# Extreme values of Hecke $L$-functions to angular characters

Daniel White  Gettysburg College, Department of Computer Science, 300 North
Washington Street, Gettysburg, PA 17325<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be an imaginary quadratic number field and let $L(s,\xi_{\ell})$
denote the Hecke $L$-function to an angular character $\xi_{\ell}$ with
frequency $\ell$. We detect values of $\log|L(\tfrac{1}{2},\xi_{\ell})|$ with
size at least $(\sqrt{2}+o_{K}(1))(\log X/\log\log X)^{1/2}$ along each dyadic
range $X\leqslant\ell\leqslant 2X$. This result relies on the resonance
method, which is applied for the first time to this family of $L$-functions,
where the classification and extraction of diagonal terms depends on the
geometry of the complex embedding of $K$.

###### Key words and phrases:

$L$-functions, extreme values, angular characters, Hecke characters, imaginary
quadratic number fields

###### 2020 Mathematics Subject Classification:

Primary 11R42; Secondary 11R04, 11R11

##  1\. Introduction

The resonance method developed by Soundararajan [11] is a flexible tool for
detecting both unusually large and unusually small values in families of
$L$-functions. In its original application to the zeta function, this method
exploits the oscillatory behavior of $\zeta(\tfrac{1}{2}+it)$ by generating a
Dirichlet polynomial

$R(t)=\sum_{n\leqslant N}r(n)n^{-it}$

that “resonates” with the “louder frequencies” of $\zeta(\tfrac{1}{2}+it)$ on
$T\leqslant t\leqslant 2T$ for some later optimized choice of real-valued
resonator coefficients $r(n)$. The argument proceeds by unpacking,
reassembling, and analyzing what is essentially the first moment of

$\zeta(\tfrac{1}{2}+it)|R(t)|^{2}\approx\sum_{k\leqslant
T}k^{-1/2}\sum_{n,m\leqslant N}r(n)r(m)\left(\frac{n}{mk}\right)^{it}$ (1)

on this interval, where one must balance the length $N$ of the resonator — a
parameter commensurate with the strength of the resulting bound (2) — against
the contribution of off-diagonal terms ($n\neq mk$) which are difficult to
manage. Comparing the mean value of (1) to the mean of $|R(t)|^{2}$ and
optimizing the resonator coefficients yields

$\max_{T\leqslant t\leqslant
2T}\log\left|\zeta(\tfrac{1}{2}+it)\right|\geqslant(1+o(1))\sqrt{\frac{\log
T}{\log\log T}}.$ (2)

The bound (2) substantially improved upon the previously best known result [1,
2] and is much larger than the bulk of the normal distribution with mean value
$0$ and variance $(\log\log T)/2$; see [13, §11.13]. This result is fairly
close to predictions from random matrix theory [7] that suggest the right hand
side of (2) may be within a factor of $(\log\log T)/\sqrt{2}$ to the true
maximum.

Bondarenko and Seip [4, 5] have since managed to substantially extend the
length of the resonator for the zeta function, thus improving (2), but at the
cost of localization; the full strength of their result requires the entire
segment $0\leqslant t\leqslant T$ and can not recover information along dyadic
ranges.

For a number field $K$, the pure adelic analogue for the set of values of
$\zeta(\tfrac{1}{2}+it)$ is the set of central values of Hecke $L$-functions
to Hecke characters that ramify only at a single place; see the upcoming
Section 2.1 for a brief recollection of Hecke characters. Over $\mathbb{Q}$
this presents the additional problem of detecting large values of
$L(\tfrac{1}{2},\chi)$ for Dirichlet characters $\chi$ over prime power
moduli. In a related but not identical direction, Hough [8] used the resonance
method to establish a result of quality similar to $\eqref{sound_zeta}$ within
prescribed angular sectors for Dirichlet characters varying on prime moduli,
while de la Bretèche and Tenenbaum [6] have since improved the quality of this
bound for such even characters (without the angular sector restriction) to the
level of Bondarenko and Seip’s result on the zeta function.

The complexity of Hecke characters expands as one moves to proper extensions
of $\mathbb{Q}$. For a fundamental example beyond both the archimedean and
Dirichlet character types, when $K$ is an imaginary quadratic number field,
_angular characters_ emerge from sole ramification at the unique complex
place. On principal ideals of the ring of integers $\mathcal{O}_{K}$, these
characters behave precisely as $\xi_{\ell}((\beta))=e(\ell\arg\beta/2\pi)$ for
integers $\ell$ divisible by $|\mathcal{O}_{K}^{\times}|$, where $\beta$ is an
element of a fixed complex embedding of $\mathcal{O}_{K}$. The integer $\ell$
is said to be the _frequency_ of the character and there are multiple angular
characters of any given permissible frequency when $\mathcal{O}_{K}$ is not a
PID; this multiplicity is exactly equal to the size of the class group. The
purpose of this article is to apply for the first time the resonance method to
Hecke $L$-functions to angular characters to address for the first time the
problem of detecting their large values, complementing the progress made on
the archimedean and Dirichlet character types over $\mathbb{Q}$. We prove the
following.

######  Theorem 1.

Let $K$ be an imaginary quadratic number field and $\mathfrak{F}_{\ell}$ be
its associated set of angular characters with frequency $\ell$. Then

$\max_{\begin{subarray}{c}X\leqslant\ell\leqslant 2X\\\
\xi_{\ell}\in\mathfrak{F}_{\ell}\end{subarray}}\log\left|L(\tfrac{1}{2},\xi_{\ell})\right|\geqslant\left(\sqrt{2}+o_{K}(1)\right)\sqrt{\frac{\log
X}{\log\log X}}.$

The most substantial contextual difference in our application of the resonance
method arises in Sections 4.1 and 4.2 when one seeks to isolate and then
analyze the diagonal terms (the analogues of $n=mk$ in (1)) of the analogous
quantity which appears, very roughly, in this article as

$\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{N}\mathfrak{k}\ll X\\\
\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\text{
principal}\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}\mathfrak{k}}}\sum_{k\asymp
X}e(k\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}/2\pi),$

where $\gamma_{\mathfrak{n}}$ is the generator of the principal ideal
$\mathfrak{n}$ nearest to the positive real line with respect to rotation
about the origin. A similar, but more relaxed, analysis precedes this in
Section 4.1 when examining the mean of our resonator polynomial. At this
point, one must consider the geometry of the embedding of $K$ into
$\mathbb{C}$. Indeed, the diagonal condition becomes characterized by whether
the product $\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}$ is principally
generated by a rational element of $\mathcal{O}_{K}$. This is the only way the
innermost sum above can avoid meaningful cancellation when $N$ is bounded
appropriately in terms of $X$.

We therefore bound length of the resonator in terms of $X$ (as is typical) so
that our analysis of the quantity illustrated above depends on an area
$\mathbb{C}$ where the embedding of $\mathcal{O}_{K}$ is “sufficiently
discretized” such that all $\gamma_{\mathfrak{n}}$ _near the real line_ are
_exactly rational_. As a consequence, the off-diagonal terms where the
$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}$ are not rationally generated
will not meaningfully contribute.

###  1.1. Notation

In this article, unless stated otherwise, we let $\varepsilon$ represent a
small, fixed positive number that may differ from line to line and let $K$ be
an imaginary quadratic number field. As usual, $f\ll g$ and $f=O(g)$ mean that
$|f|\leqslant Cg$ for some constant $C>0$ which may differ from line to line
and depends on no parameters except for possibly $\varepsilon$, the number
field $K$, or any other parameters explicitly stated. Additionally, $f=o(g)$
indicates that $f/g\to 0$ in a limit that will be clear from context. As is
customary in analytic number theory, we write $e(z)$ to mean $\exp(2\pi iz)$.

###  1.2. Acknowledgments

The author would like to thank Djordje Milićević for his thoughtful support
and our many helpful discussions during the writing of this article. The
author would like to further recognize and thank Bryn Mawr College for its
support during the writing of this article, which was completed in partial
fulfillment of the requirements for the degree of Doctor of Philosophy in the
Department of Mathematics.

##  2\. Preliminaries

###  2.1. Hecke characters

We now recall the basic properties of Hecke characters; the reader may
reference [10, Section 7.6] for additional details. Let $K$ be a general
number field and $\mathcal{O}_{K}$ its ring of integers. We will denote the
group of non-zero fractional ideals as $\mathfrak{I}$; it contains the
subgroup $\mathfrak{P}$ of non-zero fractional principal ideals. The quotient
group $\mathfrak{I}/\mathfrak{P}$ is the class group, which has finite size
$h_{K}$. For a non-zero integral ideal $\mathfrak{m}\subseteq\mathcal{O}_{K}$,
we further denote $\mathfrak{I}_{\mathfrak{m}}$ as the set of all fractional
ideals coprime to $\mathfrak{m}$.

Suppose $K$ has $r_{1}$ real embeddings and $r_{2}$ complex embeddings up to
conjugation. A Hecke character modulo $\mathfrak{m}$ is a continuous
homomorphism $\xi:\mathfrak{I}_{\mathfrak{m}}\to S^{1}$ (when interpreted as a
morphism on the idele class group of $K$) for which there are two characters

$\chi_{f}:(\mathcal{O}_{K}/\mathfrak{m})^{\times}\to
S^{1}\quad\text{and}\quad\chi_{\infty}:(\mathbb{R}^{\times})^{r_{1}}\times(\mathbb{C}^{\times})^{r_{2}}\to
S^{1}$

such that $\xi((\beta))=\chi_{f}(\beta)\chi_{\infty}(\beta)$ for every
$\beta\in\mathcal{O}_{K}$ where $((\beta),\mathfrak{m})=1$. A pair of
characters $(\chi_{f},\chi_{\infty})$ is associated to a Hecke character if
and only if $\chi_{f}(\epsilon)\chi_{\infty}(\epsilon)=1$ for every
$\epsilon\in\mathcal{O}_{K}^{\times}$. The Hecke characters associated to such
a pair differ by multiplication of a character of the class group. Let
$\mathcal{H}$ denote the set of all Hecke characters of $K$.

We now focus our attention to when $K$ is an imaginary quadratic number field
and fix an embedding into $\mathbb{C}$ so that no ambiguity may occur from
conjugation. Hence we may use $\beta\in\mathcal{O}_{K}$ to mean both the ring
element and its image under the embedding. Let
$\omega_{K}=|\mathcal{O}_{K}^{\times}|$ and for each integer $\ell$ divisible
by $\omega_{K}$ define the set

$\mathfrak{F}_{\ell}:=\left\\{\xi_{\ell}\in\mathcal{H}:\xi_{\ell}\big{(}(\beta)\big{)}=e\left(\frac{\ell\arg\beta}{2\pi}\right)\text{
for all non-zero }\beta\in\mathcal{O}_{K}\right\\}$ (3)

of angular characters with frequency $\ell$ as in Theorem 1, where
$|\mathfrak{F}_{\ell}|=h_{K}$. In the event that $\omega_{K}\nmid\ell$, we
simply take $\mathfrak{F}_{\ell}$ to be empty. It will be useful to have a
standard choice of generator for principal ideals that we encounter. For a
principal ideal $\mathfrak{n}$, let $\gamma_{\mathfrak{n}}$ denote the
generator of $\mathfrak{n}$ whose principal argument resides in
$[-\pi\omega_{K}^{-1},\pi\omega_{K}^{-1})$.

###  2.2. Functional equation

For any number field $K$, we may associate to any Hecke character its Hecke
$L$-function

$L(s,\xi)=\sum_{\mathfrak{k}\neq
0}\xi(\mathfrak{k})\mathfrak{N}\mathfrak{k}^{-s}$

which converges absolutely for $\mathrm{Re}\,s>1$ and has meromorphic
continuation to $\mathbb{C}$. Each such $L$-function possesses a functional
equation. In particular, for the $L$-functions of interest in this article,
[9, Theorem 3.8] boils down to the following.

######  Theorem 2.

Let $D$ denote the discriminant of the imaginary quadratic number field $K$
and $\xi_{\ell}$ be an angular character of frequency $\ell$. The function

$\Lambda(s,\xi_{\ell})=|D|^{s/2}(2\pi)^{-s}\Gamma(s+|\ell|/2)L(s,\xi_{\ell})$
(4)

is entire for $\xi_{\ell}$ non-trivial and satisfies the functional equation

$\Lambda(s,\xi_{\ell})=\Lambda(1-s,\overline{\xi_{\ell}}).$ (5)

When examining $L$-functions along the critical line, it is often beneficial
to express the $L$-function as a smoothly weighted Dirichlet series in what is
known as the approximate functional equation. Roughly speaking, and in
general, this weight has approximate value one until its transition point,
where it then drops rapidly to zero. One may think of the approximate
functional equation as essentially a finite sum. For Hecke $L$-functions, the
location of this point and the decay rate are both subject to the complexity
of the associated character.

######  Theorem 3.

For $K$ and $\xi_{\ell}$ as in Theorem 2,

$L\left(\tfrac{1}{2},\xi_{\ell}\right)=\sum_{\mathfrak{k}\neq(0)}\frac{\xi_{\ell}(\mathfrak{k})+\overline{\xi_{\ell}}(\mathfrak{k})}{\sqrt{\mathfrak{N}\mathfrak{k}}}W_{K}\left(\mathfrak{N}\mathfrak{k},|\ell|\right)$

where

$W_{K}\bigg{(}\frac{|D|^{1/2}y}{2\pi},2x\bigg{)}=V(y,x)=\frac{1}{2\pi
i}\int_{(1)}\rho_{s}(x)y^{-s}\frac{\mathrm{d}s}{s}$ (6)

and

$\rho_{s}(x)=\frac{\Gamma(x+s+\tfrac{1}{2})}{\Gamma(x+\tfrac{1}{2})}.$ (7)

###### Proof.

Let $\Lambda(s,\xi_{\ell})$ be as in (4) in Theorem 2. By a contour shift
permitted by the exponential decay of the gamma function along vertical
strips, we find

$\frac{1}{2\pi
i}\int_{(1)}\Lambda(s+\tfrac{1}{2},\xi_{\ell})\frac{\mathrm{d}s}{s}=\Lambda(\tfrac{1}{2},\xi_{\ell})+\frac{1}{2\pi
i}\int_{(-1)}\Lambda(s+\tfrac{1}{2},\xi_{\ell})\frac{\mathrm{d}s}{s}.$

On the right side above, applying the substitution $s\mapsto-s$ and the
functional equation (5) from Theorem 2 before substituting (4) and integrating
term-wise on each side, we obtain

$L(\tfrac{1}{2},\xi_{\ell})=\frac{1}{2\pi i}\sum_{\mathfrak{k}\neq
0}\frac{\xi_{\ell}(\mathfrak{k})+\overline{\xi_{\ell}}(\mathfrak{k})}{\sqrt{\mathfrak{N}\mathfrak{k}}}\int_{(1)}\rho_{s}(|\ell|/2)\left(2\pi|D|^{-1/2}\mathfrak{N}\mathfrak{k}\right)^{-s}\frac{\mathrm{d}s}{s},$

as desired. ∎

The following lemmata characterize the behavior of $\rho_{s}(x)$ and its
derivatives so that we may determine where $V(y,x)$, as defined in (6), is
“essentially supported” and how quickly it decays. The first of these
demonstrates that, in suitable bounded vertical strips, $\rho_{\sigma+it}(x)$
has an approximately linear phase in $t$.

######  Lemma 1.

Let $x\geqslant 1$ and $s=\sigma+it$ where $|\sigma|\leqslant x/2$. For
$\rho_{s}(x)$ as in (7), we have

$\displaystyle\rho_{s}(x)=(x/e)^{\sigma}\left(1+\sigma/x\right)^{x+\sigma}\exp\left(F_{x+\sigma}(t)+i\psi(t)\right)\left(1+O\left(x^{-1}\right)\right)$

with

$F_{\kappa}(t)=-\int_{0}^{t}\arctan\left(u/\kappa\right)\mathrm{d}u$

and real-valued phase

$\psi(t)=t\log\left(x+\sigma\right)+O\left(|t|^{3}x^{-2}\right).$

###### Proof.

We begin by recalling Stirling’s formula [12, Theorem 2.3], which states that

$\Gamma(s)=\exp(s\log
s-s)\left(2\pi/s\right)^{1/2}\left(1+O\left(|s|^{-1}\right)\right)$

as $|s|\to\infty$ within any sector of $\mathbb{C}$ which omits the negative
real axis. Applying Stirling’s formula to (7) and noting (9) shows

$\displaystyle\rho_{s}(x)$
$\displaystyle=\frac{(x+s+\tfrac{1}{2})^{x+s}}{(x+\tfrac{1}{2})^{x}}e^{-s}\left(1+O(x^{-1})\right)$
$\displaystyle=(x/e)^{s}\left(1+s/x\right)^{x+s}\left(1+O\left(x^{-1}\right)\right)$
(8)

where the expansion of $\log(1+z)$ revealed

$\displaystyle(x+s+\tfrac{1}{2})^{x+s}$
$\displaystyle=(x+s)^{x+s}\left(1+\tfrac{1}{2}(x+s)^{-1}\right)^{x+s}$
$\displaystyle=\sqrt{e}(x+s)^{x+s}\left(1+O\left(x^{-1}\right)\right),$ (9)

and one notes that the error term above is uniformly bounded for
$|\mathrm{Re}\,s|\leqslant x/2$. Toward explicating the magnitude and phase of
(8), we may factor $(1+s/x)^{x+s}$ as the product of

$\displaystyle(1+s/x)^{x+\sigma}$
$\displaystyle=(1+\sigma/x)^{x+\sigma}\left(1+\tfrac{it}{x+\sigma}\right)^{x+\sigma}$
$\displaystyle=(1+\sigma/x)^{x+\sigma}\exp\left(\tfrac{1}{2}(x+\sigma)\log\big{(}1+\left(\tfrac{t}{x+\sigma}\right)^{2}\big{)}+i\phi_{1}(t)\right),$
where $\displaystyle\phi_{1}(t)$
$\displaystyle=(x+\sigma)\arctan\left(\tfrac{t}{x+\sigma}\right)$
$\displaystyle=t+O\left(|t|^{3}x^{-2}\right),$ (10)

and

$\displaystyle(1+s/x)^{it}$
$\displaystyle=(1+\sigma/x)^{it}\left(1+\tfrac{it}{x+\sigma}\right)^{it}$
$\displaystyle=\exp\left(-t\arctan\left(\tfrac{t}{x+\sigma}\right)+i\phi_{2}(t)\right),$
where $\displaystyle\phi_{2}(t)$
$\displaystyle=t\log(1+\sigma/x)+\tfrac{t}{2}\log\big{(}1+\left(\tfrac{t}{x+\sigma}\right)^{2}\big{)}$
$\displaystyle=t\log(1+\sigma/x)+O(|t|^{3}x^{-2}),$ (11)

noting that both $\phi_{1}$ and $\phi_{2}$ are real-valued. Inserting each of
(10) and (11) back into (8) shows

$\displaystyle\rho_{s}(x)=$
$\displaystyle\left(1+O\left(x^{-1}\right)\right)(x/e)^{\sigma}\left(1+\sigma/x\right)^{x+\sigma}$
$\displaystyle\times\exp\left(\tfrac{1}{2}(x+\sigma)\log\big{(}1+\left(\tfrac{t}{x+\sigma}\right)^{2}\big{)}-t\arctan\left(\tfrac{t}{x+\sigma}\right)\right)$
$\displaystyle\times\exp\left(i\phi_{1}(t)+i\phi_{2}(t)+it(-1+\log x)\right)$

thus yielding

$\rho_{s}(x)=(x/e)^{\sigma}\left(1+\sigma/x\right)^{x+\sigma}\exp\left(\tilde{F}_{x+\sigma}(t)+i\psi(t)\right)\left(1+O\left(x^{-1}\right)\right)$

for $\psi(t)$ as defined in the statement of Lemma 1 and

$\tilde{F}_{\kappa}(t):=-t\arctan\left(\tfrac{t}{\kappa}\right)+\tfrac{1}{2}\kappa\log\left(1+t^{2}/\kappa^{2}\right).$
(12)

Note that $\tilde{F}_{\kappa}(0)=0$ and
$\tilde{F}_{\kappa}^{\prime}(t)=-\arctan(t/\kappa)$, so that an application of
the fundamental theorem of calculus shows
$F_{\kappa}(t)=\tilde{F}_{\kappa}(t)$, finishing the proof. ∎

######  Lemma 2.

For $n\geqslant 1$ and $\mathrm{Re}\,z\geqslant 1$, we have

$\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\log\Gamma(z)=\delta_{n}^{1}\log
z+O_{n}\left(\mathrm{Re}\,(z)^{-\delta_{n}^{1}-n+1}\right),$

where $\delta_{n}^{m}$ is the Kronecker delta function.

###### Proof.

We begin with the well-known identity [14, §12.31]

$\frac{\mathrm{d}}{\mathrm{d}z}\log\Gamma(z)=\log
z-\frac{1}{2z}-\int_{0}^{\infty}\left(\frac{1}{2}-\frac{1}{u}+\frac{1}{e^{u}-1}\right)e^{-uz}\,\mathrm{d}u.$

The case $n=1$ follows from the exponential decay of the integrand above,
which begins at $\mathrm{Re}\,(z)^{-1}$. For $n\geqslant 2$, see that

$\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\log\Gamma(z)=(-1)^{n}(n-1)!\left(\frac{1}{nz^{n-1}}+\frac{1}{2z^{n}}+O\left(\mathrm{Re}\,(z)^{-n}\right)\right),$

which finishes the proof. ∎

######  Lemma 3.

Let $x\geqslant 1$ and $s=\sigma+it$ with $|\sigma|\leqslant x/2$. Then, for
$n\geqslant 0$,

$\rho_{s}^{(n)}(x)=\rho_{s}(x)\log^{n}(1+s/x)\left(1+O_{n}(x^{-1})\right).$
(13)

###### Proof.

The case $n=0$ is clear and the case $n=1$ follows from the logarithmic
derivative

$\frac{\mathrm{d}}{\mathrm{d}x}\rho_{s}(x)=\rho_{s}(x)\frac{\mathrm{d}}{\mathrm{d}x}\log\rho_{s}(x)$

and Lemma 2. Proceeding inductively for $n\geqslant 2$, note that

$\displaystyle\rho_{s}^{(n)}(x)$
$\displaystyle=\frac{\mathrm{d}^{n-1}}{\mathrm{d}x^{n-1}}\left(\rho_{s}(x)\frac{\mathrm{d}}{\mathrm{d}x}\log\rho_{s}(x)\right)$
$\displaystyle=\sum_{m=0}^{n-1}\binom{n-1}{m}\rho_{s}^{(m)}(x)\frac{\mathrm{d}^{n-m}}{\mathrm{d}x^{n-m}}\log\rho_{s}(x).$

By our inductive hypothesis (13) for $m<n$ and Lemma 2, we note the main
contribution from $m=n-1$ and conclude the proof. ∎

Having established Lemmata 1 and 3, we are now equipped to understand the
general behavior of $V(y,x)$ and its derivatives in the approximate functional
equation of $L(\tfrac{1}{2},\xi_{\ell})$ as introduced in Theorem 3.

######  Lemma 4.

Let $y\geqslant 1$, $x\geqslant 3$, $n\geqslant 0$, and $V(y,x)$ be as in (6).
Then

$\frac{\partial^{n}}{\partial
x^{n}}V(y,x)=\delta_{n}^{0}+O_{n}\Big{(}x^{-n/2}(y/x)^{\sqrt{x}}\Big{)}\quad\text{and}\quad\frac{\partial^{n}}{\partial
x^{n}}V(y,x)\ll_{n}x^{-n/2}(x/y)^{\sqrt{x}},$ (14)

where $\delta_{n}^{m}$ is the Kronecker delta function.

###### Proof.

Let $|\sigma|=x^{1/2}$. Shifting the line of integration in (6) to $(\sigma)$
(perhaps picking up contribution from the pole at $s=0$), differentiating $n$
times with respect to $x$, and substituting $s=\sigma+it$ yields

$\frac{\partial^{n}}{\partial
x^{n}}V(y,x)=\delta_{n}^{0}\Delta_{\sigma}+\frac{y^{-\sigma}}{2\pi\sigma}\int_{-\infty}^{\infty}\rho^{(n)}_{\sigma+it}(x)y^{-it}G(\sigma,t)\,\mathrm{d}t,$
(15)

where $\Delta_{\sigma}\in\\{0,1\\}$ indicates whether $\sigma$ is negative and

$G(\sigma,t)=\frac{\sigma}{\sigma+it}.$

We will start by bounding the tails of the integral in (15), corresponding to
the values $|t|\geqslant x^{1/2+\varepsilon}$. Exercising Lemmata 1 and 3 with
$(1+\sigma/x)^{x+\sigma}=\exp(\sigma+O(1))$ from our choice $|\sigma|=x^{1/2}$
shows these tails are bounded by an absolute constant multiple of

$\displaystyle\int_{|t|\geqslant
x^{1/2+\varepsilon}}\left|\rho_{\sigma+it}^{(n)}(x)\right|\mathrm{d}t$
$\displaystyle\ll_{n}\int_{|t|\geqslant
x^{1/2+\varepsilon}}\left|\rho_{\sigma+it}(x)\log^{n}\left(1+\tfrac{\sigma+it}{x}\right)\right|\mathrm{d}t$
$\displaystyle\ll
x^{\sigma-n}\int_{x^{1/2+\varepsilon}}^{\infty}\exp\left(n\log
t-\int_{0}^{t}\arctan\left(\tfrac{u}{x+\sigma}\right)\
\mathrm{d}u\right)\mathrm{d}t.$

Noting that $\arctan(v)\geqslant v\pi/4$ for $0\leqslant v\leqslant 1$ and
$\arctan(v)\geqslant\pi/4$ for $v\geqslant 1$ allows us to bound the right
hand side above by

$\displaystyle\leqslant x^{\sigma-n}$
$\displaystyle\int_{x^{1/2+\varepsilon}}^{x+\sigma}\exp\left(\tfrac{-t^{2}\pi}{8x}(1+o_{n}(1))\right)\mathrm{d}t$
$\displaystyle+x^{\sigma-n}\exp\left(\tfrac{\pi}{8}(x+\sigma)\right)\int_{x+\sigma}^{\infty}\exp\left(-\tfrac{\pi}{4}t(1+o_{n}(1))\right)\mathrm{d}t$

where utilizing the bound $\int_{c}^{\infty}e^{-u^{2}}\mathrm{d}u\ll
e^{-c^{2}}/c$ then shows that the main contribution above comes from the left
summand whose integral is bounded by

$\ll_{n}\int_{x^{1/2+\varepsilon}}^{\infty}\exp\left(\frac{-t^{2}\pi}{16x}\right)\,\mathrm{d}t\ll
x^{1/2-\varepsilon}\exp(-x^{\varepsilon}\pi/16).$

We therefore find that the integral in (15) becomes

$O_{n}\left(x^{\sigma-n+1/2-\varepsilon}\exp(-x^{\varepsilon}\pi/16)\right)+\int_{-x^{1/2+\varepsilon}}^{x^{1/2+\varepsilon}}\rho^{(n)}_{\sigma+it}(x)y^{-it}G(\sigma,t)\mathrm{d}t.$
(16)

Another application of lemmata 1 and 3, keeping in mind
$(1+\sigma/x)^{x+\sigma}$ is equal to $\exp(\sigma+O(1))$ by our choice
$|\sigma|=x^{1/2}$, shows that the integral in (16) is

$\ll_{n}x^{\sigma}\int_{-x^{1/2+\varepsilon}}^{x^{1/2+\varepsilon}}\exp\left(F_{x+\sigma}(t)\right)\left|\log^{n}\left(1+\tfrac{\sigma+it}{x}\right)\right|\,\mathrm{d}t$
(17)

where, by (12), we may expand

$\displaystyle F_{x+\sigma}(t)$
$\displaystyle=-t\arctan\left(\tfrac{t}{x+\sigma}\right)+\tfrac{1}{2}(x+\sigma)\log\left(1+\tfrac{t^{2}}{(x+\sigma)^{2}}\right)$
$\displaystyle=-\frac{t^{2}}{2x}\left(1+O(x^{-1/2}+t^{2}x^{-2})\right).$ (18)

Inserting (18) into (17) yields a value

$\ll
x^{\sigma}\int_{-x^{1/2+\varepsilon}}^{x^{1/2+\varepsilon}}\exp(-\tfrac{1}{4}t^{2}/x)\left|\log^{n}\left(1+\tfrac{\sigma+it}{x}\right)\right|\,\mathrm{d}t\ll
x^{\sigma-n+1/2}\sigma^{n}$

which can be used as an upper bound for the integral in (16) resulting,
ultimately, in a bound for (15) of the form

$\displaystyle\frac{\partial^{n}}{\partial x^{n}}V(y,x)$
$\displaystyle=\delta_{n}^{0}\Delta_{\sigma}+O_{n}\left(\frac{y^{-\sigma}}{2\pi\sigma}\left(x^{\sigma-n+1/2}\sigma^{n}\right)\right)$
$\displaystyle=\delta_{n}^{0}\Delta_{\sigma}+O_{n}\left(x^{-n/2}(x/y)^{\sigma}\right)$

as desired. ∎

It will be convenient to produce lower bounds on sums of products involving
factors of $W_{K}$ by truncating the sums, where all other factors of the
summands are non-negative. To do so, we must prove that $V(y,x)$ is also non-
negative for the input $(y,x)$ under consideration. Note that the bounds in
(14) are insufficient here. We therefore prove the following lemma which
happens to also show that $V(y,x)$ can be expressed as the ratio of an
incomplete Gamma function to a complete Gamma function.

######  Lemma 5.

If $y>0$ and $x>0$, then $V(y,x)\in(0,1)$.

###### Proof.

We begin by shifting the contour in the definition of $V(y,x)$ to the piece-
wise contour
$\gamma_{\varepsilon}^{-}+\gamma_{\varepsilon}+\gamma_{\varepsilon}^{+}$,
where the $\gamma_{\varepsilon}^{\pm}$ sit along $\pm i[\varepsilon,\infty)$
and $\gamma_{\varepsilon}$ forms the right half of the circle with radius
$\varepsilon$ centered at the origin. Define

$I_{\varepsilon}^{\pm}=\frac{1}{2\pi
i}\int_{\gamma_{\varepsilon}^{\pm}}\rho_{s}(x)y^{-s}\frac{\mathrm{d}s}{s}$

and $I_{\varepsilon}$ similarly for $\gamma_{\varepsilon}$, where we denote
the limit as $\varepsilon\to 0^{+}$ of each $I_{\varepsilon}^{\pm}$ and
$I_{\varepsilon}$ as $I^{\pm}$ and $I$, respectively. Note that $I=1/2$ and

$\displaystyle I^{-}+I^{+}$
$\displaystyle=\frac{1}{\pi\Gamma(x+\tfrac{1}{2})}\int_{0}^{\infty}\mathrm{Im}\,\Gamma(x+\tfrac{1}{2}+it)y^{-it}\frac{\mathrm{d}t}{t}$
$\displaystyle=\frac{1}{2}\Gamma(x+\tfrac{1}{2})^{-1}\int_{0}^{\infty}u^{x-1/2}e^{-u}\,\mathrm{sgn}\log(u/y)\,\mathrm{d}u$

which implies that $I^{-}+I^{+}\in\mathbb{R}$ is strictly less than $1/2$ in
absolute value. The statement of the lemma follows. ∎

##  3\. The setup

Fix an imaginary quadratic number field $K$ and let $X$ be a large positive
value. We begin by introducing our notion of a resonator

$R(\xi_{\ell})=\sum_{\mathfrak{N}\mathfrak{a}\leqslant
N}\xi_{\ell}(\mathfrak{a})r(\mathfrak{a})$ (19)

where $\xi_{\ell}\in\mathfrak{F}_{\ell}$ is a Hecke character as defined in
(3) and the $r(\mathfrak{a})$ are non-negative resonator coefficients. The
length of our resonator will be set by a constraint involving the frequencies
of the $\xi_{\ell}$ under consideration, as is expected within the resonator
method, where the goal is to build $R(\xi_{\ell})$ to resonate with the
oscillatory function $L(\tfrac{1}{2},\xi_{\ell})$ for $\xi_{\ell}$ varying
among the $\mathfrak{F}_{\ell}$ with $X\leqslant\ell\leqslant 2X$. One then
expects large central values to be pronounced when examining
$L(\tfrac{1}{2},\xi_{\ell})|R(\xi_{\ell})|^{2}$. To make this idea precise,
let $0\leqslant\Phi(x)\leqslant 1$ be a smooth weight function supported in
$[1,2]$ with uniformly bounded derivatives and note that

$\max_{\begin{subarray}{c}X\leqslant\ell\leqslant 2X\\\
\xi_{\ell}\in\mathfrak{F}_{\ell}\end{subarray}}\left|L(\tfrac{1}{2},\xi_{\ell})\right|\geqslant\frac{\big{|}\sum_{\ell}\Phi(\ell/X)\sum_{\xi_{\ell}\in\mathfrak{F}_{\ell}}L(\tfrac{1}{2},\xi_{\ell})|R(\xi_{\ell})|^{2}\big{|}}{\sum_{\ell}\Phi(\ell/X)\sum_{\xi_{\ell}\in\mathfrak{F}_{\ell}}|R(\xi_{\ell})|^{2}}.$
(20)

We make provisions for the resonator coefficients before an explicit choice is
later made. Let $r\geqslant 0$ be multiplicative, $r(\mathfrak{p})=0$ on prime
ideals $\mathfrak{p}$ that are not generated by a factor of a split rational
prime, and for all prime ideals $\mathfrak{p}$,
$r(\mathfrak{p})=r(\overline{\mathfrak{p}})$ and $r(\mathfrak{p}^{m})=0$ for
$m\geqslant 2$. These choices do not affect the quality of our result and
serve to simplify many calculations that are encountered.

###  3.1. Bounding the denominator

To start, we examine the denominator of the ratio in (20), upon which we wish
to produce an upper bound. By orthogonality of the class group characters and
the definition of $\gamma_{\mathfrak{n}}$, both discussed in Section 2.1, we
have

$\displaystyle\sum_{\omega_{K}\mid\ell}\Phi(\ell/X)\sum_{\xi_{\ell}\in\mathfrak{F}_{\ell}}|R(\xi_{\ell})|^{2}$
$\displaystyle=\sum_{\omega_{K}\mid\ell}\Phi(\ell/X)\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{\xi_{\ell}\in\mathfrak{F}_{\ell}}\xi_{\ell}(\mathfrak{a}\overline{\mathfrak{b}})$
$\displaystyle=h_{K}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{a}\overline{\mathfrak{b}}\in
P\end{subarray}}r(\mathfrak{a})r(\mathfrak{b})\sum_{k}\Phi(\omega_{K}k/X)e(\omega_{K}k\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}/2\pi),$
(21)

where $P$ denotes the set of principal ideals of $\mathcal{O}_{K}$.
Application of Poisson summation with a change of variable shows that the
above equals

$\displaystyle
h_{K}\omega_{K}^{-1}X\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{a}\overline{\mathfrak{b}}\in
P\end{subarray}}r(\mathfrak{a})r(\mathfrak{b})\sum_{j}\hat{\Phi}\left(X(\omega_{K}^{-1}j-\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}/2\pi)\right).$
(22)

The rapid decay of $\hat{\Phi}$ allows us to profitably split this sum into
two cases according to whether
$\left|\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}\right|$ passes the
threshold $X^{-1+\varepsilon}$. Note the series expansion of
$\sin\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}$ in the case
$\left|\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}\right|<X^{-1+\varepsilon}$
shows that

$\displaystyle\left|\mathrm{Im}\,\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}\right|\leqslant
N\left(X^{-1+\varepsilon}+O\left(X^{-2}\right)\right).$ (23)

It is here that a constraint on the length of the resonator appears should we
wish to avoid off-diagonal contribution in (22); later, though, we further
restrict the length. Let $P_{0}$ denote the set of principal ideals generated
by rational numbers and $P^{\prime}$ be the set of ideals which contain no
$P_{0}$-factors. Enforcing $N\leqslant X^{1-\varepsilon}$ guarantees
$\left|\mathrm{Im}\,\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}\right|<1$ by
(23), so that $\mathfrak{a}\overline{\mathfrak{b}}\in P_{0}$ for $X$
sufficiently large relative to $\varepsilon$.

By considering the prime factorization of $\mathfrak{a}$ and $\mathfrak{b}$,
we may write

$\mathfrak{a}=\mathfrak{a}_{0}\mathfrak{a}^{\prime}\quad\text{and}\quad\mathfrak{b}=\mathfrak{b}_{0}\mathfrak{b}^{\prime}$
(24)

uniquely for $\mathfrak{a}_{0},\mathfrak{b}_{0}\in P_{0}$ and
$\mathfrak{a^{\prime}},\mathfrak{b}^{\prime}\in P^{\prime}$ to find that
$\mathfrak{a}\overline{\mathfrak{b}}\in P_{0}$ if and only if
$\mathfrak{a}^{\prime}=\mathfrak{b}^{\prime}$. By the bound
$\hat{\Phi}(y)\ll_{n}y^{-n}$ (derived from repeated integration by parts) and
the deduction above, the total contribution to the sum in (22) over all such
pairs $(\mathfrak{a},\mathfrak{b})$ is

$\displaystyle\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{a}\overline{\mathfrak{b}}\in
P_{0}\end{subarray}}r(\mathfrak{a})r(\mathfrak{b})\big{(}\hat{\Phi}(0)+O(X^{-1})\big{)}$
$\displaystyle\ll\hat{\Phi}(0)\sum_{\mathfrak{a}_{0},\mathfrak{b}_{0}\in
P_{0}}r(\mathfrak{a}_{0})r(\mathfrak{b}_{0})\sum_{\begin{subarray}{c}\mathfrak{a}^{\prime}\in
P^{\prime}\\\
(\mathfrak{a}^{\prime},\mathfrak{a}_{0}\mathfrak{b}_{0})=1\end{subarray}}r(\mathfrak{a}^{\prime})^{2}$
$\displaystyle=\hat{\Phi}(0)\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\left(1+4r(\mathfrak{p})^{2}+r(\mathfrak{p})^{4}\right),$
(25)

where the primed product above is indexed over arbitrary representatives of
conjugate pairs of factors of split rational primes. Again wielding the above
bound on $\hat{\Phi}(y)$ and exercising the Cauchy–Schwarz inequality, we find
that the off-diagonal contribution (_i.e._ when
$\left|\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}}\right|\geqslant
X^{-1+\varepsilon}$) to the sum in (22) is

$\ll_{m}X^{-\varepsilon m}N\sum_{\mathfrak{a}\neq
0}r(\mathfrak{a})^{2}\leqslant X^{1-\varepsilon
m}\prod_{\mathfrak{p}}(1+r(\mathfrak{p})^{2}).$ (26)

Choosing $m$ sufficiently large relative to $\varepsilon$ allows the quantity
in (25) to dominate that in (26). Combining this observation with (21) and
(22) yields

$\sum_{\ell}\Phi(\ell/X)\sum_{\xi_{\ell}\in\mathfrak{F}_{\ell}}|R(\xi_{\ell})|^{2}\ll
X\hat{\Phi}(0)\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\left(1+4r(\mathfrak{p})^{2}+r(\mathfrak{p})^{4}\right).$
(27)

##  4\. Resonator application

Application of Theorem 3 to the numerator in (20) produces

$\displaystyle\sum_{\ell}\Phi(\ell/X)\sum_{\xi_{\ell}\in\mathfrak{F}_{\ell}}|R(\xi_{\ell})|^{2}\sum_{\mathfrak{k}}\frac{\xi_{\ell}(\mathfrak{k})+\overline{\xi_{\ell}}(\mathfrak{k})}{\sqrt{\mathfrak{N}\mathfrak{k}}}W_{K}\big{(}\mathfrak{N}\mathfrak{k},|\ell|\big{)}$
(28)

upon which we wish to place a lower bound. Our analysis of the sum over
$\xi_{\ell}(\mathfrak{k})$ above will end up having built-in symmetry with the
dual sum over $\overline{\xi_{\ell}}(\mathfrak{k})$. Further, the main
contribution to the sum over $\xi_{\ell}(\mathfrak{k})$, hence also
$\overline{\xi_{\ell}}(\mathfrak{k})$, will be real-valued. We thus explicitly
develop our argument first (and only) for the sum in (28) over
$\xi_{\ell}(\mathfrak{k})$. With this in mind, we proceed as in (21) and (22)
– exchanging order of summation and using Poisson summation – to obtain

$\displaystyle
h_{K}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}\mathfrak{k}}}\sum_{k}\Phi(\omega_{K}k/X)W_{K}(\mathfrak{N}\mathfrak{k},\omega_{K}k)e\left(\omega_{K}k\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}\mathfrak{k}}/2\pi\right)$
$\displaystyle=h_{K}\omega_{K}^{-1}X\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}\mathfrak{k}}}\sum_{j}\mathcal{F}_{u}[\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k},uX)]\left(X\left(\omega_{K}^{-1}j-\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}\mathfrak{k}}/2\pi\right)\right)$
(29)

where $\mathcal{F}_{u}[f]$ denotes the Fourier transform of $f(u)$. Over the
course of this section, we will establish a lower bound on (28) by extracting
the contribution to (29) from the case

$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P_{0}\quad\text{and}\quad\mathfrak{N}\mathfrak{k}\ll X.$

For $\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus P_{0}$, the
rapid decay of the Fourier transform in (29) will be quite useful in our
management, provided that
$\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}$ is guaranteed
to not be too small. Once focused on
$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P_{0}$ (and its $j=0$ term
in (29)), we will be permitted to truncate at $\mathfrak{N}\mathfrak{k}\ll X$
by the positivity of $\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k},uX)$ which is
established in Lemma 5. Afterward, the resulting sum will undergo a somewhat
lengthy factorization process enabled by Rankin’s trick, as recalled in
Section 4.3.

We begin by forming a handle on the decay of the transform of the function

$\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k},uX)$

above. It will suffice to place bounds on its derivatives. Note that

$\displaystyle\frac{\mathrm{d}^{n}}{\mathrm{d}u^{n}}\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k},uX)=\sum_{0\leqslant
m\leqslant
n}\binom{n}{m}\Phi^{(n-m)}(u)\frac{\mathrm{d}^{m}}{\mathrm{d}u^{m}}W_{K}(\mathfrak{N}\mathfrak{k},uX)$

and recall $\Phi$ is supported in $[1,2]$ with uniformly bounded derivatives,
as stated in Section 3. Setting $c_{K}=4\pi/\sqrt{D}$ and using (14) from
Lemma 4, we find that the above is

$\ll_{n}\left\\{\begin{array}[]{ll}X^{n/2}\left(\frac{uX}{c_{K}\mathfrak{N}\mathfrak{k}}\right)^{\sqrt{uX/2}},&\text{if
}u\leqslant c_{K}\mathfrak{N}\mathfrak{k}/X;\\\
1+X^{n/2}\left(\frac{uX}{c_{K}\mathfrak{N}\mathfrak{k}}\right)^{-\sqrt{uX/2}},&\text{if
}u>c_{K}\mathfrak{N}\mathfrak{k}/X.\end{array}\right.$ (30)

Repeating integration by parts $n$ times on the Fourier transform of interest
and exercising the bound in (30) shows that, for $v\neq 0$,

$\displaystyle\mathcal{F}_{u}[\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k};uX)](v)\ll_{n}v^{-n}X^{n/2}S(\mathfrak{N}\mathfrak{k},X)+v^{-n}\max\left(0,2-\tfrac{c_{K}\mathfrak{N}\mathfrak{k}}{X}\right)$
(31)

where

$S(\mathfrak{N}\mathfrak{k},X):=\int_{\begin{subarray}{c}u\leqslant
c_{K}\mathfrak{N}\mathfrak{k}/X\\\
u\in(1,2)\end{subarray}}\left(\tfrac{uX}{c_{K}\mathfrak{N}\mathfrak{k}}\right)^{\sqrt{uX/2}}\mathrm{d}u+\int_{\begin{subarray}{c}u\geqslant
c_{K}\mathfrak{N}\mathfrak{k}/X\\\
u\in(1,2)\end{subarray}}\Big{(}\tfrac{c_{K}\mathfrak{N}\mathfrak{k}}{uX}\Big{)}^{\sqrt{uX/2}}\mathrm{d}u.$
(32)

Using the substitution $w=uX/(c_{K}\mathfrak{N}\mathfrak{k})$ in (32) and
bounding the exponents trivially based on the interval of integration reveals

$\displaystyle
S(\mathfrak{N}\mathfrak{k},X)\ll_{n}\begin{cases}X^{-1/2}\left(c_{K}\mathfrak{N}\mathfrak{k}/X\right)^{\sqrt{X/2}},&\mathfrak{N}\mathfrak{k}\leqslant
X/c_{K};\\\ X^{-1/2},&X/c_{K}<\mathfrak{N}\mathfrak{k}\leqslant 2X/c_{K};\\\
X^{-1/2}\left(\tfrac{1}{2}c_{K}\mathfrak{N}\mathfrak{k}/X\right)^{-\sqrt{X/2}},&2X/c_{K}<\mathfrak{N}\mathfrak{k},\end{cases}$
(33)

so that the insertion of (33) into (31), combined with a replacement of $n$
with any real-valued $\eta\geqslant 1$ via interpolation (a geometric mean
argument), yields

$\displaystyle\mathcal{F}_{u}[\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k};uX)](v)$
$\displaystyle\ll
v^{-\eta}\chi_{(0,2X/c_{K}]}(\mathfrak{N}\mathfrak{k})+\frac{X^{\eta/2-1/2}}{v^{\eta}}\min\left(1,\left(c_{K}\mathfrak{N}\mathfrak{k}/X\right)^{\sqrt{X/2}},\left(\tfrac{1}{2}c_{K}\mathfrak{N}\mathfrak{k}/X\right)^{-\sqrt{X/2}}\right).$
(34)

The next two subsections will break the outer sum of (29) into two cases: the
_diagonal case_ $\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P_{0}$ and
the _off-diagonal case_ $\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P\setminus P_{0}$.

###  4.1. Off-diagonal contribution

We first consider the _off-diagonal case_
$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus P_{0}$, and we
will do so with four subcases, the first of which is the _off-diagonal subcase
I_ : $\mathfrak{N}\mathfrak{k}\leqslant 2X/c_{K}$ and
$|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\geqslant
X^{-1/2}$. It will be beneficial to subdivide the sum over $\mathfrak{k}$ in
(29) according to the size of $\mathfrak{N}\mathfrak{k}$ along the dyadic
partitions

$D_{\nu}(X)=(2^{-\nu}X/c_{K},2^{-\nu+1}X/c_{K}]$

and $\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}$ according
to the intervals

$I_{m}(X)=[mX^{-1/2},(m+1)X^{-1/2}).$

By the bound in (34), the contribution to (29) for
$\mathfrak{N}\mathfrak{k}\leqslant 2X/c_{K}$ with $\eta\geqslant 1$ is

$\ll
X^{1/2-\eta/2}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{\begin{subarray}{c}\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P\\\ \mathfrak{N}\mathfrak{k}\leqslant
2X/c_{K}\end{subarray}}\mathfrak{N}\mathfrak{k}^{-1/2}\sum_{j}\left(\omega_{K}^{-1}j-\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}/2\pi\right)^{-\eta}$
(35)

where, by the definition of $\gamma_{\mathfrak{n}}$, the main contribution to
the sum over $j$ above comes from $j=0$. Precisely,

$\sum_{j}\left(\omega_{K}^{-1}j-\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}/2\pi\right)^{-\eta}\ll_{\eta}(\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}})^{-\eta}.$
(36)

By the restriction on the argument of
$\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}$ in this subcase, we
now bound the contribution to (35) as

$\displaystyle\ll_{\eta}X^{1/2-\eta/2}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})X^{\eta/2}\sum_{m\geqslant
1}m^{-\eta}\sum_{0\leqslant\nu\leqslant\log_{2}X}2^{\nu/2}X^{-1/2}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in
D_{\nu}(X)\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\in I_{m}(X)\\\
\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus
P_{0}\end{subarray}}1$
$\displaystyle\ll_{\eta}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{0\leqslant\nu\leqslant\log_{2}X}2^{\nu/2}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in
D_{\nu}(X)\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\in I_{m}(X)\\\
\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus
P_{0}\end{subarray}}1$

Bounding the inner sum above can take place by counting the number of
$\mathcal{O}_{K}$ lattice points associated to each
$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}$ (translating $\mathfrak{k}$
to a generator in $\mathbb{C}$ by one of a fixed _finite_ number of
representatives of the class group) within the annulus defined by $D_{\nu}(X)$
while restricted to the appropriate angular sector. Note that the area of this
annular sector is commensurate to $2^{-\nu}X^{1/2}$ and the perimeter to
$2^{-\nu/2}X^{1/2}$, allowing us to estimate the number of lattice points as
$\ll 2^{-\nu/2}X^{1/2}+1$.

Therefore the total contribution to (29) from _off-diagonal subcase I_ is

$\ll_{\eta}X^{1/2}\log
X\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b}).$ (37)

We next consider the _off-diagonal subcase II_ :
$\mathfrak{N}\mathfrak{k}\leqslant 2X/c_{K}$ and
$|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|<X^{-1/2}$. By
the series expansion of sine around zero, we have
$|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\gg
N^{-1}\mathfrak{N}\mathfrak{k}^{-1/2}$ for
$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus P_{0}$, and so
using (34) to again obtain (35) with its associated bound (36) on the sum over
$j$ shows that the contribution to (29) from the _off-diagonal subcase II_ is

$\displaystyle\ll_{\eta}X^{1/2-\eta/2}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{a}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{0\leqslant\nu\leqslant\log_{2}X}(X2^{-\nu})^{-1/2}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in
D_{\nu}(X)\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus
P_{0}\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|<X^{-1/2}\end{subarray}}N^{\eta}(X2^{-\nu})^{\eta/2}$
$\displaystyle\ll
N^{\eta}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{a}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{0\leqslant\nu\leqslant\log_{2}X}2^{\nu/2-\nu\eta/2}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in
D_{\nu}(X)\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus
P_{0}\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|<X^{-1/2}\end{subarray}}1.$

Bounding the inner sum above can take place by counting lattice points within
the annulus defined by $D_{\nu}(X)$ while restricted to the appropriate
angular sector, as done in the previous subcase. This again allows us to
estimate the number of lattice points as $\ll 2^{-\nu/2}X^{1/2}+1$.
Substitution to achieve an upper bound then shows the contribution to (29)
from _off-diagonal subcase II_ is

$\ll_{\eta}X^{1/2}N^{\eta}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b}).$ (38)

In the remaining two cases, we will split $\mathfrak{N}\mathfrak{k}$ according
to the intervals

$D^{\prime}_{\nu}(X)=(\nu 2X/c_{K},(\nu+1)2X/c_{K}].$

Now consider the _off-diagonal subcase III_ :
$\mathfrak{N}\mathfrak{k}>2X/c_{K}$ and
$|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\geqslant
X^{-1/2}$. With respect to (34), notice that we have entered the range where

$\mathcal{F}_{u}[\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k};uX)](v)$

decays rapidly. Again exercising (34), one finds that the contribution to (29)
with the constraint $\mathfrak{N}\mathfrak{k}>2X/c_{K}$ and observation (36)
is

$\ll_{\eta}X^{1/2-\eta/2}\left(\frac{2X}{c_{K}}\right)^{\sqrt{X/2}}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{\begin{subarray}{c}\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P\setminus P_{0}\\\
\mathfrak{N}\mathfrak{k}>2X/c_{K}\end{subarray}}\mathfrak{N}\mathfrak{k}^{-1/2-\sqrt{X/2}}(\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}})^{-\eta}.$
(39)

We may bound the inner sum above

$\displaystyle\sum_{\nu\geqslant 1}$ $\displaystyle\sum_{m\geqslant
1}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in D_{\nu}^{\prime}(X)\\\
\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\in
I_{m}(X)\end{subarray}}\mathfrak{N}\mathfrak{k}^{-1/2-\sqrt{X/2}}(\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}})^{-\eta}$
$\displaystyle\ll
X^{\eta/2}\left(\frac{2X}{c_{K}}\right)^{-1/2-\sqrt{X/2}}\sum_{\nu\geqslant
1}\nu^{-1/2-\sqrt{X/2}}\sum_{m\geqslant
1}m^{-\eta}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in
D_{\nu}^{\prime}(X)\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\in
I_{m}(X)\end{subarray}}1$ (40)

where a lattice point counting argument, explained previously in _off-diagonal
subcase I_ , can be applied where the area of the angular sector is of size
$X^{1/2}$ and the perimeter of size $\nu^{-1/2}X^{1/2}$. We therefore bound
the inner sum of (40) as $\ll X^{1/2}$. Subsequent evaluation and substitution
of (40) into (39) shows that the contribution from _off-diagonal subcase III_
to (29) is

$\ll_{\eta}X^{1/2}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b}).$ (41)

Finally, there is the _off-diagonal subcase IV_ :
$\mathfrak{N}\mathfrak{k}\geqslant 2X/c_{K}$ and
$|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|<X^{-1/2}$.
With (39), the bound
$|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|\gg\mathfrak{N}\mathfrak{k}^{-1/2}N^{-1}$,
and the same lattice point counting argument in previous cases, the
contribution to (29) from the inner sum of (39) keeping in mind (36) is

$\displaystyle\ll_{\eta}N^{\eta}\sum_{\nu\geqslant
1}\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\in D_{\nu}^{\prime}\\\
|\arg\gamma_{\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}}|<X^{-1/2}\end{subarray}}\mathfrak{N}\mathfrak{k}^{-1/2-\sqrt{X/2}+\eta/2}\ll
N^{\eta}X^{\eta/2}\left(\frac{2X}{c_{K}}\right)^{-\sqrt{X/2}}.$ (42)

Substitution of the above into (39) then bounds the contribution to (29) from
_off-diagonal subcase IV_ by

$\ll_{\eta}X^{1/2}N^{\eta}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b}).$ (43)

In consideration of the bounds from each of _off-diagonal subcases I-IV_ , it
is shown that the total contribution to (29) from the _off-diagonal subcases
I-IV_ is bounded by (43). Concisely, for $\eta\geqslant 1$ and $N\gg\log X$
(as will be chosen), we have

$\displaystyle
X\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P\setminus
P_{0}\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}\mathfrak{k}}}\sum_{j}$
$\displaystyle\mathcal{F}_{u}[\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k},uX)]\left(X\left(\omega_{K}^{-1}j-\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}\mathfrak{k}}/2\pi\right)\right)$
$\displaystyle\ll_{\eta}X^{1/2}N^{\eta}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b}).$ (44)

###  4.2. Diagonal contribution

With respect to the contribution from
$\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P_{0}$ to (29), one may
readily handle the case $j\neq 0$ of the inner sum by the approach from the
previous subsection since
$\omega_{K}^{-1}j-\arg\gamma_{\mathfrak{a}\overline{\mathfrak{b}}\mathfrak{k}}/2\pi$
here is at least as large as its counterpart in the off-diagonal case with
$j=0$. This certainly yields no more than the quantity (44) found in the off-
diagonal case. Further, for $j=0$ in the inner sum of (29), the non-negativity
of $\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k},uX)$ following from Lemma 5 allows
one use the outer sum according to $\mathfrak{N}\mathfrak{k}\leqslant
X/2c_{K}$ to produce a lower bound on (29), provided that the term resulting
from truncation is $\gg$ than the quantity in (44) with appropriate implied
constant.

Note that (14) from Lemma 4 demonstrates that
$\mathcal{F}_{u}[\Phi(u)W_{K}(\mathfrak{N}\mathfrak{k};uX)]\left(0\right)$ is
effectively constant in this range. Write
$\mathfrak{a}=\mathfrak{a}_{0}\mathfrak{a}^{\prime}$ and
$\mathfrak{b}=\mathfrak{b}_{0}\mathfrak{b}^{\prime}$ as in (24). Further
factorizing

$\mathfrak{a}^{\prime}=(\mathfrak{a}^{\prime},\mathfrak{b}^{\prime})\mathfrak{a}^{\prime\prime}\quad\text{and}\quad\mathfrak{b}^{\prime}=(\mathfrak{a}^{\prime},\mathfrak{b}^{\prime})\mathfrak{b}^{\prime\prime},$
(45)

we note that $\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in P_{0}$
precisely when
$\mathfrak{k}=\mathfrak{k}_{0}\overline{\mathfrak{a}^{\prime\prime}}\mathfrak{b}^{\prime\prime}$
for $\mathfrak{k}_{0}\in P_{0}$. The truncated sum discussed above is of
magnitude

$\displaystyle
X\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b})\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{k}\leqslant
X/2c_{K}\\\ \mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P_{0}\end{subarray}}\mathfrak{N}\mathfrak{k}^{-1/2}$
$\displaystyle=X\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}}\sum_{\mathfrak{N}\mathfrak{k}_{0}\leqslant
X/2c_{K}\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}\mathfrak{N}\mathfrak{k}_{0}^{-1/2}$
$\displaystyle\gg
XN^{-1}\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}r(\mathfrak{a})r(\mathfrak{b}).$

when $N\ll X^{1/2+\varepsilon}$. Indeed, we now finally set
$N=X^{1/4-\varepsilon}$ and $\eta=1+\varepsilon$ so that
$X^{1/2}N^{\eta}=o(X/N)$. Keeping in mind the dual sum of (28), we find that
two times (29) — thus ultimately (28) — is bounded from below in absolute
value by

$\left(2h_{K}\omega_{K}^{-1}\hat{\Phi}(0)+o(1)\right)X\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\ \mathfrak{N}\mathfrak{k}\leqslant X/2c_{K}\\\
\mathfrak{k}\mathfrak{a}\overline{\mathfrak{b}}\in
P_{0}\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}\mathfrak{k}}}.$
(46)

As done in Section 3.1, we wish to factorize (46) so that we may more readily
compare it to (27) on a subpower scale, thereby detecting large values of
$|L(\tfrac{1}{2},\xi_{\ell})|$, which are expected to be subpower with respect
to $X$; see again (20) and Theorem 1.

###  4.3. Factorization of the diagonal sum

As in (24) and (45), write
$\mathfrak{a}=\mathfrak{a}_{0}\mathfrak{a}^{\prime}$ and
$\mathfrak{b}=\mathfrak{b}_{0}\mathfrak{b}^{\prime}$ so that

$\mathfrak{a}\overline{\mathfrak{b}}=\mathfrak{a}_{0}\mathfrak{b}_{0}(\mathfrak{a}^{\prime},\mathfrak{b}^{\prime})\overline{(\mathfrak{a}^{\prime},\mathfrak{b}^{\prime})}\mathfrak{a}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}}$

where $(\mathfrak{a}^{\prime\prime},\mathfrak{b}^{\prime\prime})=1$ and
$\mathfrak{a}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}}\in
P^{\prime}$. Then for $\mathfrak{ka}\overline{\mathfrak{b}}\in P_{0}$, we must
have

$\mathfrak{k}=\mathfrak{k}_{0}\overline{\mathfrak{a}^{\prime\prime}}\mathfrak{b}^{\prime\prime}$

for $\mathfrak{k}_{0}\in P_{0}$. Moving forward, should context require it, we
will assume in our notation that $\mathfrak{a}$ and $\mathfrak{b}$, and also
$\mathfrak{a}^{\prime}$ and $\mathfrak{b}^{\prime}$, factorize as above in
relation to one another. The sum in (46) is thus

$\sum_{\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}}\sum_{\mathfrak{N}\mathfrak{k}_{0}\leqslant
X/2c_{k}\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}\mathfrak{N}\mathfrak{k}_{0}^{-1/2}\geqslant\sum_{\begin{subarray}{c}\mathfrak{N}\mathfrak{a},\mathfrak{N}\mathfrak{b}\leqslant
N\\\
(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}}.$
(47)

We now wish to complete the sum in this lower bound over all non-zero
$\mathfrak{a}$ and $\mathfrak{b}$. We do exactly this using Rankin’s trick,
which is the observation that for any non-negative sequence $(m_{n})$ and any
$\alpha\geqslant 0$, we have

$\sum_{0\leqslant n\leqslant N}m_{n}=\sum_{n\geqslant
0}m_{n}+O\left(N^{-\alpha}\sum_{n\geqslant 0}n^{\alpha}m_{n}\right).$

Using the above, first on the sum over $\mathfrak{a}$ on the right hand side
of (47), and then on the sum over $\mathfrak{b}$ (both times using the same
value of $\alpha\geqslant 0$ to be chosen later) shows that the right side of
(47) equals

$\displaystyle\sum_{\begin{subarray}{c}\mathfrak{a},\mathfrak{b}\neq 0\\\
(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}}$
$\displaystyle+O\left(\Xi(\alpha,0)+\Xi(\alpha,\alpha)\right)$ (48)

where

$\Xi(\alpha_{1},\alpha_{2})=N^{-\alpha_{1}-\alpha_{2}}\sum_{\begin{subarray}{c}\mathfrak{a},\mathfrak{b}\neq
0\\\
(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\end{subarray}}\frac{r(\mathfrak{a})r(\mathfrak{b})}{\sqrt{\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}}\mathfrak{N}\mathfrak{a}^{\alpha_{1}}\mathfrak{N}\mathfrak{b}^{\alpha_{2}}.$
(49)

We first factorize the sum in $\Xi(\alpha,\alpha)$ of (48), where the
factorization of the expected main term in (48) will follow from the
substitution $\alpha=0$. Afterward, we will factorize the sum $\Xi(\alpha,0)$
in (48) in a similar way. To begin, write the sum in $\Xi(\alpha,\alpha)$ as

$\sum_{\begin{subarray}{c}\mathfrak{a}^{\prime},\mathfrak{b}^{\prime}\in
P^{\prime}\\\
(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\end{subarray}}\frac{r(\mathfrak{a}^{\prime})r(\mathfrak{b}^{\prime})}{\sqrt{\mathfrak{N}(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})}}\mathfrak{N}(\mathfrak{a}^{\prime}\mathfrak{b}^{\prime})^{\alpha}\sum_{\begin{subarray}{c}\mathfrak{a}_{0}\in
P_{0}\\\
(\mathfrak{a}_{0},\mathfrak{a}^{\prime})=1\end{subarray}}r(\mathfrak{a}_{0})\mathfrak{N}\mathfrak{a}_{0}^{\alpha}\sum_{\begin{subarray}{c}\mathfrak{b}_{0}\in
P_{0}\\\
(\mathfrak{b}_{0},\mathfrak{b}^{\prime})=1\end{subarray}}r(\mathfrak{b}_{0})\mathfrak{N}\mathfrak{b}_{0}^{\alpha}.$
(50)

After factorizing the two inner sums of the above expression as

$\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}A_{\alpha}(\mathfrak{p})^{2}\prod_{\mathfrak{p}\mid\mathfrak{a}^{\prime}}A_{\alpha}(\mathfrak{p})^{-1}\prod_{\mathfrak{p}\mid\mathfrak{b}^{\prime}}A_{\alpha}(\mathfrak{p})^{-1},\quad
A_{\alpha}(\mathfrak{p}):=(1+r(\mathfrak{p})^{2}\mathfrak{N}\mathfrak{p}^{2\alpha}),$

we write $\mathfrak{c}^{\prime}=(\mathfrak{a}^{\prime},\mathfrak{b}^{\prime})$
so that (50) becomes

$\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}A_{\alpha}(\mathfrak{p})^{2}\sum_{\mathfrak{c}^{\prime}\in
P^{\prime}}r(\mathfrak{c}^{\prime})^{2}(\mathfrak{N}\mathfrak{c}^{\prime})^{2\alpha}\prod_{\mathfrak{p}\mid\mathfrak{c}^{\prime}}A_{\alpha}(\mathfrak{p})^{-2}\sum_{\begin{subarray}{c}(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\\\
(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime},\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}})=1\end{subarray}}\frac{r(\mathfrak{a}^{\prime\prime})r(\mathfrak{b}^{\prime\prime})}{(\mathfrak{N}\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})^{1/2-\alpha}}\prod_{\mathfrak{p}\mid\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime}}A_{\alpha}(\mathfrak{p})^{-1}.$
(51)

With
$\mathfrak{c}^{\prime\prime}=\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime}$
(recalling that $\mathfrak{a}^{\prime\prime},\mathfrak{b}^{\prime\prime}\in
P^{\prime}$ by their definitions) and consideration of the divisor function
$d$, we may write the innermost sum above as

$\sum_{\begin{subarray}{c}\mathfrak{c}^{\prime\prime}\in P^{\prime}\\\
(\mathfrak{c}^{\prime\prime},\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}})=1\end{subarray}}\frac{r(\mathfrak{c}^{\prime\prime})d(\mathfrak{c}^{\prime\prime})}{(\mathfrak{N}\mathfrak{c}^{\prime\prime})^{1/2-\alpha}}\prod_{\mathfrak{p}\mid\mathfrak{c}^{\prime\prime}}A_{\alpha}(p)^{-1}=\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}\nmid\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}}}B_{\alpha}(\mathfrak{p}),\quad
B_{\alpha}(\mathfrak{p}):=1+\frac{4r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2-\alpha}}A_{\alpha}(\mathfrak{p})^{-1},$
(52)

so that inserting (52) into the inner sum of (51) yields

$\displaystyle\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}$ $\displaystyle
A_{\alpha}(\mathfrak{p})^{2}B_{\alpha}(\mathfrak{p})\sum_{\mathfrak{c}^{\prime}\in
P^{\prime}}r(\mathfrak{c}^{\prime})^{2}(\mathfrak{N}\mathfrak{c}^{\prime})^{2\alpha}\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}\mid\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}}}A_{\alpha}(\mathfrak{p})^{-2}B_{\alpha}(\mathfrak{p})^{-1}$
$\displaystyle=\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\left(A_{\alpha}(\mathfrak{p})^{2}B_{\alpha}(\mathfrak{p})+2r(\mathfrak{p})^{2}\mathfrak{N}\mathfrak{p}^{2\alpha}\right)$
$\displaystyle=\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\left(1+\frac{4r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2}}\mathfrak{N}\mathfrak{p}^{\alpha}+4r(\mathfrak{p})^{2}\mathfrak{N}\mathfrak{p}^{2\alpha}+\frac{4r(\mathfrak{p})^{3}}{\mathfrak{N}\mathfrak{p}^{1/2}}\mathfrak{N}\mathfrak{p}^{3\alpha}+r(\mathfrak{p})^{4}\mathfrak{N}\mathfrak{p}^{4\alpha}\right)$
(53)

as the factorization of the sum in $\Xi(\alpha,\alpha)$ in (48).

Using a similar strategy, one may also factorize the sum in $\Xi(\alpha,0)$ of
(48). This quantity is

$\sum_{\begin{subarray}{c}\mathfrak{a}^{\prime},\mathfrak{b}^{\prime}\in
P^{\prime}\\\
(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\end{subarray}}\frac{r(\mathfrak{a}^{\prime})r(\mathfrak{b}^{\prime})}{\sqrt{\mathfrak{N}\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime}}}(\mathfrak{N}\mathfrak{a}^{\prime})^{\alpha}\sum_{\begin{subarray}{c}\mathfrak{a}_{0}\in
P_{0}\\\
(\mathfrak{a}_{0},\mathfrak{a}^{\prime})=1\end{subarray}}r(\mathfrak{a}_{0})\mathfrak{N}\mathfrak{a}_{0}^{\alpha}\sum_{\begin{subarray}{c}\mathfrak{b}_{0}\in
P_{0}\\\
(\mathfrak{b}_{0},\mathfrak{b}^{\prime})=1\end{subarray}}r(\mathfrak{b}_{0}).$
(54)

We may factorize the two inner sums of the above expression as

$\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}A_{\alpha}(\mathfrak{p})A_{0}(\mathfrak{p})\prod_{\mathfrak{p}\mid\mathfrak{a}^{\prime}}A_{\alpha}(\mathfrak{p})^{-1}\prod_{\mathfrak{p}\mid\mathfrak{b}^{\prime}}A_{0}(\mathfrak{p})^{-1},$

so that (54) becomes

$\displaystyle\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}A_{\alpha}(\mathfrak{p})A_{0}(\mathfrak{p})$
$\displaystyle\sum_{\mathfrak{c}^{\prime}\in
P^{\prime}}r(\mathfrak{c}^{\prime})^{2}(\mathfrak{N}\mathfrak{c}^{\prime})^{\alpha}\prod_{\mathfrak{p}\mid\mathfrak{c}^{\prime}}A_{\alpha}(\mathfrak{p})^{-1}A_{0}(\mathfrak{p})^{-1}$
$\displaystyle\times\sum_{\begin{subarray}{c}(\mathfrak{a}^{\prime\prime}\overline{\mathfrak{a}^{\prime\prime}},\mathfrak{b}^{\prime\prime}\overline{\mathfrak{b}^{\prime\prime}})=1\\\
(\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime},\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}})=1\end{subarray}}\frac{r(\mathfrak{a}^{\prime\prime})r(\mathfrak{b}^{\prime\prime})}{(\mathfrak{N}\mathfrak{a}^{\prime\prime}\mathfrak{b}^{\prime\prime})^{1/2}}\prod_{\mathfrak{p}\mid\mathfrak{a}^{\prime\prime}}A_{\alpha}(\mathfrak{p})^{-1}\mathfrak{N}\mathfrak{p}^{\alpha}\prod_{\mathfrak{p}\mid\mathfrak{b}^{\prime\prime}}A_{0}(\mathfrak{p})^{-1}.$
(55)

Rewriting the inner sum above as follows and factorizing allow us to obtain

$\displaystyle\sum_{\begin{subarray}{c}\mathfrak{c}^{\prime\prime}\in
P^{\prime}\\\
(\mathfrak{c}^{\prime\prime},\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}})=1\end{subarray}}\frac{r(\mathfrak{c}^{\prime\prime})}{(\mathfrak{N}\mathfrak{c}^{\prime\prime})^{1/2}}\prod_{\mathfrak{p}\mid\mathfrak{c}^{\prime\prime}}A_{0}(\mathfrak{p})^{-1}$
$\displaystyle\sum_{\mathfrak{a}^{\prime\prime}\mid\mathfrak{c}^{\prime\prime}}(\mathfrak{N}\mathfrak{a}^{\prime\prime})^{\alpha}\prod_{\mathfrak{p}\mid\mathfrak{a}^{\prime\prime}}A_{0}(\mathfrak{p})A_{\alpha}(\mathfrak{p})^{-1}$
$\displaystyle=\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}\nmid\mathfrak{c}^{\prime}\overline{\mathfrak{c}^{\prime}}}\left(1+\frac{2r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2}}(A_{0}(\mathfrak{p})^{-1}+\mathfrak{N}\mathfrak{p}^{\alpha}A_{\alpha}(\mathfrak{p})^{-1})\right)$

so that for

$C_{\alpha}(\mathfrak{p}):=A_{\alpha}(\mathfrak{p})A_{0}(\mathfrak{p})+\frac{2r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2}}(A_{\alpha}(\mathfrak{p})+\mathfrak{N}\mathfrak{p}^{\alpha}A_{0}(\mathfrak{p}))$

we may further factorize (55) as

$\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}C_{\alpha}(\mathfrak{p})\sum_{\mathfrak{c}^{\prime}\in
P^{\prime}}r(\mathfrak{c}^{\prime})^{2}(\mathfrak{N}\mathfrak{c}^{\prime})^{\alpha}\prod_{\mathfrak{p}\mid\mathfrak{c}^{\prime}}C_{\alpha}(\mathfrak{p})^{-1}=\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\left(C_{\alpha}(\mathfrak{p})+2r(\mathfrak{p})^{2}\mathfrak{N}\mathfrak{p}^{\alpha}\right).$

This ultimately yields for the sum in $\Xi(\alpha,0)$ a factorization of

$\displaystyle\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\bigg{(}1+\frac{2r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2}}(1+\mathfrak{N}\mathfrak{p}^{\alpha})+r(\mathfrak{p})^{2}$
$\displaystyle(1+2\mathfrak{N}\mathfrak{p}^{\alpha}+\mathfrak{N}\mathfrak{p}^{2\alpha})$
$\displaystyle+\frac{2r(\mathfrak{p})^{3}}{\mathfrak{N}\mathfrak{p}^{1/2}}(\mathfrak{N}\mathfrak{p}^{\alpha}+\mathfrak{N}\mathfrak{p}^{2\alpha})+r(\mathfrak{p})^{4}\mathfrak{N}\mathfrak{p}^{2\alpha}\bigg{)}.$
(56)

Once a choice of resonator coefficients is made, it will become necessary to
show that

$\Xi(\alpha,0)+\Xi(\alpha,\alpha)=o\left(\Xi(0,0)\right)$ (57)

so that the main term of (48) can be used as a lower bound on (46), and hence
as a lower bound on the numerator of (28).

##  5\. Choice of resonator coefficients

We will now choose explicit values for all remaining parameters in our
application of the resonance method. Recall the properties of
$r(\mathfrak{p})$ given in the beginning of Section 3 and let

$L=\sqrt{\tfrac{1}{2}\log N\log\log N};\quad
r(\mathfrak{p})=\frac{L}{\mathfrak{N}\mathfrak{p}^{1/2}\log\mathfrak{N}\mathfrak{p}},\quad
L^{2}\leqslant\mathfrak{N}\mathfrak{p}\leqslant\exp(\log^{2}L),$ (58)

and $r(\mathfrak{p})=0$ for prime ideals $\mathfrak{p}$ outside of this range.
Further, we set the Rankin’s trick parameter $\alpha=1/\log^{3}L$.

###  5.1. Ratio of error terms to main term

We will need to examine each of $\Xi(\alpha,0)$, $\Xi(\alpha,\alpha)$, and
$\Xi(0,0)$ on a logarithmic scale to demonstrate that the main term of (48)
truly dominates the error terms. Expanding

$\mathfrak{N}\mathfrak{p}^{\alpha}=1+O(\alpha\log\mathfrak{N}\mathfrak{p}),\quad\mathfrak{N}\mathfrak{p}\leqslant\exp(\log^{2}L),$

then making the crude observations

$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}$
$\displaystyle\frac{r(\mathfrak{p})^{3}\log\mathfrak{N}\mathfrak{p}}{\mathfrak{N}\mathfrak{p}^{1/2}}\ll
L^{3}\sum_{\mathfrak{N}\mathfrak{p}\geqslant
L^{2}}\mathfrak{N}\mathfrak{p}^{-2}(\log\mathfrak{N}\mathfrak{p})^{-2}\ll\frac{(\log
N)^{1/2}}{(\log\log N)^{5/2}}$
$\displaystyle\text{and}\quad\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}r(\mathfrak{p})^{4}\log\mathfrak{N}\mathfrak{p}\ll
L^{4}\sum_{\mathfrak{N}\mathfrak{p}\geqslant
L^{2}}\mathfrak{N}\mathfrak{p}^{-2}(\log\mathfrak{N}\mathfrak{p}^{-3})\ll\frac{\log
N}{(\log\log N)^{3}},$

lets us deduce from (53) and (56), for $E_{\alpha}=\log N/(\log\log N)^{3}$,
that

$\displaystyle\frac{\Xi(\alpha,\alpha)}{\Xi(0,0)}+\frac{\Xi(\alpha,0)}{\Xi(0,0)}$
$\displaystyle\ll\exp\left(-2\alpha\log
N+\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}\left((\mathfrak{N}\mathfrak{p}^{\alpha}-1)\frac{4r(\mathfrak{p})}{\sqrt{\mathfrak{N}\mathfrak{p}}}+(\mathfrak{N}\mathfrak{p}^{2\alpha}-1)4r(\mathfrak{p})^{2}\right)\right)$
$\displaystyle\qquad\qquad\times\exp\left(O(\alpha E_{\alpha})\right)$
$\displaystyle+\exp\left(-\alpha\log
N+\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}\left((\mathfrak{N}\mathfrak{p}^{\alpha}-1)\frac{2r(\mathfrak{p})}{\sqrt{\mathfrak{N}\mathfrak{p}}}+(2\mathfrak{N}\mathfrak{p}^{\alpha}+\mathfrak{N}\mathfrak{p}^{2\alpha}-3)r(\mathfrak{p})^{2}\right)\right)$
$\displaystyle\qquad\qquad\times\exp\left(O(\alpha E_{\alpha})\right).$ (59)

Consideration of the linear expansion

$\mathfrak{N}\mathfrak{p}^{\alpha}=1+\alpha\log\mathfrak{N}\mathfrak{p}+O((\alpha\log\mathfrak{N}\mathfrak{p})^{2}),\quad\mathfrak{N}\mathfrak{p}\leqslant\exp(\log^{2}L),$

in conjunction with the definitions

$\displaystyle G_{\alpha}(r)$
$\displaystyle:=\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}\left(\alpha\frac{r(\mathfrak{p})\log\mathfrak{N}\mathfrak{p}}{\mathfrak{N}\mathfrak{p}^{1/2}}+\alpha^{2}r(\mathfrak{p})^{2}\log^{2}\mathfrak{N}\mathfrak{p}\right),$
$\displaystyle H_{\alpha}(r)$ $\displaystyle:=-\alpha\log
N+\alpha\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}4r(\mathfrak{p})^{2}\log\mathfrak{N}\mathfrak{p},$
(60)

allows us to rewrite (59) as

$\exp\big{(}2H_{\alpha}(r)+O(G_{\alpha}(r)+\alpha
E_{\alpha})\big{)}+\exp\big{(}H_{\alpha}(r)+O(G_{\alpha}(r)+\alpha
E_{\alpha})\big{)}$ (61)

where, by partial summation and the prime ideal theorem,

$G_{\alpha}(r)\ll\alpha^{2}\log N\log\log N\log\log\log N.$ (62)

Note the right hand side of the above dominates $\alpha E_{\alpha}$.

###  5.2. Bounding the numerator

Bounding sums similar to the one in (60) is a general requirement of the
resonance method and is delicately computed through partial summation and the
prime ideal theorem. For example, this exact computation (after exercising the
prime ideal theorem) is used in the _proof_ of [3, Lemma 7.4] with the
parameters $\omega(\mathfrak{p})=a_{\omega}=2$ in their notation. Outsourcing
this computation, we have

$\displaystyle H_{\alpha}(r)$ $\displaystyle\geqslant-\alpha\log
N+\alpha\sum_{\mathfrak{p}}\omega(p)r(\mathfrak{p})^{2}\log\mathfrak{N}\mathfrak{p}+O(1)$
$\displaystyle=-\alpha\frac{\log N\log\log\log N}{\log\log
N}+O\left(\alpha\frac{\log N}{\log\log N}\right).$

Applying the bounds in (61) and (62), we find that the ratio of the sum of
errors to the expected main term in (48) is

$\ll\exp\left(-\alpha\frac{\log N\log\log\log N}{2\log\log N}\right).$

Therefore, in consider of (48) and its relationship to (46), we have shown
that the numerator of (20) is bounded from below by

$Xh_{K}\omega_{K}^{-1}\hat{\Phi}(0)\sideset{}{{}^{\prime}}{\prod}_{\mathfrak{p}}\left(1+\frac{4r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2}}+4r(\mathfrak{p})^{2}+r(\mathfrak{p})^{4}\right).$
(63)

##  6\. Main result

With respect to the ratio in (20), we may now utilize our upper bound on its
denominator (27) and our lower bound on its numerator (63). Doing so yields

$\max_{\begin{subarray}{c}X\leqslant\ell\leqslant 2X\\\
\xi_{\ell}\in\mathfrak{F}_{\ell}\end{subarray}}\log\left|L(\tfrac{1}{2},\xi_{\ell})\right|\geqslant
o(1)+\sideset{}{{}^{\prime}}{\sum}_{\mathfrak{p}}\frac{4r(\mathfrak{p})}{\mathfrak{N}\mathfrak{p}^{1/2}}=(2+o(1))\frac{L}{\log
L}.$

Recalling our exact choice of parameters in (58) and the value
$N=X^{1/4-\varepsilon}$ set in Section 4.2 completes the proof of Theorem 1.

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