\&+ \frac{4 \pi ^s}{\Gamma (s)} \sqrt{y} \sum _{n=1}^\infty \tau _{s - \frac{1}{2}}(n) K_{s - \frac{1}{2}} (2 \pi n y) \cos (2 \pi n x). \end{aligned}$$
We have
$$\begin{aligned} \zeta (2s) = \frac{1}{2(s - \frac{1}{2})} + \gamma + O (s - \tfrac{1}{2}) \end{aligned}$$
and
$$\begin{aligned} y^s = y^{\frac{1}{2}}\left( 1 + \log y (s - \tfrac{1}{2}) + O(s - \tfrac{1}{2})\right) , \end{aligned}$$
thus,
$$\begin{aligned} y^s \zeta (2s) = \frac{y^{\frac{1}{2}}}{2(s - {\tfrac{1}{2}})} + \gamma y^{\frac{1}{2}} + {\frac{1}{2}} y^{\frac{1}{2}} \log y + O (s - {\tfrac{1}{2}}). \end{aligned}$$
Also recall the functional equation:
$$\begin{aligned} \pi ^\frac{-s}{2} \Gamma \left( \frac{s}{2} \right) \zeta (s) = \zeta (1 - s) \Gamma \left( \frac{1-s}{2} \right) \pi ^{- \frac{1-s}{2}}. \end{aligned}$$
Making the change of variables \(s \rightarrow 2s - 1\) in this equation yields
$$\begin{aligned} \pi ^{\frac{1}{2} - s} \Gamma (s - \tfrac{1}{2}) \zeta (2s - 1) = \zeta (2 - 2s) \Gamma (1-s) \pi ^{- (1-s)}, \end{aligned}$$
and thus,
$$\begin{aligned} \sqrt{\pi }\frac{\Gamma (s - {\frac{1}{2}}) \zeta (2s - 1) y^{1 - s}}{\Gamma (s)} = \frac{\pi ^ {2s - 1} \Gamma (1-s) y^{1 - s}}{\Gamma (s)} \zeta (2 - 2s). \end{aligned}$$
We want to calculate the Taylor expansion of
$$\begin{aligned} \frac{ \pi ^{2s-1} \Gamma (1-s) y^{1-s}}{\Gamma (s)} \zeta (2-2s) \end{aligned}$$
at \(s = \tfrac{1}{2}\). We have
$$\begin{aligned} \zeta (2 - 2s) = \frac{1}{2(s - \frac{1}{2})} + \gamma + O (s - 1) \end{aligned}$$
and
$$\begin{aligned} \frac{\pi ^{2s - 1} \Gamma (1-s) y^{1 - s}}{\Gamma (s)}&= y^\frac{1}{2} + \alpha (s - \tfrac{1}{2}) + O(s - \tfrac{1}{2})^2, \end{aligned}$$
where
$$\begin{aligned} \alpha := \left. \frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{\pi ^{2s-1} \Gamma (1-s) y^{1-s}}{\Gamma (s)} \right) \right| _{s = \frac{1}{2}}. \end{aligned}$$
A straightforward calculation shows
$$\begin{aligned} \alpha = y^\frac{1}{2} (2 \log \pi - \log y + 2 \gamma + 2 \log 4). \end{aligned}$$
Putting things together, we get
$$\begin{aligned} \frac{\pi ^ {2s \!-\! 1} \Gamma (1\!-\!s) y^{1 \!-\! s}}{\Gamma (s)} \zeta (2 \!-\! 2s) \!=\! \frac{-y^\frac{1}{2}}{2(s \!-\! \tfrac{1}{2})} \!+\! \gamma y^\frac{1}{2} \!\!+\!\! (\!-\! \tfrac{1}{2}) y^\frac{1}{2} (2 \log \pi \!-\! \log y \!+\! 2 \gamma \!+ \!2 \log 4), \end{aligned}$$
which after simplification gives
$$\begin{aligned} f(z,s)&= \sqrt{y}(\log y - \log (4\pi e^{-\gamma })) \\&\quad + 4 \sqrt{y} \sum _{n=1}^\infty \tau _{s- \frac{1}{2}}(n) K_{s - \frac{1}{2}} (2 \pi n y) \cos (2 \pi n x)+ O(s - \tfrac{1}{2}). \end{aligned}$$
\(\square \)

3 Proof of Theorem 1.1

It suffices to show that
$$\begin{aligned} \frac{1}{h(-D)} \sum _{\chi \in \widehat{Cl(\mathcal {O}_D)}} L(\chi , \tfrac{1}{2}) \end{aligned}$$
is nonvanishing. By Proposition 2.1, this is equivalent to showing that \(f(z_{\mathcal {O}_D},\tfrac{1}{2}) \ne 0\) for all \(D\), and by Proposition 2.2, we can expand
$$\begin{aligned} f(z_{\mathcal {O}_D},\tfrac{1}{2})&=\sqrt{y}\big (\log (y)-\log (4\pi e^{-\gamma })\big ) + 4 \sqrt{y} \sum _{n=1}^\infty \tau _{0}(n) K_{0} (2 \pi n y) \cos (2 \pi n x), \end{aligned}$$
where \(z_{\mathcal {O}_D}=\frac{b+\sqrt{D}}{2}\) so that \(x=\frac{b}{2}\) and \(y=\frac{\sqrt{D}}{2}\). We assume that \(f(z_{\mathcal {O}_D},\tfrac{1}{2})=0\) for some \(D\) and obtain a contradiction. From [5, (8.451.6)], we have that
$$\begin{aligned} K_0(t) = \sqrt{\frac{\pi }{2t}}e^{-t}\left( 1+\frac{\theta }{2t}\right) \end{aligned}$$
for \(t > 0\) where \(|\theta |\le \tfrac{1}{4}\). This, along with \(y \ge \frac{\sqrt{3}}{2}\), allows us to bound the \(K\)-Bessel function term as
$$\begin{aligned} |K_0(2\pi n y)| \le \left( 1 + \frac{1}{8 \sqrt{3} \pi } \right) \sqrt{\frac{1}{4n y}}e^{-2\pi n y}. \end{aligned}$$
Using the elementary bound \(\tau _0(n) \le 2\sqrt{n}\), we bound the infinite sum in the Fourier expansion by
$$\begin{aligned} \left| 4\sqrt{y} \sum _{n=1}^\infty \tau _0(n)K_0(2\pi n y)\cos (2 \pi n x)\right|&\le 4 \sum _{n=1}^\infty \left| 2\sqrt{ny}K_0(2\pi n y)\right| \\&\le \left( 4 + \frac{1}{2 \sqrt{3} \pi } \right) \sum _{n=1}^\infty e^{-2\pi n y} \\&= \left( 4 + \frac{1}{2 \sqrt{3} \pi } \right) \cdot \frac{e^{-2\pi y}}{1-e^{-2\pi y}}\\&\le \left( 4 + \frac{1}{2 \sqrt{3} \pi }\right) \cdot \frac{e^{-\pi \sqrt{3}}}{1-e^{-\pi \sqrt{3}}}\\&< 0.018. \end{aligned}$$
Since \(f(z_{\mathcal {O}_D},\tfrac{1}{2})=0\), we have
$$\begin{aligned} |\sqrt{y}\left( \log (y)-\log (4\pi e^{-\gamma })\right) | = \left| 4\sqrt{y} \sum _{n=1}^{\infty } \tau _0(n)K_0(2\pi n y)\cos (2 \pi n x) \right| < 0.018. \end{aligned}$$
Therefore
$$\begin{aligned} |\log (y)-\log (4\pi e^{-\gamma }) | \le \frac{2}{\sqrt{3}}|\sqrt{y}(\log (y)-\log (4\pi e^{-\gamma }))| < 0.021. \end{aligned}$$
This implies
$$\begin{aligned} 4\pi e^{-\gamma -0.021}<y<4\pi e^{-\gamma +0.021}, \end{aligned}$$
so that \(6.90 <y<7.21\). However, using this bound, we can improve the bound on the infinite sum to
$$\begin{aligned} \left| 4\sqrt{y} \sum _{n=1}^\infty \tau _0(n)K_0(2\pi n y)\cos (2 \pi n x) \right| \le \left( 4 \!+\! \frac{1}{2 \sqrt{3} \pi } \right) \frac{e^{-\pi \cdot 13.8}}{1-e^{-\pi \cdot 13.8}} < 6.08 \cdot 10^{-19}. \end{aligned}$$
Repeating the previous argument with this much sharper bound, we find that
$$\begin{aligned} 4\pi e^{-\gamma -6.08 \cdot 10^{-19}}<y<4\pi e^{-\gamma +6.08 \cdot 10^{-19}} \end{aligned}$$
and
$$\begin{aligned} 199.12076<4y^2<199.12077. \end{aligned}$$
But since \(y=\tfrac{\sqrt{D}}{2}\), we have \(199<D<200\), so that \(D \not \in \mathbb {Z}\), a contradiction. Thus, \(f(z_{\mathcal {O}_D},\tfrac{1}{2}) \ne 0\) for all \(D\).

4 Proof of Theorem 1.2

We apply (2), and so consider \(f(z_{\mathfrak a},\tfrac{1}{2})\), where \(z_{\mathfrak a}=\frac{b+\sqrt{D}}{2N(\mathfrak {a})}\). Recall that \(\frac{\sqrt{D}}{2N(\mathfrak {a})} \ge \tfrac{\sqrt{3}}{2}\) so that \(N({\mathfrak a}) \le \sqrt{D/3}\). After applying Proposition 2.2, a calculation yields
$$\begin{aligned} L(\chi , \tfrac{1}{2}) \!&= \! \frac{2}{w}\sum \limits _{[{\mathfrak a}] \in Cl(\mathcal {O}_D)} \frac{\chi ({\mathfrak a})}{\sqrt{N({\mathfrak a})} }\nonumber \\&\times \left\{ \log \left( \frac{\beta \sqrt{D}}{N({\mathfrak a}) }\right) \!+\! 4 \sum \limits _{n=1}^{\infty } \tau _0 (n) K_0 \left( \frac{ \pi n \sqrt{D}}{N({\mathfrak a}) }\right) \cos \left( \frac{ \pi n b}{N({\mathfrak a}) }\right) \right\} ,\nonumber \\ \end{aligned}$$
(3)
where \(\beta :=e^{\gamma }/8\pi \).
Let \(\chi = \chi _0\) be the trivial character. By assumption, \(L(\chi _D, \tfrac{1}{2}) \ge 0\), so that \(L(\chi _0, \tfrac{1}{2}) \le 0\). Then, by (3) we obtain
$$\begin{aligned} \frac{2}{\omega }\sum \limits _{[{\mathfrak a}] \in Cl(\mathcal {O}_D)} \frac{1}{\sqrt{N({\mathfrak a})}} \log \left( \beta \frac{\sqrt{D}}{N({\mathfrak a})}\right) \le |E|, \end{aligned}$$
(4)
where
$$\begin{aligned} E := \frac{2}{\omega }\sum \limits _{[{\mathfrak a}] \in Cl(\mathcal {O}_D)} \frac{4}{\sqrt{N({\mathfrak a})}} \sum \limits _{n=1}^{\infty } \tau _0 (n) K_0 \left( \frac{ \pi n \sqrt{D}}{N({\mathfrak a})}\right) \cos \left( \frac{ \pi n b}{N({\mathfrak a})} \right) . \end{aligned}$$
Assume now that \(\log (\beta \sqrt{D}) \ge \varepsilon \log (D)\) for some arbitrary, fixed \(0 < \varepsilon < 1/2\) (in particular, under this assumption \(\omega =2\)). Split the sum on the left-hand side of (4) as \(S_1 + S_2\), where
$$\begin{aligned} S_1&:= \sum \limits _{1 \le N({\mathfrak a}) \le \beta \sqrt{D}} \frac{1}{\sqrt{N({\mathfrak a})}} \log \left( \beta \frac{\sqrt{D}}{N({\mathfrak a})}\right) , \\ S_2&:= \sum \limits _{\beta \sqrt{D} < N({\mathfrak a}) \le \sqrt{\frac{D}{3}}} \frac{1}{\sqrt{N({\mathfrak a})}} \log \left( \beta \frac{\sqrt{D}}{N({\mathfrak a})}\right) . \end{aligned}$$
Then, each summand in \(S_1\) is nonnegative, and we have
$$\begin{aligned} S_1 \le |E| + |S_2|. \end{aligned}$$
Using \(N({\mathfrak a}) \le \sqrt{D/3}\), we argue as in the proof of Theorem 1.1 to obtain
$$\begin{aligned} |E|&\le \left( 4 + \frac{1}{2\sqrt{3}\pi }\right) \sum _{[{\mathfrak a}] \in Cl(\mathcal {O}_D)}\sum _{n=1}^{\infty } \frac{2\sqrt{n}}{\sqrt{N({\mathfrak a})} } \left( \frac{2 n \sqrt{D}}{N({\mathfrak a}) } \right) ^{-\frac{1}{2}} \exp \left( -\frac{\pi n \sqrt{D}}{N({\mathfrak a})}\right) \\&= \left( 4\sqrt{2} + \frac{1}{\sqrt{6}\pi }\right) D^{- \frac{1}{4}} \sum _{[{\mathfrak a}] \in Cl(\mathcal {O}_D)}\sum _{n=1}^{\infty } \exp \left( -\frac{\pi n \sqrt{D}}{N({\mathfrak a})}\right) \le C_1 D^{- \frac{1}{4}} h(-D), \end{aligned}$$
where
$$\begin{aligned} C_1:= \left( 4\sqrt{2} + \frac{1}{\sqrt{6}\pi }\right) \left( \frac{e^{-\pi \sqrt{3}}}{1-e^{-\pi \sqrt{3}}} \right) . \end{aligned}$$
Next, we have, after applying the bounds in the sum,
$$\begin{aligned} |S_2| \le \sum \limits _{\beta \sqrt{D} < N({\mathfrak a}) \le \sqrt{\frac{D}{3}}} \frac{D^{- \frac{1}{4}}}{\sqrt{\beta }} \left| \log (\beta \sqrt{3} ) \right| \le C_2 D^{-\frac{1}{4}} N(-D),\\ \end{aligned}$$
where
$$\begin{aligned} C_2:= \frac{\left| \log (\beta \sqrt{3})\right| }{\sqrt{\beta }} \end{aligned}$$
and
$$\begin{aligned} N(-D): = \# \left\{ z_{{\mathfrak a}} \; | \; \frac{\sqrt{3}}{2} \le \mathfrak {Im}(z_{{\mathfrak a}}) \le \frac{1}{2 \beta }\right\} . \end{aligned}$$
Clearly, \(N(-D) \le h(-D)\), so that
$$\begin{aligned} |S_2| \le C_2 D^{- \frac{1}{4}} h(-D). \end{aligned}$$
Since each summand in \(S_1\) is nonnegative, we have (discarding every term except the one with \(N({\mathfrak a}) = 1\))
$$\begin{aligned} S_1 \ge \log (\beta \sqrt{D}) \ge \varepsilon \log (D). \end{aligned}$$
The second inequality is satisfied for all \(D \ge (8 \pi /e^{\gamma })^{(\frac{1}{2}-\varepsilon )^{-1}}\). By combining the preceding estimates, we conclude that
$$\begin{aligned} (C_1 + C_2) D^{- \frac{1}{4}} h(-D) \ge \varepsilon \log (D), \end{aligned}$$
which after a short calculation yields
$$\begin{aligned} h(-D) \ge 0.1265 \cdot \varepsilon D^{\frac{1}{4}} \log (D). \end{aligned}$$

Notes

Acknowledgments

This work was completed during the 2013 REU in Number Theory at Texas A&M University. The authors thank the NSF for their generous support. The authors also thank Riad Masri for persistent encouragement and advice. The authors thank the referee for their careful reading of the manuscript and many helpful suggestions.

References

  1. 1.
    Blomer, V.: Non-vanishing of class group \(L\)-functions at the central point. Ann. Inst. Fourier (Grenoble) 54, 831–847 (2004)Google Scholar
  2. 2.
    Duke, W.: Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73–90 (1988)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Duke, W., Friedlander, J., Iwaniec, H.: Class group \(L\)-functions. Duke Math. J. 79, 1–56 (1995)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Fouvry, E., Iwaniec, H.: Low-lying zeros of dihedral \(L\)-functions. Duke Math. J. 116, 189–217 (2003)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Gradhsteyn, I.S., Rizhik, I.M.: Table of Integrals, Series and Products, 6th edn. Academic Press, New York (2000)Google Scholar
  6. 6.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. Colloquium Publications, Am. Math. Soc., Providence (2004)Google Scholar
  7. 7.
    Iwaniec, H., Sarnak, P.: Perspectives on the analytic theory of \(L\)-functions, GAFA: (Tel Aviv, 1999). Geom. Funct. Anal. II, 705–741 (2000)Google Scholar
  8. 8.
    Michel, P., Venkatesh, A.: Heegner points and non-vanishing of Rankin/Selberg \(L\)-functions. Analytic Number Theory, vol. 7, pp. 169–183. Clay Math. Proc., Am. Math. Soc., Providence (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsBrigham Young UniversityProvoUSA
  2. 2.CambridgeUSA
  3. 3.SMC 6939Carnegie Mellon UniversityPittsburghUSA

Personalised recommendations