An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$

TL;DR

This paper constructs a new family of holomorphic modular functions for Hecke groups associated with real quadratic fields, extending the classical Dedekind eta function and analyzing their Fourier coefficients.

Contribution

It introduces an analogue of the Dedekind eta function for Hecke groups $H(\sqrt{D})$, providing new modular functions with detailed growth and sign pattern analysis.

Findings

Holomorphic modular functions vanish at the cusp at infinity.

Fourier coefficients exhibit specific asymptotic growth patterns.

Sign patterns of Fourier coefficients are characterized.

Abstract

Let $D\equiv 1\bmod{4}$ be a fundamental discriminant of a real quadratic field. We construct an analogue of the classical Dedekind eta function for the Hecke group $H(\sqrt{D})$. This gives rise to a new family of holomorphic modular functions for $H(\sqrt{D})$ which vanish at the cusp at $\infty$. We establish results on the asymptotic growth and sign patterns of the Fourier coefficients associated to these modular forms.

Tables

Table 1. Table 1. a D ​ ( N ) a_{D}(N) corresponding to D = 5 , 13 D=5,13 and 17 17 , for N = 1 N=1 to 25 25 .

$N$	$a_5\left(N\right)$	$a_{13}\left(N\right)$	$a_{17}\left(N\right)$
1	$-1-\sqrt{5}$	$-1-\sqrt{13}$	$-1-\sqrt{17}$
2	$\frac{1}{2}\left(7+\sqrt{5}\right)$	$\frac{1}{2}\left(15+\sqrt{13}\right)$	$\frac{1}{2}\left(15-\sqrt{17}\right)$
3	$-2\sqrt{5}$	$-2-4\sqrt{13}$	$19-\sqrt{17}$
4	$4-\sqrt{5}$	$27+\sqrt{13}$	$\frac{1}{2}\left(13-23\sqrt{17}\right)$
5	$6-2\sqrt{5}$	$-12\sqrt{13}$	$69-11\sqrt{17}$
6	$\frac{1}{2}\left(13-7\sqrt{5}\right)$	$66-3\sqrt{13}$	$139-23\sqrt{17}$
7	$11-5\sqrt{5}$	$27-29\sqrt{13}$	$166-70\sqrt{17}$
8	$\frac{1}{2}\left(35-13\sqrt{5}\right)$	$155-18\sqrt{13}$	$492-89\sqrt{17}$
9	$19-11\sqrt{5}$	$128-64\sqrt{13}$	$818-182\sqrt{17}$
10	$\frac{1}{2}\left(69-25\sqrt{5}\right)$	$\frac{1}{2}\left(715-119\sqrt{13}\right)$	$1242-378\sqrt{17}$
11	$37-21\sqrt{5}$	$364-148\sqrt{13}$	$2567-561\sqrt{17}$
12	$\frac{1}{2}\left(129-45\sqrt{5}\right)$	$795-164\sqrt{13}$	$4235-998\sqrt{17}$
13	$70-38\sqrt{5}$	$912-332\sqrt{13}$	$6780-1756\sqrt{17}$
14	$108-43\sqrt{5}$	$1752-393\sqrt{13}$	$\frac{1}{2}\left(23637-5515\sqrt{17}\right)$
15	$134-62\sqrt{5}$	$2160-706\sqrt{13}$	$18977-4559\sqrt{17}$
16	$176-80\sqrt{5}$	$3699-891\sqrt{13}$	$\frac{1}{2}\left(59823-14961\sqrt{17}\right)$
17	$233-103\sqrt{5}$	$4761-1467\sqrt{13}$	$48557-11627\sqrt{17}$
18	$\frac{1}{2}\left(603-267\sqrt{5}\right)$	$\frac{1}{2}\left(15069-3855\sqrt{13}\right)$	$76106-18308\sqrt{17}$
19	$377-175\sqrt{5}$	$9986-2970\sqrt{13}$	$117103-28745\sqrt{17}$
20	$\frac{1}{2}\left(1017-429\sqrt{5}\right)$	$15069-3957\sqrt{13}$	$\frac{1}{2}\left(363839-87715\sqrt{17}\right)$
21	$608-288\sqrt{5}$	$20174-5854\sqrt{13}$	$276958-67034\sqrt{17}$
22	$\frac{1}{2}\left(1625-693\sqrt{5}\right)$	$\frac{1}{2}\left(58843-15677\sqrt{13}\right)$	$417234-101814\sqrt{17}$
23	$985-455\sqrt{5}$	$39390-11286\sqrt{13}$	$629047-152045\sqrt{17}$
24	$1265-556\sqrt{5}$	$56002-15115\sqrt{13}$	$\frac{1}{2}\left(1871277-453479\sqrt{17}\right)$
25	$1564-706\sqrt{5}$	$74911-21265\sqrt{13}$	$1381414-335786\sqrt{17}$

Equations

$$T_{\lambda }:=\left(\begin{array}{cc}1& \lambda \\ 0& 1\end{array}\right),S=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right).$$

$$\mathfrak{h}:=\{x+iy\mid x\in \mathbb{R},y>0\}$$

$$\gamma z=\frac{az+b}{cz+d}.$$

$$\lambda =2\cos (\pi /q),(\text{for an integer}q>3\text{or}q=\infty ),$$

$$\eta (z):=q^{\frac{1}{24}}\prod _{n=1}^{\infty }\left(1-q^n\right)$$

$$\eta (z+1)=e^{\frac{\pi i}{12}}\eta (z),\eta (-1/z)=\sqrt{-iz}\cdot \eta (z).$$

$$\eta (z{)}^{24}=\Delta (z),$$

$${\eta }_5(z):=q^{\frac{1}{5}}\cdot \prod _{n=1}^{\infty }((1-q^n{)}^{{\chi }_{{}_5}(n)}\prod _{a=1}^4{\left(1-e^{\frac{2\pi ia}{5}}{q}^n\right)}^{{\chi }_{{}_5}(a)}).$$

$${\Delta }_5(z)=\sum _{N=1}^{\infty }{\tau }_5(N)q^N$$

$$+(1415+490\sqrt{5})q^5+\left(\frac{20565+6965\sqrt{5}}{2}\right)q^6+\cdots .$$

$${\Delta }_5\left(\frac{az+b}{cz+d}\right)={\Delta }_5(z)$$

$${\eta }_5\left(\frac{az+b}{cz+d}\right)=u_{\gamma }\cdot {\eta }_5(z)$$

$${\eta }_D(z):=q^{{\mathfrak{m}}_D}\prod _{n=1}^{\infty }\left({\left(1-q^n\right)}^{{\chi }_D(n)}\prod _{a=1}^D{\left(1-e^{\frac{2\pi ia}{D}}{q}^n\right)}^{{\chi }_D(a)}\right).$$

$${\eta }_D\left(\frac{az+b}{cz+d}\right)={\eta }_D(z)$$

$$g_{u,v}(D\tau )=-\exp (\pi iD\tau B_2(u)+2\pi iv(u-\frac{1}{2}\left)\right)\prod _{m=0}^{\infty }(1-{\zeta }_D^a{q}^{Dm+r})(1-{\zeta }_D^{-a}{q}^{Dm+D-r}),$$

$${\omega }_D=\frac{\sqrt{D}+\sqrt{D-4}}{2},\text{and}{\omega }_D^{\prime }=\frac{\sqrt{D}-\sqrt{D-4}}{2},$$

$$W_{D}\cdot\gamma_{D}\cdot W_{D}^{-1}=\left(\begin{array}[]{cc}\sqrt{\kappa}&0\\ 0&1/\sqrt{\kappa}\end{array}\right),$$

$${\eta }_D(z)=q^{-\frac{L(-1,{\chi }_D)}{2}}\sum _{N=0}^{\infty }{a}_D(N)q^N.$$

$$|\tau (p)|\le 2p^{\frac{11}{2}},\text{for all primes }p.$$

$$p(N)\sim \frac{1}{4N\sqrt{3}}\exp \left(\pi \sqrt{\frac{2}{3}}{N}^{\frac{1}{2}}\right),\text{as}N\to \infty .$$

$${\mathfrak{S}}_D(N):=\{\begin{array}{ll}(-1{)}^N,& {\mathfrak{n}}_D=2,\\ 2\sum _{u=1}^{({\mathfrak{n}}_D-1)/2}\cos (\frac{2\pi uN}{{\mathfrak{n}}_D}-{\beta }_D(u,{\mathfrak{n}}_D)),& {\mathfrak{n}}_D\ge 3,\end{array}$$

$${\beta }_D(u,{\mathfrak{n}}_D)$$

$$+\text{mathbbm}{1}_{{\mathfrak{n}}_D\equiv 3\mathrm{mod}4}\cdot \frac{2\pi }{\varphi ({\mathfrak{n}}_D)}{\chi }_{{\mathfrak{n}}_D}(u)L(0,{\chi }_{{\mathfrak{n}}_D})L(0,{\chi }_D{\chi }_{{\mathfrak{n}}_D}).$$

$$c_D=\frac{\sqrt{D}L(2,{\chi }_D)}{{\mathfrak{n}}_D^2},$$

$${\alpha }_D=\sqrt{D}L(1,{\chi }_D)+\frac{1}{4}\log c_D-\frac{1}{2}\log (4\pi ).$$

$$a_D(N)=\frac{e^{{\alpha }_D}{\mathfrak{S}}_D(N)}{N^{3/4}}\exp (2\sqrt{c_{D}N})(1+{\mathcal{O}}_D(N^{-\frac{1}{4}})),$$

$$a_D(N)=\frac{(-1{)}^N{e}^{{\alpha }_D}}{N^{3/4}}\exp \left(2\sqrt{c_{D}N}\right)(1+{\mathcal{O}}_D(N^{-\frac{1}{4}})).$$

$$a_D(N)=\frac{2\cos (\frac{2\pi }{3}N-\beta )e^{{\alpha }_D}}{N^{3/4}}\exp \left(2\sqrt{c_{D}N}\right)(1+{\mathcal{O}}_D(N^{-\frac{1}{4}})),$$

$$\beta =\pi L(0,{\chi }_3)L(0,{\chi }_D{\chi }_3)=\frac{3\sqrt{D}}{\pi }L(1,{\chi }_3)L(1,{\chi }_D{\chi }_3),$$

$$L(1,{\chi }_D)=\frac{2}{\sqrt{5}}\log \varphi ,L(2,{\chi }_D)=\frac{4{\pi }^2}{25\sqrt{5}},$$

$$a_5(N)=\frac{(-1{)}^N}{N^{3/4}}\cdot \frac{{\varphi }^2}{2\sqrt{5}}\exp \left(\frac{2\pi \sqrt{N}}{5}\right)(1+\mathcal{O}(N^{-\frac{1}{4}})).$$

Full text

Dorian Goldfeld is partially supported by grant number 567168 from the Simons Foundation.

1. Introduction

Let $\lambda>0$. Hecke [undefd] introduced the matrix groups $H(\lambda)$ generated by the two matrices

[TABLE]

These are now called Hecke groups. In the case $\lambda=1$, the Hecke group $H(\lambda)$ is just the modular group $\operatorname{SL}(2,\mathbb{Z}).$ The Hecke groups act on the upper half plane

[TABLE]

in the usual manner. If $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in H(\lambda)$ and $z\in\mathfrak{h}$, the action of $\gamma$ on $z$ is denoted $\gamma z$ and is given by

[TABLE]

Hecke proved in [undefd] that if

[TABLE]

then $H(\lambda)$ is a discrete subgroup of $\operatorname{SL}(2,\mathbb{R})$ with fundamental domain $H(\lambda)\backslash\mathfrak{h}$ having finite area, that is, $H(\lambda)$ is a Fuchsian group of the first kind. He also proved that if $\lambda$ is any real number satisfying $\lambda>2$ then $H(\lambda)$ is a discrete subgroup of $\operatorname{SL}(2,\mathbb{R})$ with fundamental domain $H(\lambda)\backslash\mathfrak{h}$ having infinite area, that is, $H(\lambda)$ is a Fuchsian group of the second kind. Moreover, he showed that these are the only values of $\lambda$ for which $H(\lambda)$ is a discrete subgroup of $\operatorname{SL}(2,\mathbb{R})$. In the case when $\lambda>2$, the fundamental domain is given in Figure 1.

Non holomorphic Eisenstein series and Maass cusp forms for the Hecke groups $H(\lambda)$ have been an area of extensive research over the years. See Kubota [undefm] for the definition of the non holomorphic Eisenstein series. In Knopp–Sheingorn [undefj], certain holomorphic Poincaré series are constructed for $H(\lambda)$ with $\lambda>2.$ The non holomorphic Maass cusp form associated to the base eigenvalue of the Laplacian for the Hecke group $H(\lambda)$ was found by Patterson [undefr] and Sullivan [undefv]. In the paper on Sarnak’s spectral gap question [undefi], Kelmer, Kontorovich, and Lutsko show that the Hecke group $H(\lambda)$ (with $\lambda>2$) has only one Maass cusp form which is square integrable on the fundamental domain. This result is originally due to Phillips and Sarnak [undefs]. This Maass cusp form is the Patterson–Sullivan base eigenfunction with non-zero Laplace eigenvalue $<\tfrac{1}{4}.$ More recently, Karlovitz [undefh] generalized Hejhal’s algorithm [undefe], for explicitly computing Maass cusp forms on congruence subgroups of $\operatorname{SL}(2,\mathbb{Z})$, to the case of the Hecke groups $H(\lambda)$ with $\lambda>2.$

The main goal of this paper is to construct holomorphic modular forms for the Hecke groups $H\big(\sqrt{D}\big)$ where $D\equiv 1\bmod{4}$ is a fundamental discriminant of a real quadratic field. Since the smallest possible such discriminant is $D=5$ (which means $\sqrt{D}$ is bigger than 2), it will not be possible to have holomorphic modular forms on $H\big(\sqrt{D}\big)$ which are square integrable over the fundamental domain which has infinite area. Let $q=e^{2\pi iz}$ with $z\in\mathfrak{h}$. Our approach is the construction of an $H\big(\sqrt{D}\big)$ analogue of the classical Dedekind eta function $\eta(z)$ defined by

[TABLE]

which satisfies the modular relations

[TABLE]

Recall that

[TABLE]

the Ramanujan cusp form of weight $12$ for $\operatorname{SL}(2,\mathbb{Z}).$ In the case of the Hecke group $H\big(\sqrt{5}\big)$, we construct an analogue of Dedekind’s eta function (1.1) which we denote by $\eta_{5}(z)$ and also an analogue of the Ramanujan cusp form $\Delta(z)$ which is denoted by $\Delta_{5}(z).$ Their definitions are as follows.

Definition 1.3**.**

Let $z\in\mathfrak{h}$ and set $q=e^{\frac{2\pi iz}{\sqrt{5}}}.$ Let $\chi_{{}_{5}}$ be the primitive real character modulo $5$. We define

[TABLE]

Furthermore, we define $\Delta_{5}(z):=\eta_{5}(z)^{5}.$

The series expansion for $\Delta_{5}(z)$ is given by

[TABLE]

In Section 3, we prove the following theorem using a generalization of Weil’s proof [undefw] of the modularity of the Dedekind eta function.

Theorem 1.6**.**

Let $z\in\mathfrak{h}$. The function $\Delta_{5}(z)$ is a holomorphic modular form for the Hecke group $H\big(\sqrt{5}\big)$, that is,

[TABLE]

for all $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in H\big(\sqrt{5}\big)$. Furthermore, $\Delta_{5}(z)$ vanishes at the cusp $\infty$, that is, $\lim\limits_{y\to\infty}\Delta_{5}(iy)=0.$

Remark 1.7*.*

We prove in Section 3 that for every $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in H\big(\sqrt{5}\big)$, we have

[TABLE]

where $u_{\gamma}$ is a certain fifth root of unity. Since $u_{\gamma}^{5}=1$, this explains why taking the fifth power of $\eta_{5}$ gives $\Delta_{5}$ which satisfies the modular relations in Theorem 1.6. In fact, we show that $\eta_{5}(z)$ is invariant under the action of $S=\left(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right)$. This is also illustrated in Figure 2.

Next, we consider discriminants $D\equiv 1\bmod{4}$ with $D>5.$ In this case, we can define $\eta_{D}(z)$ as follows.

Definition 1.8**.**

Let $D\equiv 1\bmod{4}$ with $D>5$ be a fundamental discriminant of a real quadratic field. Let $\chi_{D}$ be the primitive real character modulo $D$. Let $\mathfrak{m}_{D}\in\mathbb{N}$ be such that $L(-1,\chi_{D})=-2\mathfrak{m}_{D}$. 111See Lemma 3.1 for a proof that $L(-1,\chi_{D})$ is a negative even number for $D>5$. Let $z\in\mathfrak{h}$ and set $q=e^{\frac{2\pi iz}{\sqrt{D}}}.$ We define

[TABLE]

In Section 3, we prove the following theorem.

Theorem 1.10**.**

Let $z\in\mathfrak{h}$. The function $\eta_{D}(z)$ is a holomorphic modular form for the Hecke group $H\big(\sqrt{D}\big)$, that is,

[TABLE]

for all $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in H\big(\sqrt{D}\big)$. Furthermore, $\eta_{D}(z)$ vanishes at the cusp $\infty$, that is, $\lim\limits_{y\to\infty}\eta_{D}(iy)=0.$

At this stage, we make a few remarks.

Remark 1.11*.*

Curiously, we have not been able to find an analogue of the modular forms in the case when $D\equiv 3\bmod 4$. It would also be worthwhile to investigate connections between $\log\eta_{D}(z)$, or any other function of $\eta_{D}$, with an Eisenstein series for $H(\sqrt{D})$. These constitute two interesting research problems for the future.

Remark 1.12* (Relation to Siegel functions).*

Set $\tau=z/\sqrt{D}$, so that $q=e^{2\pi iz/\sqrt{D}}=e^{2\pi i\tau}$ is the classical $q$-parameter. After this change of variables, the residue class subproduct over $a$ appearing in $\eta_{D}$ can be rewritten as products of Siegel functions (equivalently Siegel modular units) up to explicit $q$-powers, Dedekind eta factors and roots of unity. On the other hand, the product involving the exponent $\chi_{D}(n)$ can be written as a ratio of products of Siegel functions by breaking $n$ into arithmetic progressions. More precisely, let $\zeta_{D}=e^{2\pi i/D}$ and for $0\leq r,a\leq D-1$, set $u=r/D$, $v=a/D$. Then the Siegel function $g_{u,v}$ has the product expansion

[TABLE]

where $B_{2}(u)=u^{2}-u+\tfrac{1}{6}$ is the second Bernoulli polynomial. We refer the reader to the standard $q$-product definition of Siegel functions (see Kubert–Lang [undefl, Ch.2]), which we have specialized to the arguments $D\tau$ and $(u,v)=(r/D,a/D)$, and rewritten in terms of $q=e^{2\pi i\tau}$. See also [undefa, Eq. (15), Pg. 13].

We emphasize, however, that our modularity statements in Theorems 1.6 and 1.10 are formulated for the Hecke group $H(\sqrt{D})=\langle z\mapsto z+\sqrt{D},\ z\mapsto-1/z\rangle$ in the $z$-variable. In $\tau$-coordinates, the inversion corresponds to the map $\tau\mapsto-1/(D\tau)$, which lies outside $\mathrm{SL}_{2}(\mathbb{Z})$ and thus the standard transformation law $g_{(u,v)}(\gamma\tau)\to g_{(u,v)\gamma}(\tau)$ for $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ cannot be directly applied. Moreover, we show in Lemma 2.4, the product over $a$ acts a dual function to the product with exponent $\chi_{D}(n)$ under the inversion $z\mapsto-1/z$ which plays a key role in making $\eta_{D}(z)$ modular. One could possibly attempt to establish modularity of $\eta_{D}(z)$ by studying each Siegel modular unit separately. But such an approach would likely require tracking explicitly the action of inversion on each factor arising from $(u,v)$. Our Weil style argument instead gives a direct and compact verification of the transformation laws for the Hecke group $H(\sqrt{D})$.

Remark 1.13*.*

In the Hecke group $H(\sqrt{D})$ with $D\equiv 1\bmod{4}$ and $D\geq 5$, there is a hyperbolic element $\gamma_{D}=\left(\begin{smallmatrix}\sqrt{D}&-1\\ 1&0\end{smallmatrix}\right)$ with the fixed points

[TABLE]

that is, $\gamma_{D}\omega_{D}=\omega_{D},\gamma_{D}\omega_{D}^{\prime}=\omega^{\prime}_{D}$. Consider the matrix $W_{D}=\left(\begin{smallmatrix}1&-\omega_{D}\\ 1&-\omega_{D}^{\prime}\end{smallmatrix}\right)\in\operatorname{SL}(2,\mathbb{R})$ which has the property that the conjugate of $\gamma_{D}$ by $W_{D}$ is a diagonal matrix. It follows that

[TABLE]

with $\sqrt{\kappa}+1/\sqrt{\kappa}=\sqrt{D}$ since conjugation preserves the trace of a matrix. Then if $f(z)$ is modular for the group $H(\sqrt{D})$, the function $f\left(W_{D}^{-1}z\right)$ is invariant under the transformation $z\mapsto\kappa z$. Following [undefk], a flare domain for the conjugated Hecke group $H(\sqrt{D})$ is isometric to a domain of the form $\{z\in\mathfrak{h}:1<|z|<\kappa,\,0<\arg(z)<\alpha\}$ for some angle $\alpha<\tfrac{\pi}{2}$. In this paper, the functions $\Delta_{5}(z)$ and $\eta_{D}(z)$ (with $D\equiv 1\bmod{4}$ and $D>5$) are holomorphic functions which are modular for $H(\sqrt{D})$, vanish at the cusp at $\infty$, but are not square integrable in the flare domain.

Theorems 1.6 and 1.10, alongside Remark 1.7 show that for $z\in\mathfrak{h}$, the function $\eta_{D}(z)$ admits the following Fourier expansion:

[TABLE]

Here we have noted that $L(-1,\chi_{5})=-\frac{2}{5}$. We refer the reader to Section 8 where we present a table containing the exact values of $a_{D}(N)$ in the cases $D=5,13,17$. A natural question at this stage is to understand the size of the Fourier coefficients $a_{D}(N)$. The growth of these coefficients is highly sensitive to the structure of the infinite product defining the modular form, and changes in the exponents can dramatically alter their behavior.

To appreciate the range of possibilities, it is instructive to compare two classical extremes. For the Dedekind eta function $\eta(z)$, the exponents of the factors in the product over $n$ in (1.1) are all equal to $1$. In this setting, the Fourier coefficients $\tau(N)$ that arise from $\Delta(z)=\eta(z)^{24}$ are multiplicative and satisfy Deligne’s bound [undefb], that is,

[TABLE]

In particular, the coefficients show polynomial growth.

At the opposite extreme, if the exponents are all equal to $-1$ as in the case of $\eta(z)^{-1}$, the resulting coefficients are given by $p(N)$, the number of partitions of $N$. In this context, Hardy and Ramanujan [undefc] showed that $p(N)$ exhibits exponential growth, that is,

[TABLE]

The functions $\eta_{D}(z)$ defined in (1.4) and (1.9) occupy an intermediate position between these two extremes. Their defining infinite products involve exponents taking the values $1$ and $-1$, according to the real character $\chi_{D}$. This raises the question of whether the coefficients $a_{D}(N)$ exhibit polynomial growth, or whether the presence of negative exponents in the product formula instead leads to exponential growth. As a numerical illustration, we refer the reader to Figure 3, which appears to suggest that in the case $D=5$, $\log|a_{5}(N)|$ grows linearly with $\sqrt{N}$.

A second and equally fundamental question concerns the sign behavior of $a_{D}(N)$. Even when the growth rate of coefficients is understood, their oscillatory nature often encodes deeper arithmetic information. Here the functions $\eta_{D}(z)$ are built from infinite products whose exponents take both positive and negative values, and the factors also incorporate certain roots of unity. It is therefore not a priori clear whether the resulting Fourier coefficients $a_{D}(N)$ should display persistent sign changes or exhibit a dominant sign bias. A precise formulation of this question is to determine whether $a_{D}(N)$ changes sign infinitely often, and more generally, to quantify the frequency and distribution of such sign changes. We refer the reader to Figures 3 and 4, where the plots for the sequence $a_{D}(N)$ for $D=5,13$ and $17$ display remarkably distinct patterns of oscillations and sign changes.

In what follows, our objective is to precisely understand the arithmetic behavior of the Fourier coefficients $a_{D}(N)$ and answer both the questions raised above. To state our results, we need some notation. Let $\mathfrak{n}_{D}$ be the least quadratic non-residue modulo $D$. Note that $\mathfrak{n}_{D}$ is prime. The singular series, which will determine the sign patterns of $a_{D}(N)$, is given as follows. Define

[TABLE]

where

[TABLE]

Here the primed sum indicates that we only take one representative from each pair of complex conjugate odd characters and $\chi_{\mathfrak{n}_{D}}$ is the real primitive character modulo $\mathfrak{n}_{D}$. By the functional equation, $\beta_{D}(u,\mathfrak{n}_{D})$ can be expressed in terms of the Dirichlet $L$-values at $s=1$ which is more computationally practical via the class number formula (this is given in Lemma 5.22).

We are now ready to state our results.

Theorem 1.17**.**

Let $D\equiv 1\bmod{4}$ with $D>0$ be a fundamental discriminant of a real quadratic field. Let $\chi_{D}$ be the primitive real character modulo $D$, and $a_{D}(N)$ be given by (1.14). Then $a_{D}(N)$ lies in the ring of integers $\mathcal{O}_{K}=\mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]$ of the quadratic field $K=\mathbb{Q}(\sqrt{D})$. Furthermore, define

[TABLE]

and

[TABLE]

Then if $\mathfrak{n}_{D}<\sqrt{D}$, we have for $N\geq 1$,

[TABLE]

where $\mathfrak{S}_{D}(N)$ is given by (1.15) and the implied constant in the error term depends at most on $D$.

For clarity, we separately state the result for the special cases $\mathfrak{n}_{D}=2,3$ which is an immediate corollary of Theorem 1.17. Note that when $\mathfrak{n}_{D}=3$, there is only one non-principal character modulo $3$ and hence the sum in $\beta_{D}(u,\mathfrak{n}_{D})$ is empty.

Corollary 1.19**.**

Let the notations be as in Theorem 1.17. If $\mathfrak{n}_{D}=2$, we have for $N\geq 1$,

[TABLE]

If $\mathfrak{n}_{D}=3$, we have for $N\geq 1$,

[TABLE]

where

[TABLE]

and $\chi_{3}$ is the primitive real character modulo 3.

We make the following remarks.

Remark 1.21*.*

We do not pursue the case $\mathfrak{n}_{D}\geq\sqrt{D}$ in Theorem 1.17, since we believe it to be never true in our context. In fact, when $D$ is prime, we expect substantially stronger bounds of the form $\mathfrak{n}_{p}\ll_{\varepsilon}p^{\varepsilon}$ following Vinogradov’s conjecture. Assuming the Generalized Riemann Hypothesis, it is known that $\mathfrak{n}_{p}\leq(\log p)^{2}$, see [undefn]. Unconditionally, Leveque (see [undefp], pp. 122–123) proves the inequality $\mathfrak{n}_{p}<\sqrt{p}$ for all primes $p\neq 2,3,7,23$. This settles the case when $D$ is prime in Theorem 1.17.

However, it seems difficult to establish the inequality $\mathfrak{n}_{D}<\sqrt{D}$ for all composite moduli. By [undef] and [undefg], such results are known for all $D\geq\exp(\exp({9.594})$ say. However the lower bound on $D$ for which these results are valid are somewhat astronomical and beyond computational limits, and therefore, it is not possible to verify the remaining values numerically.

Remark 1.22*.*

When $D=5$, we have $\mathfrak{n}_{D}=2$ and the special values

[TABLE]

where $\phi=(1+\sqrt{5})/2$. In this case, $c_{D}=\pi^{2}/25$ and we have the asymptotic relation

[TABLE]

We refer the reader to Figure 5 which compares the two sides of (1.23).

The sign changes of the Fourier coefficients $a_{D}(N)$ are governed by the singular series $\mathfrak{S}_{D}(N)$. We can see from the formula for $\mathfrak{S}_{D}(N)$ in (1.15), that the sign patterns are periodic modulo $\mathfrak{n}_{D}$. In specific cases, we can precisely compute the values of $\beta_{D}(u,\mathfrak{n}_{D})$ and therefore, the singular series $\mathfrak{S}_{D}(N)$ using either

[TABLE]

or by using the class number formula. This shall give a complete picture of the sign patterns. For instance, recall from Corollary 1.19 that in the case $\mathfrak{n}_{D}=3$, we have

[TABLE]

By evaluating $\beta$, we find that the sign patterns in this case takes a very simple form of a three periodic sequence which explains the nature of the graph for $a_{17}(N)$ in Figure 4 (note that $\mathfrak{n}_{17}=3$). We also compute an example case for $\mathfrak{n}_{D}=5$, the first modulus for which this occurs being $D=73$. We present these results in the following proposition.

Proposition 1.24**.**

Let the notations be as in Theorem 1.17. For $\mathfrak{n}_{D}=3$, the expression $\cos(\tfrac{2\pi}{3}N-\beta)$ takes the values

[TABLE]

periodically, where $\beta$ is given by (1.20). In particular, for $D=17,41$, we have

[TABLE]

for $N\equiv 1,2,3\bmod 3$ respectively. For $D=73$, we have for $N\geq 1$,

[TABLE]

which takes the values

[TABLE]

for $N\equiv 1,2,3,4,5\bmod 5$ respectively.

We end the introduction with a more detailed analysis of the case $D=5$. Note that the oscillations of $a_{5}(N)$ in Figure 3 take a much more jagged appearance compared to those of $a_{13}(N)$ and $a_{17}(N)$ in Figure 4. As we shall see, this is due to the influence of secondary exponential terms. When $D=5$, these secondary terms are much closer in size to the leading term and therefore, their effect is more pronounced, particularly when $N$ is relatively small. They also include an oscillatory sign factor which is out of phase with that of the leading sign factor $(-1)^{N}$, thereby accounting for the jagged appearance. The following result states an asymptotic formula for $a_{D}(5)$ which includes these influential secondary exponential terms.

Theorem 1.25** (Two term asymptotic expansion).**

Let the notations be as in Theorem 1.17. Let $N\geq 1$ and $\phi=(1+\sqrt{5})/2$. For each fixed $J\geq 1$, we have

[TABLE]

where $b_{j},\widetilde{b_{j}}$ are effectively computable absolute constants for $1\leq j\leq J$.

Remark 1.27*.*

When $D=5$, the leading exponential terms in (1.25) are

[TABLE]

As we shall show later (see Remark 6.8), the first two leading exponential terms in the asymptotic expansion of $a_{13}(N)$ are given by

[TABLE]

Therefore, in the case $D=5$, the secondary exponential term, alongside the cosine factor in the front, has a significantly more pronounced influence on the behavior of $a_{5}(N)$ than in the case $D=13$, particularly when $N$ is relatively small. This accounts for the difference in the graphs of $a_{5}(N)$ and $a_{13}(N)$ shown in Figures 3 and 4.

One can, in principle, acquire an infinite asymptotic series of decreasing exponential terms in (1.25). For $a_{D}(N)$ in general, the lower order terms become rather complex, which is why we have restricted our attention to just two terms in the $D=5$ case.

Finally, we highlight an exact formula connecting the Fourier coefficients $a_{D}(N)$ with classical partition functions, which we establish in Section 7 and which may be of independent interest. In particular, the twisted partition function

[TABLE]

plays a central role in this correspondence. Here $\lambda$ denotes a partition of $k$ and $\ell(\lambda)$ its length, that is, the number of parts in the partition. When $\theta=1$, this reduces to the ordinary partition function. In our setting, $\theta$ will be taken to be certain $D$-th roots of unity; we refer the reader to Section 7 for a detailed account of this connection. The study of recurrence relations and congruence properties of the sequence $a_{D}(N)$ similar to the case of partitions is also of interest, and may be a topic of discussion on another occasion.

Structure of the paper

In Section 2, we construct certain eta-products twisted by real characters and derive modular relations for them. Using these modular relations, we prove Theorems 1.6 and 1.10 in Section 3. Sections 4 and 5 are devoted to the study of asymptotic behavior and sign changes of $a_{D}(N)$, leading to the proofs of Theorem 1.17 and Proposition 1.24. In Section 6, we provide a sketch of the proof of Theorem 1.25, building on the ideas from Sections 4 and 5. In Section 7, we explore some interesting connections between the Fourier coefficients $a_{D}(N)$ and the theory of partitions. Finally, in Section 8, we present a table containing the exact values of $a_{D}(N)$ in the cases $D=5,13,17$ for $1\leq N\leq 25$.

General Notations.

We employ some standard notation that will be used throughout the paper.

• •

Throughout the paper, the expressions $f(X)=\mathcal{O}(g(X))$, $f(X)\ll g(X)$, and $g(X)\gg f(X)$ are equivalent to the statement that $|f(X)|\leq C|g(X)|$ for all sufficiently large $X$, where $C>0$ is an absolute constant. A subscript of the form $\ll_{\alpha}$ means the implied constant may depend on the parameter $\alpha$. Dependence on several parameters is indicated in an analogous manner, as in $\ll_{\alpha,\lambda}$.

• •

Throughout the paper, we shall let $D\equiv 1\bmod{4}$ with $D>0$ be a fundamental discriminant of a real quadratic field and $\chi_{D}$ shall denote the primitive real character modulo $D$. For $D>5$, we set $\mathfrak{m}_{D}\in\mathbb{N}$ be such that $L(-1,\chi_{D})=2\mathfrak{m}_{D}$.

• •

We denote the least quadratic non-residue modulo $D$ by $\mathfrak{n}_{D}$ and $\chi_{\mathfrak{n}_{D}}$ shall be the primitive real character modulo $\mathfrak{n}_{D}$.

• •

For a primitive character $\chi\bmod q$, the Gauss sum is denoted by $\tau(\chi)$.

• •

The function $\phi$ denotes the Euler totient function. We also use the same notation to denote the golden ratio $(1+\sqrt{5})/2$, and this should be clear from the context.

• •

We write $\zeta_{D}$ to denote the $D$-th root of unity $\zeta_{D}=e^{\frac{2\pi i}{D}}$.

• •

We set $e(x)=e^{2\pi ix}$ and use both notations interchangeably throughout the paper.

• •

We use the notation $n\equiv\square\bmod D$ to indicate that $n$ is a square modulo $D$. Analogously, the notation $n\not\equiv\square\bmod D$ means that $n$ is not a square modulo $D$.

• •

We denote by $\varepsilon$ an arbitrarily small positive quantity that may vary from one line to the next, or even within the same line. Thus we may write $X^{2\varepsilon}\leq X^{\varepsilon}$ with no reservations.

2. Modular Relations between Twisted Eta Products

Let $D\equiv 1\bmod{4}$ with $D>0$ be a fundamental discriminant of a real quadratic field. Let ${\chi_{D}}$ be the primitive real character modulo $D$. For $z\in\mathfrak{h}$, define the twisted product

[TABLE]

The Dirichlet $L$-function $L(s,{\chi_{D}})=\sum\limits_{n=1}^{\infty}{\chi_{D}}(n)n^{-s}$ satisfies the functional equation (see [undeff, Ch. 5])

[TABLE]

In this section, we prove the following theorem which gives a modular relation between $\Phi_{D}$ and a certain dual of $\Phi_{D}$, denoted by $\Phi^{\#}_{D}$, which is defined as follows. For $z\in\mathfrak{h}$, let

[TABLE]

Theorem 2.4**.**

For $z\in\mathfrak{h}$, we have

[TABLE]

where $\Phi_{D}(z)$ and $\Phi^{\#}_{D}(z)$ are defined by (2.1) and (2.3) respectively.

We break the proof of Theorem 2.4 into the following subsections.

2.1. The function $F_{D}(z)$.

By Taylor series expansion, we have

[TABLE]

For $z\in\mathfrak{h}$, define

[TABLE]

Let $s\in\mathbb{C}$ with ${\rm Re}(s)>1$. Then for $\sigma>1,$ we have the Mellin transform pair

[TABLE]

where $u>0$. We prove the following lemma.

Lemma 2.7**.**

Let $s\in\mathbb{C}$ with ${\rm Re}(s)>1$. The function $\widetilde{F}_{D}(s)$ satisfies the functional equation

[TABLE]

Proof.

We compute

[TABLE]

We have the functional equation of the Riemann zeta function

[TABLE]

It follows from (2.2) and the identity $\Gamma\left(\frac{s}{2}\right)\Gamma\left(\frac{s+1}{2}\right)=2^{1-s}\sqrt{\pi}\,\Gamma(s)$ that

[TABLE]

Then we have the functional equation

[TABLE]

which completes the proof. ∎

2.2. The dual function $F_{D}^{\#}(z)$

We recall the identity (see [undeff, Ch. 3])

[TABLE]

which gives the Fourier expansion of ${\chi_{D}}$ in terms of additive characters. For $z\in\mathfrak{h}$, define the dual function

[TABLE]

Let $s\in\mathbb{C}$ with ${\rm Re}(s)>1$. Then for $\sigma>1,$ we have the Mellin transform pair

[TABLE]

where $u>0$. We establish the following lemma.

Lemma 2.9**.**

Let $s\in\mathbb{C}$ with ${\rm Re}(s)>1$. The dual function $\widetilde{F}^{\#}_{D}(s)$ satisfies the functional equation

[TABLE]

Proof.

We compute

[TABLE]

Combining this with Lemma 2.7 completes the proof. ∎

2.3. Proof of Theorem 2.4

We now present the proof of the modular relation (2.5). Since both sides of (2.5) are analytic functions on $\mathfrak{h}$, by the Identity Theorem, it suffices to prove (2.5) in the special case when $z=iy$ for any $y>0$. It follows from Lemma 2.7 that $\widetilde{F}_{D}(s)$ has a double pole at $s=0$ with residue $L^{\prime}(0,{\chi_{D}})$ and a pole at $s=-1$ with residue $\frac{\pi\,L(-1,\,{\chi_{D}})}{\sqrt{D}}$ and no other poles. Fix $\sigma>1.$ Using inverse Mellin transform and shifting the line of integration to the left, we have

[TABLE]

Now we apply Lemma 2.9. It follows that

[TABLE]

Finally, we exponentiate both sides of (2.3). Using the identities (2.6) and (2.8), we obtain

[TABLE]

which is equivalent to

[TABLE]

This completes the proof. ∎

3. Modular Forms on $H(\sqrt{D})$: Proofs of Theorems 1.6 and 1.10

3.1. Preliminaries

Here we prove Theorems 1.6 and 1.10. We need a preliminary lemma regarding values of quadratic Dirichlet $L$-functions at $s=-1$. Some cases of this result has been alluded to before (see Zagier [undefx, Pg. 54]). We provide a complete proof for any $D\equiv 1\bmod 4$ for the reader’s convenience.

Lemma 3.1**.**

Let $D\equiv 1\bmod{4}$ with $D>0$ be a fundamental discriminant of a real quadratic field. Let $\chi_{D}$ be the primitive real character modulo $D$. Then

[TABLE]

Proof.

The case $D=5$ can be verified numerically. So we may assume $D>5$. We have

[TABLE]

where $\zeta(s,a)$ is the Hurwitz zeta function and since ${\chi_{D}}$ is even, we have noted that

[TABLE]

Writing

[TABLE]

it suffices to show that $S({\chi_{D}})$ is divisible by $4D$. Since ${\chi_{D}}$ is primitive and real, $D$ must be squarefree. We consider the following cases.

Case 1: $D$ is prime. We write

[TABLE]

Let $g$ be a primitive root of $D$. Then the non-residues are congruent to the odd powers of $g$. Therefore, their squares are congruent to $g^{2},g^{6},g^{10}$, and so on up to $g^{2q-4}$. Thus we have

[TABLE]

The right hand side of (3.2) is congruent to zero by Fermat’s Little Theorem. Since $D>5$, we have $g^{4}-1\not\equiv 0\bmod D$. This implies that $S_{2}({\chi_{D}})\equiv 0\bmod D$. Since $S_{1}({\chi_{D}})+S_{2}({\chi_{D}})$ is the sum of squares up to $D-1$, we also have $S_{1}({\chi_{D}})+S_{2}({\chi_{D}})\equiv 0\bmod D$. It follows that $S({\chi_{D}})\equiv 0\bmod D$.

To check divisibility by $4$, note that since $-1$ is a quadratic residue modulo $D$, $S_{1}({\chi_{D}})$ is odd if $D\equiv 5\bmod 8$ and even if $D\equiv 1\bmod 8$. The claim is now immediate by writing

[TABLE]

and carrying out a parity check.

Case 2: $D$ is squarefree of the form $D=\prod_{i=1}^{k}p_{i},$ for $p_{i}$ prime and $k\geq 2$. We only consider the case when $k=2$. The argument is similar for $k>2$.

We write $D=p_{1}p_{2}$. By the Chinese Remainder Theorem, we can break $S({\chi_{D}})$ as

[TABLE]

By Case 1, each term on the right hand side of (3.1) is divisible by both $p_{1}$ and $p_{2}$, and hence by $D$. It follows that $S({\chi_{D}})\equiv 0\bmod D$. As for divisibility by $4$, observe that by Case 1, the sum of the first two terms on the right-hand side of (3.1) is congruent to $0\bmod 4$. Likewise, the sum of the last two terms of (3.1) is also congruent to $0\bmod 4$. ∎

By (1.4), (1.9) and Lemma 3.1, we see that for $z\in\mathfrak{h}$,

[TABLE]

Using Theorem 2.4, we have the relations

[TABLE]

Multiplying the two relations above, we obtain

[TABLE]

3.2. Proof of Theorem 1.6

Since $z\in\mathfrak{h}$, the products in (1.4) are convergent, and thus $\eta_{5}(z)$ is holomorphic over $\mathfrak{h}$. As mentioned in Remark 1.7, we will show that for every $\gamma\in H\big(\sqrt{5}\big)$ we have

[TABLE]

where $u_{\gamma}$ is a certain fifth root of unity. In fact, we shall precisely compute $u_{\gamma}$.

By (3.4) and (3.7), we have

[TABLE]

Choose any element of $H(\sqrt{5})$. It can be expressed in the form

[TABLE]

with $k_{1},k_{2},\ldots,k_{\ell}\in\mathbb{Z}$, where

[TABLE]

We consider the following two cases.

Case 1 : $\ell$ is even. In this case, it is straightforward to check that the matrix $\gamma$ is of the form

[TABLE]

where $a,b,c,d\in\mathbb{Z}$. We claim that

[TABLE]

To prove this, we proceed by induction. First, by (3.8) and (3.9), we have

[TABLE]

For the base case, note that an element of the form $\alpha(k_{1},\ldots,k_{\ell})$ with $\ell$ even and the smallest number of $S$-terms is

[TABLE]

One immediately checks using (3.11) that (3.10) holds in this case. We additionally note the congruence conditions $bc\equiv-1\bmod{5}$ and $c^{2}\equiv 1\bmod 5$, which we shall use in our induction step.

Let

[TABLE]

where $a,b,c,d\in\mathbb{Z}$ and the congruence conditions $bc\equiv-1\bmod{5}$ and $c^{2}\equiv 1\bmod 5$ holds. Assume that (3.10) holds for $\alpha(k_{1},k_{2},\ldots,k_{2n})$. We now show that this assumption implies (3.10) holds for $\alpha(k_{1},k_{2},\ldots,k_{2n+2})$. Note that

[TABLE]

Then

[TABLE]

where we explicitly compute that

[TABLE]

It follows (using the fact that $bc\equiv-1\bmod{5}$ and $c^{2}\equiv 1\bmod{5}$) that

[TABLE]

It follows using (3.11) that (3.10) holds for $\alpha(k_{1},k_{2},\ldots,k_{2n+2})$. The desired claim follows.

Case 2 : $\ell$ is odd. In this case, the matrix $\gamma$ is of the form

[TABLE]

where $a,b,c,d\in\mathbb{Z}$. Then we have

[TABLE]

By (3.9) and Case 1, we have

[TABLE]

Therefore in this case, one obtains

[TABLE]

Combining (3.10) and (3.12), we have for any $\gamma\in H(\sqrt{5})$,

[TABLE]

where $u_{\gamma}$ is a certain fifth root of unity. Consequently, $\Delta_{5}(z)=\eta_{5}(z)^{5}$ is invariant under the action of $H(\sqrt{5})$, which establishes the modularity of $\Delta_{5}$ as stated in Theorem 1.6. Moreover from the definition in (1.4), it follows that $\Delta_{5}(z)$ vanishes at the cusp $\infty$, that is, $\lim_{y\to\infty}\Delta_{5}(iy)=0.$ ∎

3.3. Proof of Theorem 1.10: The Case $D>5$

Again since $z\in\mathfrak{h}$, the products in (1.9) are convergent, and thus $\eta_{D}(z)$ is holomorphic over $\mathfrak{h}$. For $D>5$, by (3.4) and Lemma 3.1, we have

[TABLE]

From (3.7), we immediately obtain the relations

[TABLE]

Since $H(\sqrt{D})$ generated by

[TABLE]

we see that $\eta_{D}(z)$ is invariant under the action of $H(\sqrt{D})$. This establishes the modularity of $\eta_{D}$ as stated in Theorem 1.10. Moreover from the definition in (1.9), it follows that $\eta_{D}(z)$ vanishes at the cusp $\infty$, that is, $\lim_{y\to\infty}\eta_{D}(iy)=0.$∎

4. Asymptotics for Fourier Coefficients: Part I

In this section, we establish some preliminary lemmas towards the proof of Theorem 1.17. We aim to apply Cauchy’s formula in the form

[TABLE]

for some $0<r<1$ where

[TABLE]

and isolate the integral in (4.1) around the dominant singularities of $G(q)$. Our goal is to find the location of these singularities and give a precise description of $G(q)$ around these points.

Lemma 4.3**.**

For $|q|<1$, we have

[TABLE]

where $g(x)=\sum_{a=1}^{D-1}\chi_{D}(a)x^{a}$ is the Fekete polynomial corresponding to the character $\chi_{D}$.

Proof.

From the product formula (4.2), we have

[TABLE]

Since $\chi_{D}$ is a primitive real character, $\sum_{a=1}^{D}\chi(a)\zeta_{D}^{am}=\chi_{D}(m)\tau(\chi_{D})$ for all $m$ where $\tau(\chi_{D})$ is the Gauss sum. Thus the above is

[TABLE]

Since $\tau(\chi_{D})=\sqrt{D}$, evaluating the sum over $n$ gives us the desired result. ∎

Recall that in the statement of Theorem 1.17, we set $c_{D}=\sqrt{D}L(2,\chi_{D})/\mathfrak{n}_{D}^{2}$. The dominant singularity of $G(q)$ will roughly be of size $\exp(c_{D}/(1-|q|))$. We now give a minor arc estimate demonstrating that when $q$ is sufficiently far away from a low root of unity, $G(q)$ is substantially smaller than this.

Lemma 4.6**.**

For $A>0$ sufficiently large in terms of $D$, we have

[TABLE]

In particular, there exists $B>0$ depending at most on $A$ and $D$ such that if $q$ satisfies $|1-q^{m}|\geq B(1-|q|)$ for all $1\leq m\leq AD$, then

[TABLE]

Proof.

The left hand side of (4.7) is less than

[TABLE]

Since

[TABLE]

for $|q|\leq 1$, (4.9) is

[TABLE]

Taking $A$ sufficiently large depending on $D$, the bound in (4.7) follows. On taking $B$ sufficiently large depending on $A$ and $D$, the bound in (4.8) follows immediately by the triangle inequality. ∎

To set ideas, we note that we will eventually take $|q|=r\sim 1-\sqrt{c_{D}/N}$ in (4.1). For such $q$, the condition $|1-q^{m}|\leq B(1-|q|)$ then necessitates that $\arg q=2\pi u/v+\theta$ for some $1\leq u\leq v\leq m$ and $|\theta|\ll 1/\sqrt{N}\ll 1-r$. Our next proposition gives a more precise description of $G(q)$ around these remaining low roots of unity. Our results involve the periodic zeta function defined for $\alpha\in\mathbb{R}$ by

[TABLE]

when $\operatorname{Re}(s)>1$ (or $\operatorname{Re}(s)>0$ when $\alpha\in\mathbb{R}\backslash\mathbb{Z}$) and extended to all of $\mathbb{C}$ by analytic continuation. If $\alpha\in\mathbb{Z},$ we have $P(s,\alpha)=\zeta(s)$; otherwise it is entire. In either case, it is of at most polynomial growth in fixed vertical strips.

Proposition 4.10**.**

Let $\varepsilon>0$. Fix $u,v\in\mathbb{N}$ satisfying $1\leq u\leq v$ and $(u,v)=1$. Set $q=re^{2\pi i{u}/{v}+i\theta}$ where $0<r<1$ and $|\theta|\leq B(1-r)$ for some fixed constant $B>0$. If $D\nmid v$, we have

[TABLE]

where

[TABLE]

If $D|v$, we have

[TABLE]

for some computable constants $a_{D}(u,v),b_{D}(u,v)$ depending only on $u,v$ and $D$.

Proof.

For notational simplicity, let us replace $q$ by $qe(u/v)$ where we set $q=re^{i\theta}$ and $|\theta|\leq B(1-r)$. Recall from the proof of Lemma 4.3 that we may write

[TABLE]

with

[TABLE]

We break our proof into the following parts.

$S_{2}$** Analysis.** We look at $S_{2}$ first and consider the sum over $m$ modulo $v$. Write $S_{2}=S_{21}+S_{22}$ with

[TABLE]

Applying the identity

[TABLE]

which is valid for $\operatorname{Re}(z)>0$, we have

[TABLE]

The integrand has simple poles at $s=1,0,-1$ and so we would like to shift the integral past these points.

When estimating the integral in the left half-plane note that since $|\theta|\leq B(1-r)$,

[TABLE]

for some fixed $\delta=\delta(B)\asymp\min(1,B^{-1})$. In particular, by Stirling’s formula we have

[TABLE]

Hence the integral over the new line is absolutely convergent and $\ll_{D,B}|\log(1/q)|^{-\operatorname{Re}(s)}$ since $\zeta(s)$ and $L(1+s,\chi_{D})$ are of polynomial growth at most. Thus shifting to $\operatorname{Re}(s)=-3/2$, say, we find that

[TABLE]

on recalling that $\zeta(0)=-1/2$.

By the orthogonality of Dirichlet characters,

[TABLE]

On applying (4.15), this is equal to

[TABLE]

where $P(s,\alpha)$ is the periodic zeta function. Since $au/v\in\mathbb{Q}\backslash\mathbb{Z}$, on shifting to the line $\operatorname{Re}(s)=-3/2$, we only encounter poles at $s=0,-1$. If $\chi_{D}\psi$ is principal, we have a double pole at $s=0$; otherwise it is simple. Note that if $\chi_{D}\psi$ is principal, we must have $D\mid v$. Distinguishing between the two cases and accounting for the double pole when $D|v$, we have

[TABLE]

for some computable constants $a_{D}(u,v),b_{D}(u,v)$ depending only on $u,v$ and $D$. In total, we find that

[TABLE]

where

[TABLE]

$S_{1}$** Analysis : The Case $D\nmid v$.** We consider the sum over $n$ modulo $v$ in $S_{1}$ and write $S_{1}=S_{11}+S_{12}$ with

[TABLE]

By (4.15), we have

[TABLE]

By the functional equation (2.2), $L(0,\chi_{D})=0$ since $\chi_{D}$ is even. Thus the pole of the integrand at $s=0$ is simple. Shifting to $\operatorname{Re}(s)=-1-\varepsilon$ say, we see that

[TABLE]

after another application of the functional equation.

For $S_{12}$, by orthogonality of Dirichlet characters, we have

[TABLE]

As noted before, the character $\chi_{D}\psi$ can only be principal if $D|v$ which we are currently assuming is not the case. Hence we only have simple poles at $s=0,-1$ and on shifting contours, we see

[TABLE]

$S_{1}$** Analysis : The Case $D\mid v$.** Here we use a different decomposition of the sum. Let $v=Dv^{\prime}$. We write

[TABLE]

Now we consider the sum over $m$ modulo $v^{\prime}$ giving

[TABLE]

with

[TABLE]

and

[TABLE]

Applying Mellin inversion and writing $v^{\prime}(a+Dn)=v(n+a/D)$, we have

[TABLE]

where $\zeta(s,\alpha)=\sum_{n\geq 0}(n+\alpha)^{-s}$ is the Hurwitz zeta function. This can be continued to a meromorphic function on $\mathbb{C}$ with only a simple pole at $s=1$ with residue $1$, and with at most polynomial growth in fixed vertical strips. Since $au/D$ is not integral, the periodic zeta function in the integrand is entire and so the poles at $s=1,0,-1$ are all simple. Shifting contours to $\operatorname{Re}(s)=-1-\varepsilon$ gives

[TABLE]

since $vv^{\prime}=v^{2}/D$. The constant in the leading term here, that is, the sum over $a$ involving $P(2,au/D)$ can be evaluated using Gauss sums as follows:

[TABLE]

For $S_{12}^{\prime}$, (4.15) gives

[TABLE]

The Dirichlet series in the integrand here are given by Lerch zeta functions. Since neither $au/v$ or $ub/v^{\prime}$ are integral, these are entire functions and we only pick up poles at $s=0,-1$. Therefore, we obtain

[TABLE]

for some computable constant $c_{D}(u,v)$ depending only on $u,v$ and $D$.

Conclusion. We now put together the cases $D|v$ and $D\nmid v$ for $S_{1}$. Combining (4.18)–(4.21), we have

[TABLE]

for some computable constant $c_{D}^{\prime}(u,v)$ where

[TABLE]

For clarity, we mention that the second term on the right hand side of (4) has been absorbed into $c_{D}^{\prime}(u,v)$. Finally, we note that in our range of $\theta$, $\lvert 1-q\rvert\asymp_{B}1-r.$ Therefore, combining (4) with (4) in (4.14) and readjusting the constants $a_{D}(u,v),b_{D}(u,v)$ accordingly gives the desired result. ∎

We pause briefly to detail the most dominant singularities. Observe that the constant factors in the leading terms of (4.11) and (4.13) can be made positive by choosing $u,v$ such that $\chi_{D}(v)$ or $\chi_{D}(u)=-1$, respectively. In the case of (4.11), the constant is largest when $v$ is smallest. In other words, for those $v$ not divisible by $D$, the dominant singularity occurs at the points where $q\to e(u/v)$ and $v$ is the least quadratic non-residue modulo $D$, that is $v=\mathfrak{n}_{D}$. In this case, we have

[TABLE]

for all $1\leq u<v,(u,v)=1$. We have noted here that

[TABLE]

For those $v$ divisible by $D$, a similar heuristic shows that

[TABLE]

at points for which $\chi_{D}(u)=-1$. Clearly, the constant here is largest when $v=D$ and thus we see that the singularity for $D\nmid v$ is dominant provided $\mathfrak{n}_{D}<\sqrt{D}$ which as mentioned in Remark 1.21, we believe to be always the case. All other singularities are sub-dominant, in fact, they are exponentially smaller.

Our final lemma in this section allows us to localize the variable $\theta$ in Proposition 4.10 yet further (as well as set some notation). This is necessary for an application of the method of stationary phase in Section 5.

Lemma 4.23**.**

Let $0<r<1$, $\theta\in(-\pi,\pi]$, $N\geq 1$ be fixed and

[TABLE]

Then

[TABLE]

Proof.

We have

[TABLE]

The result then follows on distinguishing the size of $|\theta|$. ∎

5. Asymptotics for Fourier Coefficients: Part II

Our main objective in this section is to prove Theorem 1.17.

5.1. Initial preparations

Let $N\geq 1$ and $q=re^{i\theta}$ with $\theta\in[0,2\pi)$. We set $r$ such that

[TABLE]

where we recall that $c_{D}=\sqrt{D}L(2,\chi_{D})/\mathfrak{n}_{D}^{2}$. Note that $r=1-\sqrt{{c_{D}}/{{N}}}+\mathcal{O}_{D}(1/N)$ which we may use interchangeably.

By Cauchy’s formula, we have

[TABLE]

Assuming that $\mathfrak{n}_{D}<\sqrt{D}$, on applying Lemma 4.6 and Proposition 4.10, along with the remarks after it, we see that the dominant singularity of $G$ occurs around the points $e(u/\mathfrak{n}_{D})$, $1\leq u<\mathfrak{n}_{D}$, $(u,\mathfrak{n}_{D})=1$. More precisely, for $B$ sufficiently large in terms of $D$, we have

[TABLE]

for some fixed $\varepsilon>0$ depending only on $D$. This is exponentially smaller than our target bound $\exp(2\sqrt{c_{D}N})$.

By the asymptotic (4.11) in Proposition 4.10, the remaining integrals are

[TABLE]

where $\gamma_{D}(u,\mathfrak{n}_{D})$ is as defined in the statement of Proposition 4.10. Since $1-r\asymp_{D}1/\sqrt{N}$, we have $\exp(1-r)=1+\mathcal{O}_{D}(1/\sqrt{N})$. Therefore, the contribution from the error term in (5.1) is

[TABLE]

Here we have noted that the real part of $(\log(1/r)-i\theta)^{-1}$ is maximized when $\theta=0$, and the sum over $u\bmod\mathfrak{n}_{D}$ which involves $\gamma_{D}(u,\mathfrak{n}_{D})$ is $\ll_{D}1$ because none of the periodic zeta functions arising from (4.10) contain fractions $au/v$ which lie in $\mathbb{Z}$. Therefore, we arrive at the following relation:

[TABLE]

where

[TABLE]

and

[TABLE]

Note here that $f(\theta)$ and $\mathfrak{S}^{0}(N)$ both depend on $D$ as well, but we omit it from the notation for brevity.

5.2. The singular integral

We now focus on the integral in (5.3), localizing it further. By Lemma 4.23,

[TABLE]

which is exponentially smaller than our target bound $\exp(2\sqrt{c_{D}N})$. Also, we have

[TABLE]

for any $J>0$, and therefore this contribution is smaller than our target bound by any arbitrary power of $N$.

The remaining integral is given by

[TABLE]

In preparation for a Taylor expansion, we note that

[TABLE]

and

[TABLE]

for $\theta$ in the range of integration. Hence the integral (5.6) is

[TABLE]

because the integral involving the term $\theta^{3}$ vanishes due to symmetry. Since the worst error we have so far other than (5.2) is from (5.1) and is of the form $N^{-1}\exp(2\sqrt{c_{D}N})$, in total then we have

[TABLE]

The constant $c_{D}^{1/4}/2\sqrt{\pi}$ is contained in the expression for $\alpha_{D}$ in Theorem 1.17. Thus it suffices to show that the remaining constants arise from the singular series $\mathfrak{S}^{0}(N)$.

5.3. The singular series

The asymptotic (1.18) in Theorem 1.17 will follow from the following lemma.

Lemma 5.9**.**

Let $\mathfrak{S}^{0}(N)$ be as defined in (5.5). We have

[TABLE]

where $\mathfrak{S}_{D}(N)=(-1)^{N}$ when $\mathfrak{n}_{D}=2,$ and

[TABLE]

with $\beta_{D}(u,\mathfrak{n}_{D})$ as defined in (1).

Proof.

Recall that

[TABLE]

where by Proposition 4.10, we have

[TABLE]

Contribution from the principal character. Let us isolate the contribution from the principal character $\psi_{0}$. We use throughout that $\mathfrak{n}_{D}$ must be prime. Since both $\chi_{D}$ and $\psi_{0}$ are even, we have $L(0,\chi_{D}\psi_{0})=0$. Hence the second term in the sum in (5.3) disappears. For the first term, note that

[TABLE]

Since $\mathfrak{n}_{D}$ is prime, for those $n$ not divisible by $\mathfrak{n}_{D}$ the above inner sum is $\sum_{a=1}^{\mathfrak{n}_{D}-1}e(a/\mathfrak{n}_{D})=-1$ since $u$ is also coprime to $\mathfrak{n}_{D}$. If $n$ is divisible by $\mathfrak{n}_{D}$, this sum is $\phi(\mathfrak{n}_{D})$. Hence, we obtain

[TABLE]

Thus, the contribution from the principal character to the sum on the right hand side of (5.3) is

[TABLE]

since $L(0,\psi_{0})=0$, $\zeta(0)=-\frac{1}{2}$ and $\chi_{D}(\mathfrak{n}_{D})=-1$. Putting together this contribution with the first term in (5.3), we have

[TABLE]

where

[TABLE]

It remains to compute the contribution from the non-principal characters.

Contribution from the non-principal characters. Consider $\psi\bmod\mathfrak{n}_{D}$ with $\psi\neq\psi_{0}$. Note that $\psi$ is primitive since $\mathfrak{n}_{D}$ is prime. Thus we have

[TABLE]

Hence we obtain

[TABLE]

Since $\chi_{D}(-1)=1$, the parity of $\chi_{D}\psi$ equals the parity of $\psi$. Since the value at $s=0$ of even Dirichlet $L$-functions is zero, the above sum can be restricted to odd characters. The functional equation for odd characters $\chi\bmod q$ (see [undeff, Ch. 5]) reads

[TABLE]

Applying this to (5.14) gives

[TABLE]

We now use the following fact about Gauss sums: for characters $\chi_{1},\chi_{2}$ of moduli $q_{1},q_{2}$ with $(q_{1},q_{2})=1$,

[TABLE]

(see [undefq, Thm. 9.6]). Since $(\mathfrak{n}_{D},D)=1$, we may apply this to obtain

[TABLE]

Also since $\psi$ is odd and primitive, $\tau(\overline{\psi})=\psi(-1)\overline{\tau(\psi)}=-\overline{\tau(\psi)}$, so that $\tau(\overline{\psi})\tau(\psi)=-\mathfrak{n}_{D}$. Therefore, we arrive at

[TABLE]

We now isolate the real character and pair up the complex conjugate characters. Since $\mathfrak{n}_{D}$ is prime, there is only one non-principal real character $\psi(\cdot)=(\tfrac{\cdot}{\mathfrak{n}_{D}})$. This appears in the sum, that is, the character is odd, if and only if $\mathfrak{n}_{D}\equiv 3\bmod 4$ and in this case, we have

[TABLE]

by quadratic reciprocity along with the facts that $D\equiv 1\bmod 4$ and $(\frac{\mathfrak{n}_{D}}{D})=-1$. Hence, we obtain

[TABLE]

where $\chi_{\mathfrak{n}_{D}}$ denotes the real non-principal character modulo $\mathfrak{n}_{D}$.

To finish our proof, we note the following. Since the remaining odd characters occur in complex conjugate pairs, we may pair these terms up to obtain

[TABLE]

with $\beta_{D}(u,\mathfrak{n}_{D})$ as defined in (1). Therefore, we have

[TABLE]

Since $\psi$ and $\chi_{\mathfrak{n}_{D}}$ are both odd, $\psi(\mathfrak{n}_{D}-u)=-\psi(u)$ and $\chi_{\mathfrak{n}_{D}}(\mathfrak{n}_{D}-u)=-\chi_{\mathfrak{n}_{D}}(u)$. Hence we get $\beta_{D}(\mathfrak{n}_{D}-u,\mathfrak{n}_{D})=-\beta_{D}(u,\mathfrak{n}_{D}).$ It follows that $e(uN/\mathfrak{n}_{D})e^{-i\beta_{D}(u,\mathfrak{n}_{D})}$ and $e((\mathfrak{n}_{D}-u)N/\mathfrak{n}_{D})e^{-i\beta_{D}(\mathfrak{n}_{D}-u,\mathfrak{n}_{D})}=e(-uN/\mathfrak{n}_{D})e^{i\beta_{D}(u,\mathfrak{n}_{D})}$ are conjugates. Pairing up these terms in the sum for $\mathfrak{S}^{0}(N)$ in (5.10) gives

[TABLE]

for $\mathfrak{n}_{D}\geq 3$. If $\mathfrak{n}_{D}=2$, we can only have the term $u=1$ in (5.10) and $\beta_{D}(1,2)=0$. So we obtain

[TABLE]

This completes the proof. ∎

5.4. Proof of Theorem 1.17

We first show that $a_{D}(N)\in\mathbb{Q}(\sqrt{D})$. Recall that $q=e^{\frac{2\pi iz}{\sqrt{D}}}$ and $\zeta_{D}=e^{\frac{2\pi i}{D}}$. By interchanging the products in (1.9), we have the relation

[TABLE]

Therefore, it suffices to study only the coefficients arising from the products over $a$ in (5.4). We treat the case where the product runs over the quadratic residues modulo $D$; the argument for the quadratic non-residues is concomitant.

Since $D\equiv 1\bmod{4}$, we have $\mathbb{Q}(\sqrt{D})\subseteq\mathbb{Q}(\zeta_{D})$. Upon expanding the product

[TABLE]

the resulting coefficients are elementary symmetric polynomials in $\zeta_{D}^{a}$. We consider the Galois group $\mathrm{Gal}(\mathbb{Q}(\zeta_{D})/\mathbb{Q})$, which has order $\phi(D)$. Let $H$ denote the subgroup defined by

[TABLE]

This subgroup $H$ has index $2$ and its fixed field is quadratic. Moreover, the Gauss sum $\tau(\chi_{D})=\sum_{a=1}^{D}\left(\frac{a}{D}\right)\zeta_{D}^{a}$ is equal to $\sqrt{D}$ and is fixed by all elements of $H$. Therefore the fixed field of $H$ is $\mathbb{Q}(\sqrt{D})$. Since symmetric polynomials in $\zeta_{D}^{a}$ (with $a$ ranging over quadratic residues modulo $D$) are fixed under $H$, the coefficients lie in $\mathbb{Q}(\sqrt{D})$. Note that the expressions involve only roots of unity, so they are algebraic integers. Hence the coefficients lie in the ring of integers $\mathcal{O}_{K}$ of $K=\mathbb{Q}(\sqrt{D})$, which is $\mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]$.

Finally, the asymptotic relation (1.18) follows immediately from (5.8) and Lemma 5.9. ∎

5.5. Proof of Proposition 1.24

Recall the definition of $\beta_{D}(u,\mathfrak{n}_{D})$ from (1). We need the following lemma, which allows us to write $\beta_{D}(u,\mathfrak{n}_{D})$ in terms of values of $L$-functions at $s=1$.

Lemma 5.22**.**

Let $\beta_{D}(u,\mathfrak{n}_{D})$ as defined in (1). We have

[TABLE]

with the same convention for the primed sum as defined in (1).

Proof.

For an odd character $\chi\bmod q$, by (5.15) we have

[TABLE]

By (5.17), we also have

[TABLE]

Therefore, we obtain

[TABLE]

A similar treatment for the products $L(0,\psi)L(0,\chi_{D}\overline{\psi})$ $L(0,\chi_{\mathfrak{n}_{D}})L(0,\chi_{D}\chi_{\mathfrak{n}_{D}})$ gives us the desired result. For clarity, we mention that we have used (5.19) while evaluating $\chi_{\mathfrak{n}_{D}}(D)$. ∎

We are now ready to prove Proposition 1.24.

Proof.

In the case $\mathfrak{n}_{D}=3$, it is easier to use the formula for $\beta_{D}(u,\mathfrak{n}_{D})$ in terms of $L$-values at $s=1$ as in Lemma 5.22. The characters $\chi_{3}$ and $\chi_{D}\chi_{3}$ correspond to the imaginary quadratic fields $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{-3D})$ respectively. We shall use the class number formula for imaginary quadratic fields which states

[TABLE]

where $h(d)$ is the class number of $\mathbb{Q}(\sqrt{d})$ and $w$ is the number of roots of unity in this field. We have

[TABLE]

Since $D\equiv 1\bmod 4$ is squarefree and $(3,D)=1$ since $\mathfrak{n}_{D}=3$, $h(-3D)$ is even. In particular, $h(-3D)\equiv 0,2,4\bmod 6$ which shows that the pattern of the cosine factor takes the form

[TABLE]

When $D=17$ or $41$, we have $h(-3D)=2$ and this sequence occurs exactly when $N\equiv 1,2,3\bmod 3$.

The only other possible sign pattern is $-1,\frac{1}{2},\frac{1}{2}$ which can only occur when $h(-3D)\equiv 1,3,5\bmod 6$. But since $h(-3D)$ is even, this never happens.

When $\mathfrak{n}_{D}=5$, the real character in $\beta_{D}(u,\mathfrak{n}_{D})$ has no contribution. We need to consider a pair of complex conjugate characters. Let $\psi$ be the character with the sequence of values $\psi(n)=1,i,-i,-1$ for $n$ increasing. Then we have

[TABLE]

where

[TABLE]

By the formula

[TABLE]

valid for $\chi\bmod q$, we find that

[TABLE]

Now, the first moduli $D\equiv 1\bmod 4$ which has $\mathfrak{n}_{D}=5$ is $D=73$. In this case, $\psi(D)=-i$ and a short computation using (5.24) gives

[TABLE]

Inputting these values, we find

[TABLE]

Putting this in (5.23) followed by a numerical check yields the desired conclusion. ∎

6. Proof of Theorem 1.25

Here we give a brief sketch of how one can refine the methods involved in the proof of Theorem 1.17 to obtain Theorem 1.25.

We first note that with a little extra effort, the asymptotic given in Corollary 1.19 can be improved to the series expansion

[TABLE]

for some computable constants $b_{j}$ and $\phi=(1+\sqrt{5})/2$. Indeed, by shifting contours arbitrarily far to the left in the proof of Proposition 4.10, we acquire an asymptotic expansion of $\log G(q)$ as $q\to-1$ in terms of powers of $\log(-1/q)$ to any required degree of accuracy. We combine this with an arbitrarily high precision Taylor expansion of $f(\theta)$ in the stationary phase argument. Expanding the resulting sub-constant terms in the exponential into a series and integrating them against the Gaussian then yields the desired result. In particular, we note that after Taylor expanding $f(\theta)$, the terms

[TABLE]

are $o(1)$. Therefore using the relation $e^{x}=1+x+\cdots$, the contribution from $m$ odd shall vanish due to the symmetry of the integral.

For the secondary terms, we note that from Proposition 4.10, the next largest constant in the leading coefficient of the expansion of $\log G(q)$ occurs around the points $q\to\zeta_{5}^{2},\zeta_{5}^{3}$. This is the regime $D\mid v$. Therefore using (4.13), we have

[TABLE]

for $u=2,3$ and computable constants $a_{5}(u,5)$ and $b_{5}(u,5)$. Again, the lower order terms here are computable to any required degree of accuracy by shifting contours further to the left in Proposition 4.10.

From a more detailed analysis of the arguments of Proposition 4.10, we find that

[TABLE]

To see this, first note that we are in the case $D\mid v$; in fact, we have $v=D=5$. There is no contribution from $S_{21}$, and we only need to evaluate the contribution from $S_{22}$. The coefficient of $\log(\log(1/q))$ arises from the contribution of the double pole at $s=0$ in (4.16). This only happens when $\psi=\chi_{5}$ in which case, $L(1+s,\chi_{5}\psi)=L(1+s,\psi_{0})$ where $\psi_{0}$ is the principal character modulo $5$. A quick computation shows that the residue from the double pole of the integrand in (4.16) when $\psi=\chi_{5}$ is

[TABLE]

Summing over $a$ in (4.16), it suffices to check that

[TABLE]

A revisit to (5.13) and the fact $L(0,\chi_{5})=0$ since $\chi_{5}$ is even gives the desired claim.

Then, in-keeping with the saddle point method, as before we choose the contour around the points $\zeta_{5}^{2},\zeta_{5}^{3}$ to have radius such that the derivative of the function

[TABLE]

vanishes at $\theta=0$, namely $\log(1/r)=\sqrt{4\pi^{2}N/125}=2\pi\sqrt{N}/(5\sqrt{5})$. Then combining the contributions from both points $\zeta_{5}^{2},\zeta_{5}^{3}$, the methods of stationary phase give a leading order contribution of

[TABLE]

since

[TABLE]

Therefore, all it remains is to show that

[TABLE]

where $\phi=(1+\sqrt{5})/2.$ To prove (6.3), we need to evaluate $b_{5}(u,5)$ for which we need a more detailed analysis of Proposition 4.10.

Firstly, let’s treat the contribution from $S_{1}$ in Proposition 4.10 to $b_{5}(u,5)$. Our case is $D\mid v$, in fact, $v=D=5$. Therefore $v^{\prime}=1$ and only $S_{11}^{\prime}$ contributes. The contribution from $S_{11}^{\prime}$ is evaluated as follows:

[TABLE]

because $\chi_{5}(-a)=\chi_{5}(a)$ and $\zeta(0,(5-a)/5)=-\zeta(0,a/5)$ by using the relation $\zeta(0,\alpha)=\tfrac{1}{2}-\alpha$, $0<\alpha<1$. Now, we use the relation $P(1,\alpha)=\textrm{Li}_{1}(e^{2\pi i\alpha})=-\log(1-e^{2\pi i\alpha})$ where $\textrm{Li}_{s}(z)$ is the polylogarithm function. This yields

[TABLE]

Hence we obtain

[TABLE]

Therefore, we record the following contributions from $S_{1}$:

[TABLE]

Next, we look at the contribution from $S_{2}$. This is given by the contributions to the integral (4.16). For the double pole which happens when $\psi=\chi_{5}$, following (6) and (6.2), the contribution is

[TABLE]

because the $P(0,au/5)$ term vanishes after summing over $a$. We can again use the relation (5.13). By analytic continuation of both sides (note that $au/5\notin\mathbb{Z}$ and $\psi$ is non-principal), we can take derivatives with respect to $s$. Applying this here, the contribution becomes

[TABLE]

Since $\chi_{5}$ is even, an application of the functional equation shows that $L^{\prime}(0,\chi_{5})=\tfrac{\sqrt{5}}{2}L(1,\chi_{5})$. Recall that $L(1,\chi_{5})=\tfrac{2}{\sqrt{5}}\log\phi$. Hence we record the contribution from the double pole case:

[TABLE]

Finally, we again look at the contribution from $S_{2}$ but this time, the contributions are from the simple pole in the integral (4.16). These occur when $\psi\neq\chi_{5}$. We separate into $\psi=\psi_{0}$ principal, and $\psi\neq\psi_{0}$.

When $\psi=\psi_{0}$, we obtain

[TABLE]

similar to how we evaluated in (5.12). Therefore, the contribution from the principal character $\psi_{0}$ is

[TABLE]

Our last contribution is from the complex characters. Let $\psi$ be the character with the sequence of values $\psi(n)=1,i,-i,-1$ for $n$ increasing. Using (5.13) and (5.15), the contribution here is

[TABLE]

Note that $\chi_{5}\psi=\overline{\psi}$. Thus the above simplifies to

[TABLE]

Therefore it suffices to evaluate $\tau(\overline{\psi})^{2}$ and $L(0,\psi)^{2}$. We already calculated before in the proof of Proposition 1.24 that $L(0,{\psi})=\tfrac{3+i}{5}$. A quick evaluation of the Gauss sum gives $\tau(\overline{\psi})^{2}=-\sqrt{5}(1-2i)$. So we obtain the following contributions:

[TABLE]

Combining all the contributions from (6.4), (6.5), (6.6), (6.7), we arrive at

[TABLE]

It follows that

[TABLE]

which completes the proof of (6.3). In conclusion, we obtain the desired secondary term

[TABLE]

again, with lower order terms calculable if desired.

To complete the proof, we note from Proposition 4.10 that the next largest exponential term arises when $v=3$, the second least quadratic non-residue modulo $5$. A quick calculation shows that this contribution, as well as all other lower order terms is

[TABLE]

which completes the proof. ∎

Remark 6.8*.*

We provide a brief comparison between the cases $D=5$ and $D=13$. A short computation using Proposition 4.10 and the fact that $L(2,\chi_{13})=4\pi^{2}/(13\sqrt{13})$ shows that the first two leading exponential terms in the asymptotic expansion of $a_{13}(N)$ arises from the cases $v=\mathfrak{n}_{13}=2$ and $v=13$. The precise exponential terms are

[TABLE]

Comparing it with the asymptotic (1.25), we see that the secondary exponential term is much closer to the leading exponential term when $D=5$ compared to $D=13$.

7. Exact Identities between Fourier Coefficients and Partitions

In this section, we explore several interesting connections between the Fourier coefficients $a_{D}(N)$ of $\eta_{D}(z)$ and the theory of partitions. Our starting point is the relation (5.4). Let $\theta\in\mathbb{C}$ with $\lvert\theta\rvert\leq 1$. We define the following three products:

[TABLE]

In the special case when $D$ is prime, (5.4) gives

[TABLE]

By Euler’s Pentagonal Number Theorem, we have for $\lvert q\rvert<1$,

[TABLE]

where

[TABLE]

Note that the sequence $\{g(k)\}$ forms the generalized pentagonal numbers. Next, we write $\lambda\vdash k$ to denote that $\lambda$ is a partition of $k$. Then we have

[TABLE]

where

[TABLE]

and $\ell(\lambda)$ denotes the number of parts of the partition $\lambda$. In particular, when $\theta=1$, we obtain $p_{\textrm{ord}}(k,1)=p(k)$, the ordinary partition function. Finally, we also have

[TABLE]

where $p_{\textrm{NR}}(k,D)$ denotes the number of partitions of $k$ into parts which are all quadratic non-residues modulo $D$. Putting together (7.1)–(7.4), we can write

[TABLE]

Here we set $a_{0}=b_{0}=1$ and $\{a_{i}\}$ and $\{b_{j}\}$ denote the $i$-th quadratic residue and the $j$-th quadratic non-residue modulo $D$, respectively, in the interval $[1,D]$, where $1\leq i,j\leq\phi(D)/2$. The sum in (7.5) is taken subject to the following conditions:

[TABLE]

In particular, the relation (7.5) provides an exact expression for the Fourier coefficients $a_{D}(N)$ in terms of certain classical functions related to partitions. By Rademacher’s work [undeft, undefu], we have

[TABLE]

where $A_{m}(k)$ is a certain Kloosterman sum and $I_{\nu}$ is the modified Bessel function of first kind. There are many results in the literature regarding exact formulas for $p_{\textrm{NR}}(k,D)$. When $D=5$, Lehner [undefo] proved that

[TABLE]

where $\widetilde{A}_{m}(k)$ and $\widetilde{B}_{m}(k)$ are certain Dedekind sums and $I_{1}$ is the modified Bessel function of the first kind. It would be interesting to obtain similar exact formulas for $a_{D}(N)$ in terms of Kloosterman sums and Bessel functions.

8. Table of Fourier Coefficients

The following table contains the precise values of $a_{D}(N)$ for $D=5,13$ and $17$.

9. Acknowledgements

We thank Alex Kontorovich, Don Zagier and Wadim Zudilin for valuable discussions and helpful comments on the subject of the paper.