Dedekind sums arising from newform Eisenstein series

TL;DR

This paper explores new Dedekind sums derived from Eisenstein series associated with primitive Dirichlet characters, providing explicit constructions and a proof of their reciprocity formula.

Contribution

It introduces a finite-term expression for Dedekind sums from Eisenstein series and constructs elements of cohomology groups explicitly.

Findings

Explicit finite-term formulas for Dedekind sums

Construction of cohomology elements from Eisenstein series

Proof of reciprocity formula for these Dedekind sums

Abstract

For primitive non-trivial Dirichlet characters $\chi_1$ and $\chi_2$, we study the weight zero newform Eisenstein series $E_{\chi_1,\chi_2}(z,s)$ at $s=1$. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of $H^1(\Gamma_0(N), \mathbb{C})$. We also give a short proof of the reciprocity formula for this Dedekind sum.

Equations

$$E_{{\chi }_1,{\chi }_2}(z,s)=\frac{1}{2}\sum _{(c,d)=1}\frac{(q_{2}y{)}^s{\chi }_1(c){\chi }_2(d)}{|c{q}_{2}z+d{|}^{2s}},\text{Re}(s)>1.$$

$$E_{{\chi }_1,{\chi }_2}(\gamma z,s)=\psi (\gamma )E_{{\chi }_1,{\chi }_2}(z,s),$$

$$\eta (z)=e^{\pi iz/12}\prod _{n=1}^{\infty }(1-e^{2\pi inz}).$$

$$\log{\eta(\gamma z)}=\log{\eta(z)}+\pi i\Big{(}\frac{a+d}{12c}+s(-d,c)\Big{)}+\tfrac{1}{2}\log(-i(cz+d)),$$

$$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\Big{(}\frac{hr}{k}-\Bigl{\lfloor}\frac{hr}{k}\Bigr{\rfloor}-\frac{1}{2}\Big{)}.$$

$$E_{{\chi }_1,{\chi }_2}^{\ast }(z,s)=\frac{(q_2/\pi {)}^s}{\tau ({\chi }_2)}\Gamma (s)L(2s,{\chi }_1{\chi }_2)E_{{\chi }_1,{\chi }_2}(z,s).$$

$$E_{{\chi }_1,{\chi }_2}^{\ast }(z,s)=2\sqrt{y}\sum _{n\ne 0}{\lambda }_{{\chi }_1,{\chi }_2}(n,s)e(nx)K_{s-\frac{1}{2}}(2\pi |n|y),$$

$${\lambda }_{{\chi }_1,{\chi }_2}(n,s)={\chi }_2(\text{sgn}(n))\sum _{ab=|n|}{\chi }_1(a)\overline{{\chi }_2}(b){\left(\frac{b}{a}\right)}^{s-\frac{1}{2}}.$$

$$E_{{\chi }_1,{\chi }_2}^{\ast }(z,1)=f_{{\chi }_1,{\chi }_2}(z)+{\chi }_2(-1){\overline{f}}_{\overline{{\chi }_1},\overline{{\chi }_2}}(z),$$

$$f_{{\chi }_1,{\chi }_2}(z)=\sum _{n=1}^{\infty }\frac{{\lambda }_{{\chi }_1,{\chi }_2}(n,1)}{\sqrt{n}}e(nz),$$

$${\varphi }_{{\chi }_1,{\chi }_2}(\gamma ,z)={\varphi }_{{\chi }_1,{\chi }_2}(\gamma )=f_{{\chi }_1,{\chi }_2}(\gamma z)-\psi (\gamma )f_{{\chi }_1,{\chi }_2}(z),$$

$$S_{{\chi }_1,{\chi }_2}(\gamma )=\frac{\tau (\overline{{\chi }_1})}{\pi i}{\varphi }_{{\chi }_1,{\chi }_2}(\gamma ).$$

$$B_1(x)=\{\begin{array}{ll}x-\lfloor x\rfloor -\frac{1}{2}& \text{if}x\in \mathbb{R}\backslash \mathbb{Z}\\ 0& \text{if}x\in \mathbb{Z}.\end{array}$$

$$S_{\chi_{1},\chi_{2}}(\gamma)=\sum_{j\negthickspace\negthickspace\negthickspace\pmod{c}}\sum_{n\negthickspace\negthickspace\negthickspace\pmod{q_{1}}}\overline{\chi_{2}}(j)\overline{\chi_{1}}(n)B_{1}\Big{(}\frac{j}{c}\Big{)}B_{1}\left(\frac{n}{q_{1}}+\frac{aj}{c}\right).$$

$$S_{{\chi }_1,{\chi }_2}(\gamma )=S_{{\chi }_2,{\chi }_1}({\gamma }^{\prime }).$$

$$S_{{\chi }_1,{\chi }_2}(\gamma )=-S_{{\chi }_2,{\chi }_1}({\gamma }^{\prime })+(1-\psi (\gamma ))\frac{\tau (\overline{{\chi }_1})\tau (\overline{{\chi }_2})}{(\pi i{)}^2}L(1,{\chi }_1)L(1,{\chi }_2).$$

$${\varphi }_{{\chi }_1,{\chi }_2}(\gamma ,z)=-{\chi }_2(-1){\overline{\varphi }}_{\overline{{\chi }_1},\overline{{\chi }_2}}(\gamma ,z).$$

$${\varphi }_{{\chi }_1,{\chi }_2}(\gamma )=\frac{1}{2}({\varphi }_{{\chi }_1,{\chi }_2}(\gamma )-{\chi }_2(-1){\overline{\varphi }}_{\overline{{\chi }_1},\overline{{\chi }_2}}(\gamma )).$$

$$S_{{\chi }_1,{\chi }_2}({\gamma }_1{\gamma }_2)=S_{{\chi }_1,{\chi }_2}({\gamma }_1)+\psi ({\gamma }_1)S_{{\chi }_1,{\chi }_2}({\gamma }_2).$$

$${\varphi }_{{\chi }_1,{\chi }_2}({\gamma }_1{\gamma }_2)=\underset{{\varphi }_{{\chi }_1,{\chi }_2}({\gamma }_1)}{\underbrace{f_{{\chi }_1,{\chi }_2}({\gamma }_1{\gamma }_{2}z)-\psi ({\gamma }_1)f_{{\chi }_1,{\chi }_2}({\gamma }_{2}z)}}+\psi ({\gamma }_1)\underset{{\varphi }_{{\chi }_1,{\chi }_2}({\gamma }_2)}{\underbrace{(f_{{\chi }_1,{\chi }_2}({\gamma }_{2}z)-\psi ({\gamma }_2)f_{{\chi }_1,{\chi }_2}(z))}}.\text{qed}$$

$$(T_{n}f)(z)=\frac{1}{\sqrt{n}}\sum_{ad=n}\chi(a)\sum_{b\negthickspace\negthickspace\negthickspace\pmod{d}}f\Big{(}\frac{az+b}{d}\Big{)}.$$

$$T_n{E}_{{\chi }_1,{\chi }_2}^{\ast }(z,s)={\lambda }_{{\chi }_1,{\chi }_2}(n,s)E_{{\chi }_1,{\chi }_2}^{\ast }(z,s),$$

$$T_n{f}_{{\chi }_1,{\chi }_2}={\lambda }_{{\chi }_1,{\chi }_2}(n,1)f_{{\chi }_1,{\chi }_2},$$

$$\phi_{\chi_{1},\chi_{2}}(\gamma)=\lim_{u\rightarrow 0^{+}}\big{[}f_{\chi_{1},\chi_{2}}(\tfrac{a}{c}+iu)-\psi(\gamma)f_{\chi_{1},\chi_{2}}(\tfrac{-d}{c}+\tfrac{i}{c^{2}u})\big{]}.$$

$${\varphi }_{{\chi }_1,{\chi }_2}(\gamma )=\underset{u\to 0^{+}}{\lim }{f}_{{\chi }_1,{\chi }_2}(\frac{a}{c}+iu).$$

$$f_{{\chi }_1,{\chi }_2}(z)=\sum _{k=1}^{\infty }\sum _{l=1}^{\infty }\frac{{\chi }_1(l)\overline{{\chi }_2}(k)}{l}e(klz).$$

$$f_{{\chi }_1,{\chi }_2}(z)=\sum _{l=1}^{\infty }\frac{{\chi }_1(l)}{l}{\theta }_{{\chi }_2}(z,l),\text{where}{\theta }_{\chi }(z,l):=\sum _{k=1}^{\infty }\overline{\chi }(k)e(klz).$$

$$\sum _{j(\mathrm{mod}c)}\overline{\chi }(j)e_c(alj)=0.$$

$$\sum _{j(\mathrm{mod}c)}\overline{\chi }(j)e_c(alj)=\sum _{A(\mathrm{mod}q)}\overline{\chi }(A)e_c(alA)\sum _{B(\mathrm{mod}c/q)}{e}_{c/q}(alB).$$

$${\theta }_{\chi }\left(\frac{a}{c}+iu,l\right)=\sum _{j=1}^{c-1}\overline{\chi }(j)e_c(alj)\frac{x^j-1}{1-x^c},\text{where}x=e(iul).$$

Full text

This work was conducted in summer 2018 during an REU conducted at Texas A&M University. The authors thank the Department of Mathematics at Texas A&M and the NSF for supporting the REU. In addition, this material is based upon work supported by the National Science Foundation under agreement No. DMS-170222 (M.Y.). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

1. Introduction

1.1. Background and statement of result

Let $\chi_{1},\chi_{2}$ be primitive Dirichlet characters modulo $q_{1},q_{2}$, respectively, with $\chi_{1}\chi_{2}(-1)=1$. The weight zero newform Eisenstein series attached to $\chi_{1}$ and $\chi_{2}$ is defined (initially) as

[TABLE]

Here $E_{\chi_{1},\chi_{2}}$ is an automorphic form on the congruence subgroup $\Gamma_{0}(q_{1}q_{2})$ with central character $\psi=\chi_{1}\overline{\chi_{2}}$. Precisely, for all $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(q_{1}q_{2})$,

[TABLE]

where $\psi(\gamma)=\psi(d)$. Moreover, $E_{\chi_{1},\chi_{2}}$ is an eigenfunction of all the Hecke operators (see (2.4) below), which indicates why it is called a newform. We refer to [Y] for the properties of the newform Eisenstein series used in this paper.

The classical Kronecker limit formula relates the constant term in the Laurent expansion of $E_{1,1}(z,s)$ at $s=1$ to $\log{\eta}$, where $\eta$ is the Dedekind $\eta$-function given by

[TABLE]

For $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(1)$, with $c>0$, $\log\eta$ obeys the transformation formula

[TABLE]

where $s(h,k)$ is the classical Dedekind sum given by

[TABLE]

See [A] for more background on the $\eta$-function and the classical Dedekind sums.

Consider the “completed” Eisenstein series defined by

[TABLE]

Here $\tau$ denotes the Gauss sum given by $\tau(\chi)=\sum_{n\negthickspace\negmedspace\pmod{q}}\chi(n)e_{q}(n)$, where $e_{q}(n)=e(n/q)$, $e(x)=\exp(2\pi ix)$, and $\chi$ is a Dirichlet character modulo $q$. The Fourier expansion for $E^{*}_{\chi_{1},\chi_{2}}$ is conveniently stated in [Y] (see also [H]). When $q_{1},q_{2}\neq 1$, the Fourier expansion simplifies as

[TABLE]

where $K_{\nu}$ is the $K$-Bessel function and

[TABLE]

The Fourier expansion gives the analytic continuation of $E_{\chi_{1},\chi_{2}}^{*}(z,s)$ to $s\in\mathbb{C}$. In particular, there is no pole at $s=1$, and (1.3) specializes as

[TABLE]

where

[TABLE]

using $K_{1/2}(2\pi y)=2^{-1}y^{-1/2}\exp(-2\pi y)$. Because $E^{*}_{\chi_{1},\chi_{2}}(z,s)$ has no pole at $s=1$, (1.5) is the analogue of the Kronecker limit formula and the function $f_{\chi_{1},\chi_{2}}$ is the analogue of $\log\eta$.

Define

[TABLE]

for $\gamma\in\Gamma_{0}(q_{1}q_{2})$ and $z\in\mathbb{H}$; in Lemma 2.1 below, we show that $\phi_{\chi_{1},\chi_{2}}$ is independent of $z$.

Definition 1.1**.**

Let $\chi_{1},\chi_{2}$ be primitive Dirichlet characters of conductors $q_{1},q_{2}$, respectively, with $q_{1},q_{2}>1$, and $\chi_{1}\chi_{2}(-1)=1$. For $\gamma\in\Gamma_{0}(q_{1}q_{2})$, define the Dedekind sum $S_{\chi_{1},\chi_{2}}$ associated to the newform Eisenstein series $E_{\chi_{1},\chi_{2}}$ by

[TABLE]

Let $B_{1}$ denote the first Bernoulli function given by

[TABLE]

The first main result in this paper is an evaluation of $S_{\chi_{1},\chi_{2}}$ in finite terms:

Theorem 1.2**.**

Let $\chi_{1},\chi_{2}$ be primitive Dirichlet characters of conductors $q_{1},q_{2}$, respectively, with $q_{1},q_{2}>1$, and $\chi_{1}\chi_{2}(-1)=1$. Let $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(q_{1}q_{2})$. For $c\geq 1$, then

[TABLE]

Our second main result gives a simple proof of the following reciprocity formula:

Theorem 1.3**.**

For $\gamma=(\begin{smallmatrix}a&b\\ cq_{1}q_{2}&d\end{smallmatrix})\in\Gamma_{0}(q_{1}q_{2})$, let $\gamma^{\prime}=(\begin{smallmatrix}d&-c\\ -bq_{1}q_{2}&a\end{smallmatrix})\in\Gamma_{0}(q_{1}q_{2})$. If $\chi_{1}$ and $\chi_{2}$ are even, then

[TABLE]

If $\chi_{1}$ and $\chi_{2}$ are odd, then

[TABLE]

The main step in the proof of Theorem 1.3 is to study the action of the Fricke involution $\omega_{q_{1}q_{2}}=(\begin{smallmatrix}0&-1\\ q_{1}q_{1}&0\end{smallmatrix})$. Since $E_{\chi_{1},\chi_{2}}$ is a pseudo-eigenvector of all the Atkin-Lehner operators (see [Y, Section 9]), it seems plausible that an adaptation of the proof can give a family of reciprocity formulas, one for each Atkin-Lehner operator.

Many authors have investigated generalized Dedekind sums arising from various types of Eisenstein series. Goldstein [G] studies the Eisenstein series attached to cusps for the principal congruence subgroup $\Gamma(N)$. Nagasaka [N] and Goldstein and Razar [GR] investigate functions essentially equivalent, in our notation, to $f_{\chi,\chi}$; they derive the transformation properties of $f_{\chi,\chi}$ (including the reciprocity formula) by relation to the Mellin transform of the product of Dirichlet $L$-functions instead of via properties of Eisenstein series.

The generalized Dedekind sums attached to pairs of Dirichlet characters have appeared in the literature in connection with certain Eisenstein-type series. Berndt [B2, Section 6] defines generalized Dedekind sums which essentially correspond to the right hand side of (1.8) when $q_{1}=1$ or $q_{2}=1$. Berndt derives properties of his Dedekind sums using a different variant of Eisenstein series than what is used in this paper; Berndt’s Eisenstein-type series have more complicated transformation properties than $E_{\chi_{1},\chi_{2}}$ (compare [B1, Theorem 2] to (1.1)). Many authors have studied generalized Dedekind sums, such as [M] [S] [CCK] [DC], based ultimately on Berndt’s transformation formulas.

Reciprocity formulas for variants of $S_{\chi_{1},\chi_{2}}$, with general pairs of characters $\chi_{1},\chi_{2}$ have appeared in [DC]. However, it appears that Theorem 1.3 is new (e.g. [DC, Theorem 1] excludes the case $p=1$ which would correspond to Theorem 1.3).

In Section 5 we connect $S_{\chi_{1},\chi_{2}}$ to the Eisenstein component of the Eichler-Shimura isomorphism in weight $2$.

1.2. Acknowledgements

The third author thanks Riad Masri and Ian Petrow for thoughtful comments.

2. Basic properties of $S_{\chi_{1},\chi_{2}}$

Lemma 2.1**.**

The function $\phi_{\chi_{1},\chi_{2}}$ is independent of $z$.

Proof.

Since $E_{\chi_{1},\chi_{2}}^{*}(\gamma z,1)=\psi(\gamma)E_{\chi_{1},\chi_{2}}^{*}(z,1)$ and $E^{*}_{\chi_{1},\chi_{2}}(z,1)=f_{\chi_{1},\chi_{2}}(z)+\chi_{2}(-1)\overline{f}_{\overline{\chi_{1}},\overline{\chi_{2}}}(z)$, it immediately follows that

[TABLE]

Since $\phi_{\chi_{1},\chi_{2}}$ is holomorphic and $\overline{\phi}_{\overline{\chi_{1}},\overline{\chi_{2}}}$ is antiholomorphic, $\phi_{\chi_{1},\chi_{2}}$ must be constant in $z$. ∎

For later reference, we point out a symmetrized form for $\phi_{\chi_{1},\chi_{2}}$ following from (2.1):

[TABLE]

Lemma 2.2**.**

Let $\gamma_{1},\gamma_{2}\in\Gamma_{0}(q_{1}q_{2})$. Then

[TABLE]

Remarks. It is obvious from the definition that $S_{\chi_{1},\chi_{2}}(\gamma)=0$ if $\gamma=(\begin{smallmatrix}1&n\\ 0&1\end{smallmatrix})$, for $n\in\mathbb{Z}$, and consequently $S_{\chi_{1},\chi_{2}}(\gamma)$ only depends on the lower row of $\gamma$ (or, alternatively, the first column of $\gamma$).

Let $G=\Gamma_{0}(q_{1}q_{2})$ and $M=\mathbb{C}$, and consider the action $\gamma.z$ of $G$ on $M$ given by $\gamma.z=\psi(\gamma)z$. Note $G$ acts via automorphisms on $M$ (as a module). With this notation, Lemma 2.2 shows that $S_{\chi_{1},\chi_{2}}$ is a $1$-cocycle (or a crossed homomorphism) for this group action of $G$ on $M$. Hence, $S_{\chi_{1},\chi_{2}}$ gives rise to an element of $H^{1}(G,M)$. In particular, if $\psi$ is trivial then $H^{1}(G,M)=\operatorname{Hom}(\Gamma_{0}(q_{1}q_{2}),\mathbb{C})$ (i.e., $S_{\chi_{1},\chi_{2}}$ is a group homomorphism). Note also that $\psi$ is trivial on $\Gamma_{1}(q_{1}q_{2})$ so $S_{\chi_{1},\chi_{2}}$ may always be viewed as an element of $\operatorname{Hom}(\Gamma_{1}(q_{1}q_{2}),\mathbb{C})$.

Proof.

Since $\psi$ is multiplicative, and by Lemma 2.1, we have

[TABLE]

Let $T_{n}$ be the Hecke operator acting on weight [math] periodic functions, with character $\chi$ (cf. [I, (6.13)]), defined by

[TABLE]

It is easy to check that

[TABLE]

for any $n\geq 1$. We remark in passing that

[TABLE]

which follows immediately from (1.5) and the fact that the Hecke operators preserve holomorphicity (and anti-holomorphicity).

3. Proof of Theorem 1.2

Our goal for the proof of Theorem 1.2 is to use properties of $f_{\chi_{1},\chi_{2}}$ in order to simplify $\phi_{\chi_{1},\chi_{2}}$ and write it in finite terms. Our process loosely follows the methodology of Goldstein [G]. Let $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(q_{1}q_{2})$, with $c\geq 1$, and let $z=\frac{-d}{c}+\frac{i}{c^{2}u}$ for some $u>0$. Then $\gamma z=\frac{a}{c}+iu$, and

[TABLE]

From the Fourier expansion of $E_{\chi_{1},\chi_{2}}^{*}$, it is clear that $\lim_{u\rightarrow 0^{+}}f_{\chi_{1},\chi_{2}}\left(\frac{-d}{c}+\frac{i}{c^{2}u}\right)=0$. Thus,

[TABLE]

This is the “constant term” in the Fourier expansion of $f_{\chi_{1},\chi_{2}}$ around the cusp $a/c$.

To evaluate this limit, we begin by writing $f_{\chi_{1},\chi_{2}}$ as

[TABLE]

Then

[TABLE]

The following lemma will be used in several of the proofs below.

Lemma 3.1**.**

Let $\chi$ be a character of conductor $q$. Let $a,c,l\in\mathbb{Z}$ with $c\geq 1$, $c\equiv 0\pmod{q}$, $(a,c)=1$, and $l\not\equiv 0\pmod{\frac{c}{q}}$. Then

[TABLE]

Proof.

Let $j=A+qB$ where $A$ runs modulo $q$ and $B$ runs modulo $c/q$. Then

[TABLE]

Since $\frac{c}{q}\nmid al$, the sum over $B$ vanishes. ∎

Lemma 3.2**.**

Let $\chi$ be a character of conductor $q$. Let $a,c,l\in\mathbb{Z}$ with $c\geq 1$, $c\equiv 0\pmod{q}$, $(a,c)=1$, and $l\not\equiv 0\pmod{\frac{c}{q}}$. Then

[TABLE]

Proof.

We have

[TABLE]

Now let $k=j+mc$ where $0\leq j<c$ and $m$ runs over non-negative integers. Then

[TABLE]

Using Lemma 3.1 and adding $0=\sum_{j=1}^{c-1}\overline{\chi}(j)e_{c}(ajl)\frac{-1}{1-x^{c}}$ to (3.3) completes the proof. ∎

Corollary 3.3**.**

Under the same assumptions as Lemma 3.2,

[TABLE]

Proof.

As $u$ approaches 0, $x=e(iul)$ approaches 1, and $\lim_{x\rightarrow 1}\frac{x^{j}-1}{1-x^{c}}=\frac{-j}{c}.$ Thus,

[TABLE]

Note $\overline{\chi}(j)\left(\frac{j}{c}-\left\lfloor\frac{j}{c}\right\rfloor-\frac{1}{2}\right)=\overline{\chi}(j)B_{1}\left(\frac{j}{c}\right)$, since $\overline{\chi}(j)=0$ when $\frac{j}{c}\in\mathbb{Z}$, so using Lemma 3.1 again finishes the proof. ∎

Remark. We need a definition of the generalized Bernoulli function for a (primitive) Dirichlet character $\chi$ modulo $q$, which is stated in [B3, Definition 1]. One may easily unify Berndt’s formulas as

[TABLE]

$\boldsymbol{\mathbf{Proof\ of\ Theorem\ \ref{theorem}}}$.

We apply (3.2) to (3.1). Provided that we can interchange the limits (see Lemma 3.4 below),

[TABLE]

Then by Corollary 3.3,

[TABLE]

Applying (2.2), we obtain

[TABLE]

Changing variables $l\rightarrow-l$ and using $\chi_{1}\chi_{2}(-1)=1$, this simplifies as

[TABLE]

Letting $c=c^{\prime}q_{1}$ and substituting (3.4), we obtain

[TABLE]

Next we use [B3, Theorem 3.1] which states

[TABLE]

Substituting (3.7) into (3.6) completes the proof. ∎

Lemma 3.4**.**

The interchange of limits in (3.5) is justified.

Proof.

Applying Lemma 3.2 to the left hand side of (3.5), we have

[TABLE]

Let $R(x)=R_{j,c}(x)=\frac{x^{j}-1}{1-x^{c}}.$ Note that $R$ is a rational function (in $x$) with no poles on $0\leq x\leq 1$, so it is smooth on this interval.

Let $a_{l}=\chi_{1}(l)e_{c}(alj)$, $b_{l}=\frac{1}{l}R(e^{-2\pi lu})$, and $S(N)=\sum_{l=1}^{N}a_{l}$. By Lemma 3.1, $\sum_{l\negthickspace\negmedspace\pmod{c}}a_{l}=0$ (since we may assume $(j,q_{2})=1$ whence $j\not\equiv 0\pmod{c/q_{1}}$), so $S(N)$ is bounded (independently of $u$, of course). Therefore, by partial summation, $\sum_{l=1}^{\infty}a_{l}b_{l}=\sum_{l=1}^{\infty}S(l)(b_{l}-b_{l+1})$. We claim $|b_{l}-b_{l+1}|=O(l^{-2})$, with an implied constant independent of $u$. Given this claim, the Weierstrass $M$-test shows the sum converges uniformly in $u$ which justifies the interchange of limits.

Now we show the claim. We have

[TABLE]

Here $\frac{|R(e^{-2\pi(l+1)u})|}{l+1}\leq\frac{C_{1}}{l}$ for some constant $C_{1}$ independent of $l$ and $u$. By the mean value theorem,

[TABLE]

for some $t\in[0,1]$. Since $R(t)$ is smooth on $[0,1]$, then $|R^{\prime}(t)|\leq C_{2}$ for some constant $C_{2}$ independent of $l$ and $u$. Additionally,

[TABLE]

for some constants $C_{3}$, $C_{4}$, since $xe^{-x}$ is bounded for $0\leq x<\infty$. Putting everything together proves the claim. ∎

4. Proof of Theorem 1.3

Let $\omega=\omega_{q_{1}q_{2}}=(\begin{smallmatrix}0&-1\\ q_{1}q_{1}&0\end{smallmatrix})$ be the Fricke involution. An easy calculation shows that if $\gamma=(\begin{smallmatrix}a&b\\ cq_{1}q_{2}&d\end{smallmatrix})\in\Gamma_{0}(q_{1}q_{2})$, then

[TABLE]

where $\gamma^{\prime}=(\begin{smallmatrix}d&-c\\ -bq_{1}q_{2}&a\end{smallmatrix})\in\Gamma_{0}(q_{1}q_{2})$. Note the map $\gamma\rightarrow\gamma^{\prime}$ is an involution. The newform Eisenstein series is a generalized eigenfunction of the Fricke involution, precisely it satisfies (see [Y, Section 9.2])

[TABLE]

For the completed Eisenstein series, using (1.2) we deduce

[TABLE]

Define $\phi_{\chi_{1},\chi_{2}}(\omega)=f_{\chi_{1},\chi_{2}}(wz)-\delta_{\chi_{1},\chi_{2}}f_{\chi_{2},\chi_{1}}(z)$, and similarly define

[TABLE]

An easy modification of the proof of Lemma 2.1 shows that $\phi_{\chi_{1},\chi_{2}}(\omega)$ is independent of $z$ (justifying the notation).

Lemma 4.1**.**

Let $\chi_{1},\chi_{2}$ be primitive Dirichlet characters of conductors $q_{1},q_{2}$, respectively, with $q_{1},q_{2}>1$, and $\chi_{1}\chi_{2}(-1)=1$. Then

[TABLE]

Proof.

The ideas are similar to the proof of Theorem 1.3, so we will be brief. We have $\phi_{\chi_{1},\chi_{2}}(\omega)=\lim_{u\rightarrow 0^{+}}f_{\chi_{1},\chi_{2}}(iu)$. Then following the idea of proof in Lemma 3.2, we have

[TABLE]

Letting $u\rightarrow 0^{+}$ (using a variant on Lemma 3.4 to change the limits) gives

[TABLE]

using (3.7). Finally, we use (3.4) to complete the proof. ∎

Now we calculate $f_{\chi_{1},\chi_{2}}(\omega\gamma z)-\delta_{\chi_{1},\chi_{2}}\overline{\psi}(\gamma)f_{\chi_{2},\chi_{1}}(z)$ in two ways. One expression is

[TABLE]

Alternatively, using (4.1), it equals

[TABLE]

where we have used $\psi(\gamma^{\prime})=\psi(a)=\overline{\psi}(d)=\overline{\psi}(\gamma)$. Equating the two expressions, we derive

[TABLE]

Converting the notation using (1.7), and using $\delta_{\chi_{1},\chi_{2}}\frac{\tau(\overline{\chi_{1}})}{\tau(\overline{\chi_{2}})}=\chi_{1}(-1)$, we derive

[TABLE]

Using Lemma 4.1 and switching the roles of $\gamma$ and $\gamma^{\prime}$ completes the proof of Theorem 1.3.

5. Remarks on the Eichler-Shimura isomorphism

Let $E_{2,\chi_{1},\chi_{2}}(z)$ be the holomorphic weight $2$ Eisenstein series attached to the primitive non-trivial characters $\chi_{1},\chi_{2}$, defined by (using the notation (1.4))

[TABLE]

See [DS, Section 4.6] for more details. The Eichler-Shimura map applied to $E_{2,\chi_{1},\chi_{2}}$ is defined by

[TABLE]

for $\gamma\in\Gamma_{0}(q_{1}q_{2})$. By direct calculation with (1.6), we have

[TABLE]

Therefore the Eichler-Shimura map applied to $E_{2,\chi_{1},\chi_{2}}$ is precisely $\tau(\overline{\chi_{1}})S_{\chi_{1},\chi_{2}}$.