# Moments of central values of quartic Dirichlet $L$-functions

Peng Gao and Liangyi Zhao

###### Abstract.

In this paper, we study moments of the central values of quartic Dirichlet
$L$-functions and establish quantitative non-vanishing result for these
$L$-values.

Mathematics Subject Classification (2010): 11A15, 11L05, 11M06, 11N37

Keywords: quartic Dirichlet character, quartic large sieve, moments of
$L$-functions

##  1\. Introduction

Moments of $L$-functions at the central point over a family of characters of a
fixed order have been extensively studied in the literature. The first and
second moments of quadratic Dirichlet $L$-functions were evaluated by M.
Jutila [Jutila]. The error term for the first moment in [Jutila] was improved
in [DoHo, MPY, ViTa]. Asymptotic formulas for the second and third moments for
the same family with power savings in the error terms were obtained by K.
Soundararajan [sound1]. The error term for the third moment was subsequently
improved in [DGH] and [Young1]. In the number field case, the authors studied
in [G&Zhao6] the first and second moments of quadratic Hecke $L$-functions in
$\mathbb{Q}(i)$ and $\mathbb{Q}(\omega)$, where
$\omega=\frac{-1+\sqrt{3}i}{2}$. In [Luo], W. Luo considered the first two
moments of cubic Hecke $L$-functions in $\mathbb{Q}(\omega)$. Moments of
various families of higher order Hecke $L$-functions were studied in [FaHL,
FHL, Diac, BFH], using the method of double Dirichlet series.

Although many results are available on moments of quadratic Dirichlet
$L$-functions, less is known for the moments of higher order Dirichlet
$L$-functions. For the family of cubic Dirichlet $L$-functions, this is
investigated by S. Baier and M. P. Young in [B&Y] and they also mentioned that
it is plausible to extend their methods to study moments of quartic and sextic
Dirichlet $L$-functions. Motivated by this, it is our goal in this paper to
investigate the quartic case. Writing $\chi_{0}$ for the principal character
and letting $w:(0,\infty)\rightarrow\mathbb{R}$ be any smooth, compactly
supported function whose Fourier transform is denoted by $\widehat{w}$, we
begin with a result that establishes an asymptotic expression for the first
moment.

######  Theorem 1.1.

With notations as above and assuming the truth of the Lindelöf hypothesis, we
have

$\sum_{(q,2)=1}\;\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\bmod{q}\\\
\chi^{4}=\chi_{0}\end{subarray}}L(\tfrac{1}{2},\chi)w\left(\frac{q}{Q}\right)=CQ\widehat{w}(0)+O(Q^{9/10+\varepsilon}),$

where $C$ is a positive explicit constant given in (3.5) and the asterisk on
the sum over $\chi$ restricts the sum to primitive characters $\chi$ such that
$\chi^{2}$ remains primitive.

It is a conjecture that goes back to S. Chowla [chow] that $L(1/2,\chi)\neq 0$
for all primitive Dirichlet characters $\chi$. As a consequence of Theorem
1.1, a partial answer can be given for this conjecture for the character under
our consideration. We argue similar to the proof of [B&Y, Corollary 1.2],
using Hölder’s inequality together with the well-known bound for the eighth
moment of Dirichlet $L$-functions and the lower bound implied by the
asymptotic formula in Theorem 1.1, to obtain following non-vanishing result on
the central values of quartic Dirichlet $L$-functions.

######  Corollary 1.2.

Assume that the Lindelöf hypothesis is true. There exist infinitely many
primitive Dirichlet characters $\chi$ of order $4$ such that $L(1/2,\chi)\neq
0$. More precisely, the number of such characters with conductor $\leq Q$ is
$\gg Q^{6/7-\varepsilon}$.

For $m\in\mathbb{Z},n\in\mathbb{Z}[i],(n,2)=1$, let
$\psi_{m}((n))=\left(\frac{m}{n}\right)_{4}$ where
$\left(\frac{\cdot}{\cdot}\right)_{4}$ is the quartic residue symbol in
$\mathbb{Q}(i)$ defined in Section 2.1. It is also shown there that $\psi_{m}$
is a quartic Hecke character of trivial infinite type modulo $16m$. Our next
result concerns bounds on the second moment.

######  Theorem 1.3.

We have for any $Q\geq 1$ that

(1.1) $\sum\limits_{\begin{subarray}{c}q\leq Q\end{subarray}}\
\sideset{}{{}^{*}}{\sum}\limits_{\begin{subarray}{c}\chi\bmod q\\\
\chi^{4}=\chi_{0}\end{subarray}}\left|L(\tfrac{1}{2}+it,\chi)\right|^{2}\ll
Q^{7/6+\varepsilon}(1+|t|)^{1/2+\varepsilon}.$

Further denote for any rational integer $m$,

(1.2) $L(s,\psi_{m})=\sum_{\begin{subarray}{c}n\in\mathbb{Z}[i]\\\ n\equiv
1\bmod{(1+i)^{3}}\end{subarray}}\psi_{m}(n)N(n)^{-s}.$

Then we have

(1.3) $\sideset{}{{}^{\prime}}{\sum}_{m\leq
M}\left|L(1/2+it,\psi_{m})\right|^{2}\ll
M^{3/2+\varepsilon}(1+|t|)^{5/4+\varepsilon},$

where $\sum^{\prime}$ means that the sum runs over square-free elements of
$\mathbb{Z}$.

As the proof of the above result is similar to that of [B&Y, Theorem 1.3]
(using the approximate functional equation and then the appropriate version of
the large sieve inequality Lemmas 2.9 and 2.10), we shall therefore omit it in
the paper.

We point out here that the proof of (1.1) uses a large sieve inequality for
quartic Dirichlet characters given in Lemma 2.10 by using the term
$Q^{7/6}+Q^{2/3}M$ in the minimum there. Similarly, the proof of (1.3) follows
from recalling the definition of $C_{1}(M,Q)$ in [G&Zhao, p. 907] and the
bound $C_{1}(M,Q)\ll(QM)^{\varepsilon}(Q^{5/4}+Q^{1/2}M)$ by Lemma 2.10. A
variant of this for the cubic case plays a key role in the treatment of the
error term in the work of Baier and Young [B&Y] on the first moment of cubic
Dirichlet $L$-functions. However, our Lemma 2.10 is just short of producing
admissible error term in our first moment result, even after incorporating the
recursive technique of Baier and Young (see third arXiv version of the paper
[B&Y] arXiv:0804.2233v3). This observation drives us to seek for bounds of
individual quartic Dirichlet $L$-functions in order to control the error term
and leads us to derive our Theorem 1.1 under the Lindelöf hypothesis since the
current best known subconvexity bounds for the $L$-functions under our
consideration do not provide a satisfactory error term either.

###  1.4. Notations

The following notations and conventions are used throughout the paper.
$e(z)=\exp(2\pi iz)=e^{2\pi iz}$.
$f=O(g)$ or $f\ll g$ means $|f|\leq cg$ for some unspecified positive constant
$c$.
$\mu_{[i]}$ denotes the Möbius function on $\mathbb{Z}[i]$.
$\zeta_{\mathbb{Q}(i})(s)$ is the Dedekind zeta function for the field
$\mathbb{Q}(i)$.

##  2\. Preliminaries

In this section we provide various tools used throughout the paper.

###  2.1. Quartic symbol and primitive quartic Dirichlet characters

The symbol $(\left(\frac{\cdot}{n}\right))_{4}$ is the quartic residue symbol
in the ring $\mathbb{Z}[i]$. For a prime $\varpi\in\mathbb{Z}[i]$ with
$N(\varpi)\neq 2$, the quartic residue symbol is defined for
$a\in\mathbb{Z}[i]$, $(a,\varpi)=1$ by
$\left(\frac{a}{\varpi}\right)_{4}\equiv a^{(N(\varpi)-1)/4}\pmod{\varpi}$,
with $\left(\frac{a}{\varpi}\right)_{4}\in\\{\pm 1,\pm i\\}$. When $\varpi|a$,
we define $\left(\frac{a}{\varpi}\right)_{4}=0$. Then the quartic residue
symbol can be extended to any composite $n$ with $(N(n),2)=1$
multiplicatively. We further set $\left(\frac{\cdot}{n}\right)_{4}=1$ when $n$
is a unit in $\mathbb{Z}[i]$. We also define the quadratic residue symbol
$\left(\frac{\cdot}{n}\right)$ so that
$\left(\frac{\cdot}{n}\right)=\left(\frac{\cdot}{n}\right)^{2}_{4}$.

Note that in $\mathbb{Z}[i]$, every ideal co-prime to $2$ has a unique
generator congruent to 1 modulo $(1+i)^{3}$. Such a generator is called
primary. Observe that $n=a+bi,a,b\in\mathbb{Z}$ in $\mathbb{Z}[i]$ is
congruent to $1\bmod{(1+i)^{3}}$ if and only if $a\equiv 1\pmod{4},b\equiv
0\pmod{4}$ or $a\equiv 3\pmod{4},b\equiv 2\pmod{4}$ by [I&R, Lemma 6, p. 121].
It follows from this that we have

(2.1) $\displaystyle a\equiv(-1)^{\frac{N(n)-1}{4}}\quad\pmod{4},\quad b\equiv
1-(-1)^{\frac{N(n)-1}{4}}\quad\pmod{4}.$

Recall that [Lemmermeyer, Theorem 6.9] the quartic reciprocity law states that
for two primary integers $m,n\in\mathbb{Z}[i]$,

$\displaystyle\left(\frac{m}{n}\right)_{4}=\left(\frac{n}{m}\right)_{4}(-1)^{((N(n)-1)/4)((N(m)-1)/4)}.$

Also, the supplement laws to the quartic reciprocity law states that for
$n=a+bi$ being primary,

(2.2)
$\displaystyle\left(\frac{i}{n}\right)_{4}=i^{(1-a)/2}\qquad\mbox{and}\qquad\hskip
7.22743pt\left(\frac{1+i}{n}\right)_{4}=i^{(a-b-1-b^{2})/4}.$

This implies that for any $n\equiv 1\pmod{16}$, we have

$\displaystyle\
\left(\frac{i}{n}\right)_{4}=\left(\frac{1+i}{n}\right)_{4}=1.$

Hence the functions $n\rightarrow\left(\frac{i}{n}\right)_{4}$ and
$n\rightarrow\left(\frac{1+i}{n}\right)_{4}$ can be regarded as Hecke
characters $\pmod{16}$ of trivial infinite type. From this we see that
$\psi_{m}$ is a quartic Hecke character of trivial infinite type modulo $16m$
for every $m\in\mathbb{Z}$.

Using the quartic residue symbols, we have the following classification of all
the primitive quartic Dirichlet characters of conductor $q$ co-prime to $2$:

######  Lemma 2.2.

The primitive quartic Dirichlet characters of conductor $q$ coprime to $2$
such that their squares remain primitive are of the form
$\chi_{n}:m\mapsto\left(\frac{m}{n}\right)_{4}$ for some $n\in\mathbb{Z}[i]$,
$n\equiv 1\pmod{(1+i)^{3}}$, $n$ square-free and not divisible by any rational
primes, with norm $N(n)=q$.

The above result is given in [G&Zhao, Section 2.2], but the correspondence
given there is not exact. We take this opportunity to thank Francesca
Balestrieri and Nick Rome for some helpful discussions on this topic.

###  2.3. The Gauss sums

For any $n\in\mathbb{Z}[i]$, $(n,2)=1$, the quartic Gauss sum $g(n)$ is
defined by

$\displaystyle
g(n)=\sum_{x\bmod{n}}\left(\frac{x}{n}\right)_{4}\widetilde{e}\left(\frac{x}{n}\right),$

where $\widetilde{e}(z)=\exp\left(2\pi
i(\frac{z}{2i}-\frac{\overline{z}}{2i})\right)$.

The following well-known formula (see [P, p. 195]) holds for all $n$:

(2.3) $\displaystyle|g(n)|$
$\displaystyle=\begin{cases}\sqrt{N(n)}\qquad&\text{if $n$ is square-free},\\\
0\qquad&\text{otherwise}.\end{cases}$

More generally, we define for $n,k\in\mathbb{Z}[i],(n,2)=1$,

$\displaystyle
g(k,n)=\sum_{x\bmod{n}}\left(\frac{x}{n}\right)_{4}\widetilde{e}\left(\frac{kx}{n}\right).$

We note two properties of $g(r,n)$ that can be found in [Diac]:

(2.4) $\displaystyle g(rs,n)$
$\displaystyle=\overline{\left(\frac{s}{n}\right)_{4}}g(r,n),\quad(s,n)=1,$
(2.5) $\displaystyle g(r,n_{1}n_{2})$
$\displaystyle=\left(\frac{n_{2}}{n_{1}}\right)_{4}\left(\frac{n_{1}}{n_{2}}\right)_{4}g(r,n_{1})g(r,n_{2}),\quad(n_{1},n_{2})=1.$

We further define the Gauss sum $\tau(\chi)$ associated to a primitive
Dirichlet character $\chi\pmod{q}$ by

$\tau(\chi)=\sum_{1\leq x\leq q}\chi(x)e\left(\frac{x}{q}\right).$

When $\chi$ is a primitive quartic Dirichlet character identified with
$\chi_{n}$ by Lemma 2.2, we have ([G&Zhao, p. 894])

(2.6) $\tau(\chi_{n})=\left(\frac{\overline{n}}{n}\right)_{4}g(n).$

Here we have $n\equiv 1\pmod{(1+i)^{3}}$ and $n$ free of rational prime
divisors. If we write $n=a+bi$ with $a,b\in\mathbb{Z}$, then we deduce from
this that $(a,b)=1$ so that

(2.7)
$\displaystyle\left(\frac{\overline{n}}{n}\right)_{4}=\left(\frac{a-bi}{a+bi}\right)_{4}=\left(\frac{2a}{a+bi}\right)_{4}=\left(\frac{2(-1)^{\frac{N(n)-1}{4}}}{a+bi}\right)_{4}\left(\frac{(-1)^{\frac{N(n)-1}{4}}a}{a+bi}\right)_{4}.$

As $(-1)^{\frac{N(n)-1}{4}}a$ is primary according to (2.1), we have by the
quartic reciprocity law,

(2.8)
$\displaystyle\left(\frac{(-1)^{\frac{N(n)-1}{4}}a}{a+bi}\right)_{4}=(-1)^{((N(a)-1)/4)((N(n)-1)/4)}\left(\frac{a+bi}{a}\right)_{4}=\left(\frac{a+bi}{a}\right)_{4}=\left(\frac{bi}{a}\right)_{4}=\left(\frac{b}{a}\right)_{4}\left(\frac{i}{a}\right)_{4}=\left(\frac{i}{a}\right)_{4},$

where the last equality follows from [I&R, Proposition 9.8.5], which states
that for $a,b\in\mathbb{Z},(a,2b)=1$ ,

$\left(\frac{b}{a}\right)_{4}=1.$

It follows from (2.2) that

(2.9)
$\displaystyle\left(\frac{i}{a}\right)_{4}=i^{(1-(-1)^{\frac{N(n)-1}{4}}a)/2}=(-1)^{\frac{a^{2}-1}{8}}.$

On the other hand, note that by the definition that

(2.10)
$\displaystyle\left(\frac{(-1)^{\frac{N(n)-1}{4}}}{a+bi}\right)_{4}=(-1)^{\frac{N(n)-1}{4}\cdot\frac{N(n)-1}{4}}=(-1)^{\frac{N(n)-1}{4}}=\left(\frac{-1}{n}\right)_{4}.$

We then deduce that, putting together (2.7), (2.8), (2.9) and (2.10),

$\displaystyle\left(\frac{\overline{n}}{n}\right)_{4}=\left(\frac{-2}{n}\right)_{4}(-1)^{\frac{a^{2}-1}{8}}=\begin{cases}\left(\frac{-2i}{n}\right)_{4}=\overline{\left(\frac{(-2i)^{3}}{n}\right)}_{4}\qquad&\text{if
$\left(\frac{-1}{n}\right)_{4}=1$},\\\ \\\
i^{-1}\left(\frac{-2i}{n}\right)_{4}=i^{-1}\overline{\left(\frac{(-2i)^{3}}{n}\right)}_{4}\qquad&\text{if
$\left(\frac{-1}{n}\right)_{4}=-1$ }.\end{cases}$

Thus from this and (2.6),

(2.11)
$\displaystyle\tau(\chi_{n})=\begin{cases}\overline{\left(\frac{(-2i)^{3}}{n}\right)}_{4}g(n)\qquad&\text{if
$\left(\frac{-1}{n}\right)_{4}=1$},\\\ \\\
i^{-1}\overline{\left(\frac{(-2i)^{3}}{n}\right)}_{4}g(n)\qquad&\text{if
$\left(\frac{-1}{n}\right)_{4}=-1$ }.\end{cases}$

###  2.4. The approximate functional equation

Let $G(s)$ be any even function which is holomorphic and bounded in the strip
$-4<\Re(s)<4$ satisfying $G(0)=1$. It follows from [iwakow, Theorem 5.3] that
we have the following approximate functional equation for Dirichlet
$L$-functions.

######  Proposition 2.5.

Let $\chi$ be a primitive Dirichlet character of conductor $q$. For any
$\alpha\in\mathbb{C},j\in\\{\pm 1\\}$, let

(2.12)
$a_{j}=\frac{1-j}{2},\quad\epsilon(\chi)=i^{-a_{\chi(-1)}}q^{-1/2}\tau(\chi),\quad
X_{\alpha,j}=\left(\frac{q}{\pi}\right)^{-\alpha}\frac{\Gamma\left(\tfrac{1/2+a_{j}-\alpha}{2}\right)}{\Gamma\left(\tfrac{1/2+a_{j}+\alpha}{2}\right)}.$

We define

$V_{\alpha,j}(x)=\frac{1}{2\pi
i}\int\limits_{(2)}\frac{G(s)}{s}\gamma_{\alpha,j}(s)x^{-s}\mathrm{d}s,\quad\text{where}\quad\gamma_{\alpha,j}(s)=\pi^{-s/2}\frac{\Gamma\left(\tfrac{1/2+a_{j}+\alpha+s}{2}\right)}{\Gamma\left(\tfrac{1/2+a_{j}+\alpha}{2}\right)}.$

Furthermore, let $A$ and $B$ be positive real numbers such that $AB=q$. Then
for any $|\Re(\alpha)|<1/2$ we have

$L(1/2+\alpha,\chi)=\sum_{m=1}^{\infty}\frac{\chi(m)}{m^{1/2+\alpha}}V_{\alpha,\chi(-1)}\left(\frac{m}{A}\right)+\epsilon(\chi)X_{\alpha,\chi(-1)}\sum_{m=1}^{\infty}\frac{\overline{\chi}(m)}{m^{1/2-\alpha}}V_{-\alpha,\chi(-1)}\left(\frac{m}{B}\right).$

For $\alpha=0$ we set $\gamma_{1}=\gamma_{0,1}$, $\gamma_{-1}=\gamma_{0,-1}$,
$V_{1}=V_{0,1}$, $V_{-1}=V_{0,-1}$.

With a suitable $G(s)$ (for example $G(s)=e^{s^{2}}$), we have for any $c>0$
(see [HIEK, Proposition 5.4]):

$\displaystyle
V_{\alpha,j}\left(\xi\right)\ll\left(1+\frac{\xi}{1+|\alpha|}\right)^{-c}.$

On the other hand, when $G(s)=1$, we have (see [sound1, Lemma 2.1]) that
$V_{\pm 1}(\xi)$ is real-valued and smooth on $[0,\infty)$ and for the $j$-th
derivative of $V_{\pm 1}(\xi)$,

(2.13) $V_{\pm
1}\left(\xi\right)=1+O(\xi^{1/2-\epsilon})\;\mbox{for}\;0<\xi<1\quad\mbox{and}\quad
V^{(j)}_{\pm 1}\left(\xi\right)=O(e^{-\xi})\;\mbox{for}\;\xi>0,\;j\geq 0.$

###  2.6. Analytic behavior of Dirichlet series associated with Gauss sums

For any Hecke character $\chi\pmod{16}$ of trivial infinite type, we let

$\displaystyle h(r,s;\chi)=\sum_{\begin{subarray}{c}(n,r)=1\\\ n\equiv
1\bmod{(1+i)^{3}}\end{subarray}}\frac{\chi(n)g(r,n)}{N(n)^{s}}.$

The following lemma gives the analytic behavior of $h(r,s;\chi)$ on
$\Re(s)>1$.

######  Lemma 2.7.

[G&Zhao1, Lemma 2.5] The function $h(r,s;\chi)$ has meromorphic continuation
to the entire complex plane. It is holomorphic in the region $\sigma=\Re(s)>1$
except possibly for a pole at $s=5/4$. For any $\varepsilon>0$, letting
$\sigma_{1}=3/2+\varepsilon$, then for
$\sigma_{1}\geq\sigma\geq\sigma_{1}-1/2$, $|s-5/4|>1/8$, we have for
$t=\Im(s)$,

$h(r,s;\chi)\ll
N(r)^{(\sigma_{1}-\sigma)/2+\varepsilon}(1+t^{2})^{3(\sigma_{1}-\sigma)/2+\varepsilon}.$

Moreover, the residue satisfies

$\mathop{\mathrm{Res}}_{s=5/4}h(r,s;\chi)\ll N(r)^{1/8+\varepsilon}.$

###  2.8. The large sieve with quartic Dirichlet characters

The following large sieve inequality for quadratic residue symbols is needed
in the proof.

######  Lemma 2.9.

[Onodera, Theorem 1] Let $M,N$ be positive integers, and let
$(a_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence of complex numbers, where
$n$ runs over $\mathbb{Z}[i]$. Then we have for any $\varepsilon>0$,

$\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}m\in\mathbb{Z}[i]\\\
(m,2)=1\\\ N(m)\leq M\end{subarray}}\left|\
\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}n\in\mathbb{Z}[i]\\\
(n,2)=1\\\ N(n)\leq
N\end{subarray}}a_{n}\left(\frac{n}{m}\right)\right|^{2}\ll_{\varepsilon}(MN)^{\varepsilon}(M+N)\sum_{N(n)\leq
N}|a_{n}|^{2},$

where $\sum^{\prime}$ means that the sum runs over square-free elements of
$\mathbb{Z}[i]$ and $(\frac{\cdot}{m})$ is the quadratic residue symbol.

Besides the notation $C_{1}(M,Q)$ mentioned in the Introduction, we also
recall the norm $C_{2}(M,Q)$ defined in [G&Zhao, p. 907]. Notice that for
square-free $n\in\mathbb{Z}[i]$ satisfying $(n,2)=1$, the square of the
quartic symbol $\left(\frac{\cdot}{n}\right)_{4}^{2}$ becomes the quadratic
residue symbol $\left(\frac{\cdot}{n}\right)$. Thus, using Lemma 2.9 in
[G&Zhao, (39)] and proceed to the estimate [G&Zhao, (40)], we see that the
estimation given in [G&Zhao, (24)] can be replaced by

(2.14) $C_{2}(M,Q)\ll(QM)^{\varepsilon}\left(M+Q^{3/2}\right),$

As we have by [G&Zhao, (23)] that $C_{1}(M,Q)\leq C_{2}(M,Q)$, we see that the
bound given in (2.14) is also valid for $C_{1}(M,Q)$, which leads to the
following large sieve inequality for quartic Dirichlet characters.

######  Lemma 2.10.

Let $(a_{m})_{m\in\mathbb{N}}$ be an arbitrary sequence of complex numbers.
Then

$\begin{split}\sum\limits_{\begin{subarray}{c}Q<q\leq 2Q\end{subarray}}\
\sideset{}{{}^{*}}{\sum}\limits_{\begin{subarray}{c}\chi\bmod q\\\
\chi^{4}=\chi_{0}\end{subarray}}&\left|\
\sideset{}{{}^{*}}{\sum}\limits_{\begin{subarray}{c}M<m\leq
2M\end{subarray}}a_{m}\chi(m)\right|^{2}\\\
\ll&(QM)^{\varepsilon}\min\left\\{Q^{3/2}+M,Q^{5/4}+Q^{1/2}M,Q^{7/6}+Q^{2/3}M,Q+Q^{1/3}M^{5/3}+M^{7/3}\right\\}\sideset{}{{}^{*}}{\sum}\limits_{\begin{subarray}{c}M<m\leq
2M\end{subarray}}\left|a_{m}\right|^{2},\end{split}$

where the asterisk on the sum over $\chi$ restricts the sum to primitive
characters whose square remain primitive and the asterisks attached to the sum
over $m$ indicates that $m$ runs over square-free integers.

##  3\. Proof of Theorem 1.1

We start by setting

$\mathcal{M}:=\sum_{(q,2)=1}\;\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\bmod{q}\\\
\chi^{4}=\chi_{0}\end{subarray}}L(\tfrac{1}{2},\chi)w\left(\frac{q}{Q}\right).$

Applying Proposition 2.5 (the approximate functional equation) with
$G(s)=1,A_{q}B=q$ yields $\mathcal{M}=\mathcal{M}_{1}+\mathcal{M}_{2}$, where

$\displaystyle\mathcal{M}_{1}$
$\displaystyle=\sum_{(q,2)=1}\;\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\bmod{q}\\\
\chi^{4}=\chi_{0}\end{subarray}}\
\sum_{m=1}^{\infty}\frac{\chi(m)}{\sqrt{m}}V_{\chi(-1)}\left(\frac{m}{A_{q}}\right)w\left(\frac{q}{Q}\right),$
$\displaystyle\mathcal{M}_{2}$
$\displaystyle=\sum_{(q,2)=1}\;\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\bmod{q}\\\
\chi^{4}=\chi_{0}\end{subarray}}\epsilon(\chi)\sum_{m=1}^{\infty}\frac{\overline{\chi}(m)}{\sqrt{m}}V_{\chi(-1)}\left(\frac{m}{B}\right)w\left(\frac{q}{Q}\right).$

For each primitive quartic Dirichlet character $\chi$ whose square remains
primitive, we write it as $\chi_{n}$ via the correspondence given in Lemma
2.2. It then follows from (2.6), (2.12) and this correspondence that

$\displaystyle\epsilon(\chi_{n})=\epsilon(\chi)=i^{-a_{\chi_{n}(-1)}}N(n)^{-1/2}\left(\frac{\bar{n}}{n}\right)_{4}g(n).$

The above allows us to further decompose $M_{1}$ and $M_{2}$ as
$\mathcal{M}_{1}=\mathcal{M}^{+}_{1}+\mathcal{M}^{-}_{1}$ and
$\mathcal{M}_{2}=\mathcal{M}^{+}_{2}+\mathcal{M}^{-}_{2}$, where

$\displaystyle\mathcal{M}^{\pm}_{1}$
$\displaystyle=\sideset{}{{}^{\prime\prime}}{\sum}_{\begin{subarray}{c}n\equiv
1\bmod(1+i)^{3}\end{subarray}}\frac{1\pm\chi_{n}(-1)}{2}\
\sum_{m=1}^{\infty}\frac{\chi_{n}(m)}{\sqrt{m}}V_{\chi_{n}(-1)}\left(\frac{m}{A_{n}}\right)w\left(\frac{N(n)}{Q}\right),$
$\displaystyle\mathcal{M}^{\pm}_{2}$
$\displaystyle=\sideset{}{{}^{\prime\prime}}{\sum}_{\begin{subarray}{c}n\equiv
1\bmod(1+i)^{3}\end{subarray}}\frac{1\pm\chi_{n}(-1)}{2}\epsilon(\chi_{n})\sum_{m=1}^{\infty}\frac{\overline{\chi}_{n}(m)}{\sqrt{m}}V_{\chi_{n}(-1)}\left(\frac{m}{B}\right)w\left(\frac{N(n)}{Q}\right).$

Here $\sum^{\prime\prime}$ means that the sum runs over square-free elements
$n\in\mathbb{Z}[i]$ that have no rational prime divisors. Also, the parameters
$A_{n},B$ sastify that $A_{n}B=N(n)$. We now introduce another parameter $A$
defined by $AB=Q$ and we note that we have $A_{n}=AN(n)/Q\asymp A$ for all $n$
under consideration because of the compact support of $w$.

It remains to evaluate $\mathcal{M}^{\pm}_{1}$ and $\mathcal{M}^{\pm}_{2}$. As
the arguments are similar, we will only evaluate $\mathcal{M}^{+}_{1}$ and
$\mathcal{M}^{+}_{2}$ in what follows. We summarize the results in the
following lemma.

######  Lemma 3.1.

We have

(3.1) $\displaystyle\mathcal{M}^{\pm}_{1}=$
$\displaystyle\frac{1}{2}CQ\widetilde{w}(1)+O\left(Q^{1/2+\varepsilon}A^{1/2+\varepsilon}+QA^{-1/4+\varepsilon}\right),$
(3.2) $\displaystyle\mathcal{M}^{\pm}_{2}\ll$ $\displaystyle
Q^{3/4+\varepsilon}B^{3/4+\varepsilon},$

where the constant $C$ is given explicitly in (3.5).

Theorem 1.1 follows from Lemma 3.1 by setting $B=Q^{1/5}$ and $A=Q^{4/5}$ in
Lemma 3.1. Thus the remainder of the paper is devoted to the proof of this
lemma.

###  3.2. Evaluating $\mathcal{M}^{+}_{1}$, the main term

We write $\mu$ for Möbius function and we define
$\mu_{\mathbb{Z}}(d)=\mu(|d|)$ for $d\in\mathbb{Z}$. Then for any $n\equiv
1\pmod{(1+i)^{3}}$, we note the following relation

(3.3) $\sum_{\begin{subarray}{c}d|n,d\in\mathbb{Z}\\\ d\equiv 1\bmod
4\end{subarray}}\mu_{\mathbb{Z}}(d)=\begin{cases}1,\quad\text{$n$ has no
rational prime divisors},\\\ 0,\quad\text{otherwise}.\end{cases}$

We apply the above and change variables $n\rightarrow dn$ to the sum over $n$.
Note that any square-free $d\in\mathbb{Z},d\equiv 1\pmod{4}$ is also square-
free when regarded as an element of $\mathbb{Z}[i]$ and this implies that the
condition that $dn$ is square-free then is equivalent to $n$ being square-free
and $(d,n)=1$. We then deduce that $M^{+}_{1}=M^{+}_{1,1}+M^{+}_{1,2}$, with

$\displaystyle\mathcal{M}^{+}_{1,1}$
$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\equiv
1\bmod
4\end{subarray}}\mu_{\mathbb{Z}}(d)\sum_{m=1}^{\infty}\frac{\left(\frac{m}{d}\right)_{4}}{\sqrt{m}}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}n\equiv
1\bmod(1+i)^{3}\\\
(n,d)=1\end{subarray}}\left(\frac{m}{n}\right)_{4}V_{1}\left(\frac{m}{A}\frac{Q}{N(nd)}\right)w\left(\frac{N(nd)}{Q}\right),$
$\displaystyle\mathcal{M}^{+}_{1,2}$
$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\equiv
1\bmod
4\end{subarray}}\mu_{\mathbb{Z}}(d)\sum_{m=1}^{\infty}\frac{\left(\frac{-m}{d}\right)_{4}}{\sqrt{m}}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}n\equiv
1\bmod(1+i)^{3}\\\
(n,d)=1\end{subarray}}\left(\frac{-m}{n}\right)_{4}V_{1}\left(\frac{m}{A}\frac{Q}{N(nd)}\right)w\left(\frac{N(nd)}{Q}\right),$

where, as before, $\sum^{\prime}$ denotes a sum over the square-free elements
of $\mathbb{Z}[i]$.

We first evaluate $\mathcal{M}^{+}_{1,1}$ by using Möbius inversion to detect
the square-free condition that $n$. So

$\mathcal{M}^{+}_{1,1}=\frac{1}{2}\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\
d\equiv 1\bmod
4\end{subarray}}\mu_{\mathbb{Z}}(d)\sum_{\begin{subarray}{c}l\equiv
1\bmod(1+i)^{3}\\\
(l,d)=1\end{subarray}}\mu_{[i]}(l)\sum_{m=1}^{\infty}\frac{\left(\frac{m}{dl^{2}}\right)_{4}}{\sqrt{m}}\mathcal{M}_{1}(d,l,m),$

where

$\mathcal{M}_{1}(d,l,m)=\sum_{\begin{subarray}{c}n\equiv 1\bmod(1+i)^{3}\\\
(n,d)=1\end{subarray}}\left(\frac{m}{n}\right)_{4}V_{1}\left(\frac{m}{A}\frac{Q}{N(ndl^{2})}\right)w\left(\frac{N(ndl^{2})}{Q}\right).$

Now Mellin inversion gives

$V_{1}\left(\frac{m}{A}\frac{Q}{N(ndl^{2})}\right)w\left(\frac{N(ndl^{2})}{Q}\right)=\frac{1}{2\pi
i}\int\limits_{(2)}\left(\frac{Q}{N(ndl^{2})}\right)^{s}\widetilde{f}(s)\mathrm{d}s,$

where

$\widetilde{f}(s)=\int\limits_{0}^{\infty}V_{1}\left(\frac{m}{A}x^{-1}\right)w(x)x^{s-1}\mathrm{d}x.$

Integration by parts and using (2.13) shows that $\widetilde{f}(s)$ is a
function satisfying the bound

$\widetilde{f}(s)\ll(1+|s|)^{-E}\left(1+m/A\right)^{-E},$

for any $\Re(s)>0$ and any integer $E>0$.

Thus we deduce from the above bound that

$\mathcal{M}_{1}(d,l,m)=\frac{1}{2\pi
i}\int\limits_{(2)}\left(\frac{Q}{N(dl^{2})}\right)^{s}L(s,\psi_{md^{4}})\widetilde{f}(s)\mathrm{d}s,$

with $L(s,\psi_{md^{4}})$ given in (1.2).

We estimate $\mathcal{M}^{+}_{1,1}$ by moving the contour to the line $\Re
s=1/2$. Observe that the Hecke $L$-function has a pole at $s=1$ when $m$ is a
fourth power. We write $\mathcal{M}_{0}$ for the contribution to
$\mathcal{M}^{+}_{1,1}$ from these residues and $\mathcal{M}_{1}^{\prime}$ for
the remainder.

We evaluate $\mathcal{M}_{0}$ by noting that

$\mathcal{M}_{0}=\frac{1}{2}\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\equiv
1\bmod{4}\end{subarray}}\mu_{\mathbb{Z}}(d)\sum_{\begin{subarray}{c}l\equiv
1\bmod(1+i)^{3}\\\
(l,d)=1\end{subarray}}\mu_{[i]}(l)\sum_{m=1}^{\infty}\frac{\left(\frac{m}{dl^{2}}\right)_{4}}{\sqrt{m}}\frac{Q}{N(dl^{2})}\widetilde{f}(1)\mathop{\text{Res}}_{s=1}L(s,\psi_{md^{4}}),$

where using the Mellin convolution formula shows that

(3.4)
$\widetilde{f}(1)=\int\limits_{0}^{\infty}V_{1}\left(\frac{m}{A}x^{-1}\right)w(x)\mathrm{d}x=\frac{1}{2\pi
i}\int\limits_{(2)}\left(\frac{A}{m}\right)^{s}\widetilde{w}(1+s)\frac{G(s)}{s}\gamma_{1}(s)\mathrm{d}s,\quad\mbox{with}\;\widetilde{w}(s)=\int\limits_{0}^{\infty}w(x)x^{s-1}\mathrm{d}x.$

From the discussions in Section 2.1, we see that $\psi_{md^{4}}$ is the
principal character only when $m$ is a fourth power, in which case

$L(s,\psi_{md^{4}})=\zeta_{\mathbb{Q}(i)}(s)\prod_{\varpi|2dm}\left(1-N(\varpi)^{-s}\right),$

where $\varpi$ denotes a prime in $\mathbb{Z}[i]$ in this section.

Let $C_{0}=\frac{\pi}{4}$, the residue of $\zeta_{\mathbb{Q}(i)}(s)$ at $s=1$,
so that

$\mathcal{M}_{0}=\frac{1}{2}C_{0}Q\sum_{m=1}^{\infty}\frac{\widetilde{f}(1)}{m^{2}}\prod_{\varpi|2m}\left(1-N(\varpi)^{-1}\right)\sum_{\begin{subarray}{c}d\in\mathbb{Z},(d,m)=1\\\
d\equiv
1\bmod{4}\end{subarray}}\frac{\mu_{\mathbb{Z}}(d)}{d^{2}}\prod_{\varpi|d}\left(1-N(\varpi)^{-1}\right)\sum_{\begin{subarray}{c}(l,md)=1\\\
l\equiv 1\bmod{(1+i)^{3}}\end{subarray}}\frac{\mu_{[i]}(l)}{N(l^{2})}.$

Computing the sum over $l$ explicitly, we obtain that

$\displaystyle\mathcal{M}_{0}$
$\displaystyle=\frac{1}{2}C_{0}\zeta^{-1}_{\mathbb{Q}(i)}(2)Q\sum_{m=1}^{\infty}\frac{\widetilde{f}(1)}{m^{2}}\prod_{\varpi|2m}\left(1-N(\varpi)^{-1}\right)\sum_{\begin{subarray}{c}d\in\mathbb{Z},(d,m)=1\\\
d\equiv
1\bmod{4}\end{subarray}}\frac{\mu_{\mathbb{Z}}(d)}{d^{2}}\prod_{\varpi|d}\left(1-N(\varpi)^{-1}\right)\prod_{\varpi|2md}\left(1-N(\varpi)^{-2}\right)^{-1}$
$\displaystyle=\frac{1}{2}C_{0}\zeta^{-1}_{\mathbb{Q}(i)}(2)Q\sum_{m=1}^{\infty}\frac{\widetilde{f}(1)}{m^{2}}\prod_{\varpi|2m}\left(1+N(\varpi)^{-1}\right)^{-1}\sum_{\begin{subarray}{c}d\in\mathbb{Z},(d,m)=1\\\
d\equiv
1\bmod{4}\end{subarray}}\frac{\mu_{\mathbb{Z}}(d)}{d^{2}}\prod_{\varpi|d}\left(1+N(\varpi)^{-1}\right)^{-1}.$

We define

$\displaystyle C_{1}=\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\equiv
1\bmod{4}\end{subarray}}\frac{\mu_{\mathbb{Z}}(d)}{d^{2}}\prod_{\varpi|d}\left(1+N(\varpi)^{-1}\right)^{-1}.$

It is clear that $C_{1}$ is a constant. Using this, we have that

$\displaystyle\mathcal{M}_{0}=\frac{1}{2}C_{0}C_{1}\zeta^{-1}_{\mathbb{Q}(i)}(2)Q\sum_{m=1}^{\infty}\frac{\widetilde{f}(1)}{m^{2}}\prod_{\varpi|2m}\left(1+N(\varpi)^{-1}\right)^{-1}\prod_{p|m/(m,2)}\left(1-p^{-2}\prod_{\varpi|p}(1+N(\varpi)^{-1})^{-1}\right)^{-1}.$

Let

$Z(u)=\sum_{m=1}^{\infty}m^{-u}\prod_{\varpi|2m}\left(1+N(\varpi)^{-1}\right)^{-1}\prod_{p|m/(m,2)}\left(1-p^{-2}\prod_{\varpi|p}(1+N(\varpi)^{-1})^{-1}\right)^{-1},$

which is holomorphic and bounded for $\Re(u)\geq 1+\delta>1$. Then

$\mathcal{M}_{0}=\frac{1}{2}C_{0}C_{1}\zeta^{-1}_{\mathbb{Q}(i)}(2)Q\frac{1}{2\pi
i}\int\limits_{(1)}A^{s}Z(2+4s)\widetilde{w}(1+s)\frac{G(s)}{s}\gamma_{1}(s)\mathrm{d}s.$

We move the contour of integration to $-1/4+\varepsilon$, crossing a pole at
$s=0$ only. The integral over the new contour is $O(A^{-1/4+\varepsilon}Q)$,
while residue of the pole at $s=0$ gives

(3.5) $\frac{1}{2}CQ\widetilde{w}(1),\quad\text{where}\quad
C=C_{0}C_{1}\zeta^{-1}_{\mathbb{Q}(i)}(2)Z(2).$

Note that $Z(u)$ converges absolutely at $u=2$ so that one may express $Z(2)$
explicitly as an Euler product if interested. We then conclude that

(3.6)
$\displaystyle\mathcal{M}_{0}=\frac{1}{2}CQ\widetilde{w}(1)+O\left(QA^{-1/4+\varepsilon}\right).$

###  3.3. Evaluating $\mathcal{M}^{+}_{1}$, the remainder term

In this section, we estimate $\mathcal{M}^{\prime}_{1}$ and
$\mathcal{M}^{+}_{1,2}$. Since the arguments are similar, we shall only
estimate $\mathcal{M}^{\prime}_{1}$ here. We bound everything with absolute
values to see that for some large $E\in\mathbb{N}$,

$|\mathcal{M}_{1}^{\prime}|\ll\sum_{d\leq c_{1}\sqrt{Q}}\sum_{N(l)\leq
c_{2}\sqrt{Q}}\frac{1}{\sqrt{N(dl^{2})}}\sum_{m}\frac{\sqrt{Q}}{\sqrt{m}}(1+m/A)^{-E}\int\limits^{\infty}_{0}\left|L(1/2+it,\psi_{m})\right|\left(1+|t|\right)^{-E}\mathrm{d}t.$

Here $c_{1}$ and $c_{2}$ are constants, chosen according to the size of the
support the weight function $w$. In view of the factor $(1+m/A)^{-E}$, we may
truncate the sum over $m$ above to $m\leq M\ll A^{1+\varepsilon}$ for
$\varepsilon>0$ with a small error.

Assuming the truth of the Lindelöf hypothesis, we have for any $\varepsilon>0$
(see [iwakow, Corollary 5.20]),

$\displaystyle|L(1/2+it,\psi_{m})|\ll(|m|(1+|t|))^{\epsilon}.$

We apply this to bound the sum over $m$ as

$\sum_{m\leq M}\frac{1}{\sqrt{m}}|L(1/2+it,\psi_{m})|\ll
M^{1/2+\varepsilon}(1+|t|)^{\varepsilon}.$

Since $d\in\mathbb{Z}$, we have $N(d)=d^{2}$ so that summing trivially over
$d$ and $l$, we obtain

$|\mathcal{M}_{1}^{\prime}|+|\mathcal{M}^{+}_{1,2}|\ll
Q^{1/2+\varepsilon}A^{1/2+\varepsilon}.$

This combined with (3.6) gives (3.1).

###  3.4. Estimating $\mathcal{M}^{+}_{2}$

Using (2.11), we have $M^{+}_{2}=M^{+}_{2,1}+M^{+}_{2,2}$, where

$\displaystyle\mathcal{M}^{+}_{2,1}$
$\displaystyle=\frac{1}{2}\sum_{m=1}^{\infty}\frac{1}{\sqrt{m}}V_{1}\left(\frac{m}{B}\right)\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}n\equiv
1\bmod(1+i)^{3}\end{subarray}}\frac{\overline{\chi}_{n}(8im)g(n)}{\sqrt{N(n)}}w\left(\frac{N(n)}{Q}\right),$
$\displaystyle\mathcal{M}^{+}_{2,2}$
$\displaystyle=\frac{1}{2}\sum_{m=1}^{\infty}\frac{1}{\sqrt{m}}V_{1}\left(\frac{m}{B}\right)\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}n\equiv
1\bmod(1+i)^{3}\end{subarray}}\frac{\overline{\chi}_{n}(-8im)g(n)}{\sqrt{N(n)}}w\left(\frac{N(n)}{Q}\right).$

To estimate the above expressions, we need the following result.

######  Lemma 3.5.

For any $l\in\mathbb{Z}[i]$, we have

$H^{\prime}(l,Q):=\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}n\in\mathbb{Z}[i]\\\
n\equiv
1\bmod(1+i)^{3}\end{subarray}}\frac{\overline{\chi}_{n}(l)g(n)}{\sqrt{N(n)}}w\left(\frac{N(n)}{Q}\right)\ll
Q^{3/4+\varepsilon}N(l)^{1/8+\varepsilon}.$

To prove Lemma 3.5, we first use (3.3) to remove the condition that $n$ has no
rational prime divisors. Note that it follows from (2.5) and the quartic
reciprocity that for $d\in\mathbb{Z},n\in\mathbb{Z}[i],(dn,2)=1,(d,n)=1$ and
$d$ primary,

$\displaystyle
g(dn)=\left(\frac{n}{d}\right)_{4}\left(\frac{d}{n}\right)_{4}g(d)g(n)=\overline{\chi}_{n}(d^{2})g(d)g(n).$

We use the notation $\widetilde{g}(d)=g(d)N(d)^{-1/2}$ so that
$|\widetilde{g}(d)|\leq 1$ by (2.3). It follows further from (2.3) that
$g(n)=0$ unless $n$ is square-free. This gives that

$H^{\prime}(l,Q)=\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\equiv
1\bmod{4}\end{subarray}}\mu_{\mathbb{Z}}(d)\tilde{g}(d)\overline{\chi}_{d}(l)H(d^{2}l,Q/d^{2}),$

where

$H(d^{2}l,X)=\sum_{\begin{subarray}{c}n\in\mathbb{Z}[i]\\\ n\equiv
1\bmod{(1+i)^{3}}\end{subarray}}\frac{\overline{\chi}_{n}(d^{2}l)g(n)}{\sqrt{N(n)}}w\left(\frac{N(n)}{X}\right).$

We estimate $H$ with the next lemma.

######  Lemma 3.6.

For any $l\in\mathbb{Z}[i]$, write $l=l_{0}l_{1}$ where $l_{0}$ is a unit
times a power of $1+i$, and $l_{1}\equiv 1\pmod{(1+i)^{3}}$. Then we have

$H(l,X)\ll
X^{1/2+\varepsilon}N(l_{1})^{1/4}+X^{3/4}N(l_{1})^{1/8+\varepsilon}.$

###### Proof.

Writing $l=l_{0}l_{1}$ as above, we set

$\overline{\left(\frac{l}{n}\right)}_{4}=\overline{\left(\frac{l_{1}}{n}\right)}_{4}\cdot\overline{\left(\frac{l_{0}}{n}\right)}_{4}.$

From the discussion in Section 2.1, the function
$\lambda(n)=\overline{\left(\frac{l_{0}}{n}\right)}_{4}$ is a Hecke character
$\pmod{16}$ of trivial infinite type. Thus

$H(l,X)=\sum_{\begin{subarray}{c}n\in\mathbb{Z}[i]\\\ n\equiv
1\bmod{(1+i)^{3}}\end{subarray}}\frac{\lambda(n)\overline{\left(\frac{l_{1}}{n}\right)}_{4}g(n)}{\sqrt{N(n)}}w\left(\frac{N(n)}{X}\right).$

Note that the identity (2.4) implies
$\overline{\left(\frac{l_{1}}{n}\right)}_{4}g(n)=g(l_{1},n)$ for
$(n,l_{1})=1$. Upon introducing the Mellin transform of $w$, we get

(3.7) $H(l,X)=\frac{1}{2\pi
i}\int\limits_{(2)}\widetilde{w}(s)X^{s}h(l_{1},1/2+s;\lambda)\mathrm{d}s,$

where $\widetilde{w}(s)$ is defined as in (3.4).

We move the line of integration in (3.7) to $\Re(s)=1/2+\varepsilon$, crossing
a pole at $s=3/4$, which contributes

$\ll X^{3/4}N(l_{1})^{1/8+\varepsilon}.$

The contribution from the new line of integration is

$\ll X^{1/2+\varepsilon}N(l_{1})^{1/4}.$

This completes the proof of Lemma 3.6. ∎

Now, to prove Lemma 3.5, we treat $|d|\leq Y$ and $|d|>Y$ separately, where
$Y$ is a parameter to be chosen later. For $|d|\leq Y$ we use Lemma 3.6, while
for $|d|>Y$ we use the trivial bound $H(l,X)\ll X$. Thus

$H^{\prime}(l,Q)\ll\sum_{|d|\leq
Y}\left(\frac{Q}{d^{2}}\right)^{1/2+\varepsilon}N(d^{2}l)^{1/4}+\sum_{|d|\leq
Y}\left(\frac{Q}{d^{2}}\right)^{3/4}N(d^{2}l)^{1/8+\varepsilon}+\sum_{|d|>Y}\frac{Q}{d^{2}},$

which simplifies as

$H^{\prime}(l,Q)\ll
Q^{1/2+\varepsilon}Y^{1-2\varepsilon}N(l)^{1/4}+QY^{-1}+Q^{3/4}N(l)^{1/8+\varepsilon}Y^{\varepsilon}.$

The optimal choice of $Y$ is $Y=Q^{1/4}N(l)^{-1/8}$ and gives Lemma 3.5.
Applying this lemma and summing trivially over $m$ in the expressions for
$M^{+}_{2,1}$ and $M^{+}_{2,2}$, one easily deduces (3.2). This completes the
proof of Lemma 3.1.

Acknowledgments. P. G. is supported in part by NSFC grant 11871082 and L. Z.
by the FRG grant PS43707 and the Faculty Goldstar Award PS53450. Parts of this
work were done when P. G. visited the University of New South Wales (UNSW). He
wishes to thank UNSW for the invitation, financial support and warm
hospitality during his pleasant stay.

## References

School of Mathematical Sciences |  School of Mathematics and Statistics
---|---
Beihang University |  University of New South Wales
Beijing 100191 China |  Sydney NSW 2052 Australia
Email<EMAIL_ADDRESS>|  Email<EMAIL_ADDRESS>


# Navigating Cultural Chasms: Exploring and Unlocking the Cultural POV of
Text-To-Image Models

Mor Ventura1 Eyal Ben-David2 Anna Korhonen3 Roi Reichart1
1Faculty of Data and Decision Sciences, Technion, IIT
2Google Research
3Language Technology Lab, University of Cambridge, UK
<EMAIL_ADDRESS>
###### Abstract

Text-To-Image (TTI) models, exemplified by DALL-E and StableDiffusion, have
recently gained prominence for their remarkable zero-shot capabilities in
generating images guided by textual prompts. Language, as a conduit of
culture, plays a pivotal role in these models’ multilingual capabilities,
which in turn shape their cultural agency. In this study, we explore the
cultural perception embedded in TTI models by characterizing culture across
three hierarchical tiers: cultural dimensions, cultural domains, and cultural
concepts. We propose a comprehensive suite of evaluation techniques, including
intrinsic evaluations using the CLIP space, extrinsic evaluations with a
Visual-Question-Answer (VQA) model, and human assessments, to discern TTI
cultural perceptions. To facilitate our research, we introduce the CulText2I
dataset, derived from four diverse TTI models and spanning ten languages. Our
experiments reveal insights into these models’ cultural awareness, cultural
distinctions, and the unlocking of cultural features, releasing the potential
for cross-cultural applications.111Our code is available at
https://github.com/venturamor/CulText-2-I

##  1 Introduction

"We seldom realize, for example that our most private thoughts and emotions
are not actually our own. For we think in terms of languages and images which
we did not invent, but which were given to us by our society.”
(watts1989taboo)

Figure 1: StableDiffusion v2.1 images of ’A photo of <city>’ , while ’city’ is
translated to (top: left to right) Arabic, Chinese, German (bottom: left to
right) English, Russian, Spanish.

In the last couple of years, generative Text-To-Image (TTI) models have
witnessed a surge in popularity, accompanied by remarkable advancements in
their zero-shot capabilities. Prominent examples of these models include
DALL-E (ramesh2021zero_delle; ramesh2022hierarchical_dalle2) and
StableDiffusion (rombach2021highresolution_sd). These models exhibit the
ability to generate images while being guided by textual prompts, offering a
distinctive visual representation of their interpretation of linguistic
concepts.

Language, as a medium, transcends mere linguistic expression; it serves as a
vessel encapsulating and transmitting the very core of cultural identity and
heritage. This perspective gains added relevance in light of recent insights
(yiu2023imitation) suggesting that AI models can be viewed as cultural
technologies that play a pivotal role in augmenting the process of cultural
transmission. TTI models, characterized mainly by their text encoders, possess
multilingual capabilities, whether acquired deliberately or incidentally
during their exposure to various languages in diverse domains. This
multilingual exposure is the stem of their cultural agency.

TTI models bridge the linguistic boundaries and convey cultural nuances
through the most universally comprehensible formats - visually. The immense
potential exhibited by these models across various domains, including art,
education, communication, and more, has led to their widespread adoption.
Consequently, they have become an integral tool for nearly every content
creator. This widespread usage has generated significant social impact and
exposure, leaving an indelible mark on various facets of our society.

In this study, we aim to gain insights into the cultural perception inherent
in TTI models. We embark on a novel characterization, dissecting the complex
relationship that exists between language, culture, and TTI models. Our
approach draws inspiration from well-established cultural research, allowing
us to systematically deconstruct the wide notion of culture across three
hierarchical tiers, progressing from the broader to the finer levels of
abstractness: cultural dimensions, cultural domains, and cultural elements.

Moreover, we put forward a comprehensive suite of evaluation techniques
designed to harness the capacity for distinguishing between the TTI cultural
perceptions. These methods consist of intrinsic evaluations using the CLIP
space, extrinsic evaluations with a Visual-Question-Answer (VQA) model, and
human assessments.

In the absence of an appropriate dataset, our research commences with the
creation of the CulText2I dataset. This dataset is constructed using images
generated by four distinct TTI models, namely StableDiffusion, AltDiffusion,
DeepFloyd, and DALL-E. These models vary in terms of their multilingual
capabilities and their architectures. The generated images originate from
various prompts that represent identical cultural concepts, spanning ten
different languages. Our study is structured around three specific research
inquiries that lay the groundwork for our overarching objectives. Firstly, we
conduct experiments to ascertain whether these models exhibit cultural or
national associations, thereby gauging their cultural awareness. Secondly, we
delve into cultural distinctions by measuring cultural dissimilarities and
projecting them onto higher-level cultural dimensions such as modernity or
tradition. Lastly, we explore a method to extract and utilize cultural
features. Our empirical results reveal six key findings: (1) Firstly, our
research underscores that TTI models go beyond constrained image generation;
they possess a unique cultural perception that permeates their visual outputs.
(2) Second, we observe that the images generated by these models transcend
linguistic boundaries, exhibiting cross-cultural similarities. However,
beneath this harmonious tapestry lies the revelation that (3) cultural
representation varies, raising questions about biases within the models’
components and the equitable representation of diverse cultures. (4) Fourth,
TTI models reflect cultural dimensions, with language acting as a conduit for
the encoding of cultural nuances. (5) The fifth insight emphasizes that
language itself wields immense power; it has a profound impact on cultural
perception, shaping how these models interpret and depict cultural elements
(as shown in Figure 1), (6) Lastly, our pioneering approach in unlocking
cultural features through alphabet characters offers a tantalizing glimpse
into the potential of TTI models to bridge cultural divides and hopefully to
gain new insight into the tangled fabric of our human experience as cultural
creators.

In the sections ahead, we embark on a rigorous examination of these insights,
delving deep into the heart of TTI models, their multilingual encoders, and
their cultural agency. Our objective is not merely to uncover the complexity
of TTI’s cultural cognition but to initiate a broader dialogue about their
ethical and social dimensions. By interrogating the cultural nuances within
TTI models, we hope to challenge existing NLP paradigms and inspire innovative
applications that harness the models’ potential for cross-cultural
communication and understanding. As we navigate this evolving landscape, we
forge a path toward a more culturally conscious and inclusive future, where
language, culture, and AI models harmoniously converge.

Figure 2: "Indian Wedding". Cultural Concept: "Wedding", Language: "Hindi". StableDiffusion images generated from all the prompt templates (PTs). Left to right: (First row) English Reference | Fully Translated PT, Translated Concept, English with Nation PT, English with Short Gibberish PT and English with Long Gibberish PT. Low language coverage in the Translated PTs.

##  2 Related Work

###  2.1 Generative Text To Image Models

Text-to-image (TTI) models have witnessed considerable progress in recent
years, trained from large text-image pairs corpora and generating high-quality
images based on textual descriptions. These models typically incorporate three
core components: an LM-based text encoder, which interprets and processes
linguistic inputs; an image generator that synthesizes corresponding visual
representations; and, optionally, an image encoder, which can further analyze
and integrate information from generated images.

Incorporating multilingual capabilities marked a significant milestone in the
evolution of TTI models. The introduction of foundational architectures of
Transformer (vaswani2017attention) and ViT (vit_han2021transformer) served as
powerful catalysts for developing versatile multilingual text encoders,
including mBERT, mT5, and XLM models (mbert; xue2020mt5; xlmr_goyal2021larger;
scao2022bloom). Moreover, the performance of TTI models saw substantial
improvements owing to the integration of contrastive objectives
(weston2011wsabie; jia2021scaling; radford2021learning) ,which fostered the
learning of enriched joint representations(clip_radford2021learning;
openclip_cherti2022reproducible), and the introduction of diffusion models,
consequently enhancing the quality and sophistication of their generative
capabilities.

Exposure to multilingual text opened the door to a wide range of cultural
influences on TTI models, which are at the heart of this study.
StableDiffusion v2 (rombach2021highresolution_sd), based on OpenCLIP
(openclip_cherti2022reproducible), leverages data from LAION’s datasets
(schuhmann2022laion), filtering content based on quality and language,
hovering over English and non-English languages, namely Russian, French,
German, Spanish, and Chinese. Concurrently, AltCLIP (chen2022altclip) has
successfully amalgamated language-image representations of the CLIP model with
XLM-R, thereby enabling a visually enhanced multilingual understanding.
Meanwhile, Dall-E (ramesh2022hierarchical_dalle2), a leading off-the-shelf TTI
system, also presents multilingual properties, but its training corpus is not
publicly disclosed.

DeepFloyd IF is based of a frozen text encoder T5-XXL (raffel2020exploring),
an architecture that was inspired by Imagen (saharia2022photorealistic), and
presents partial multilingual capabilities, namely Spanish, Russian, German
and French. While this work experiments with the prominent multilingual models
presented above, extensive research in the field introduces additional models.
Finally, we find that monolingual TTI models also display a profound ability
to embody cross-lingual characteristics, thereby opening a window to the
diverse cultural influences embedded in various languages.

#### Foundational Frameworks in Cultural Studies

rokeach1967rokeach introduced the Rokeach Value Survey (RVS), illuminating how
classified values shape behaviors at both individual and societal levels.
Building on this, hofstede1983dimensions laid groundbreaking work in studying
cultural variances by introducing key cultural dimensions like individualism
versus collectivism, fostering a systematic approach to exploring cultural
differences. bond1988finding identified 36 "universal values", such as love
and freedom, underscoring the shared human values across diverse cultures and
regions. schwartz1994beyond further refined our understanding by proposing a
universal framework of ten fundamental human values, revealing how they are
prioritized and interpreted diversely across cultures, impacting behaviors and
attitudes. Later advancements by triandis1998converging and mccrae2002five
further enriched the domain of cross-cultural psychology. Triandis emphasized
the nuances of individualism and collectivism, while McCrae delved into the
variances in the manifestations of the Big Five personality traits across
different cultural settings. Complementing these, the World Values Survey
(WSV, WVS_Survey) introduced an innovative cultural map, portraying the global
shift towards more secular and self-expression values as societies advance and
prosper.

This work synthesizes the cultural literature presented above to establish a
comprehensive repository of cultural concepts and dimensions. In
§LABEL:sec:culture, we introduce our foundational culture characterization
that allows meticulous analyses of the representation and perception of
cultural perspectives within TTI models.

###  2.2 Culture in language and TTI models

With the burgeoning interest in PLMs and TTI generation models, there is
increasing scrutiny on the cultural gaps and biases inherent within these
models. These gaps manifest as discrepancies in the representation of norms,
values, beliefs, and practices across diverse cultures
(Cultural_Incongruencies; struppek2022biased).

Research on the cultural implications of PLMs has garnered considerable
attention, focusing on the nuances of cultural biases and diversities inherent
in these models (DBLP:journals/natmi/AbidFZ21; DBLP:conf/emnlp/AhnO21;
touileb2022occupational; smith-etal-2022-im). arora2022probing and
ramezani2023knowledge explore the cross-cultural values and moral norms in
PLMs and assess their alignment with theoretical frameworks, revealing a
conspicuous inclination towards Western norms. Similarly, other studies
demonstrate the challenges with English probes and monolingual LMs, which
diminish the representation of non-Western (e.g., Arab) norms in model
responses (cao2023assessing; naous2023having). In particular,
masoud2023cultural highlighted the pronounced alignment of GPT-3.5 and GPT-4
with U.S. cultural values and their inadequate reflection of the values from
countries like Saudi Arabia, China, and Slovakia. Lastly, (atari2023humans)
corroborates the overrepresentation of Western, Educated, Industrialized,
Rich, and Democratic populations in the training data of LLMs, emphasizing the
profound impact of such biases on models’ perceptions and representations of
culture.

While these works focus on the cultural implications of PLMs, we explore the
cultural aspect of TTI models. We develop a methodology to systematically
investigate the cultural values of TTI models. In addition, the above works
focus on detecting and mitigating cultural biases. Instead, our methodology
points to inspect the cultural values of TTI models, and seek to understand
how TTI models represent different cultures.

The exploration of culture is comparatively less traversed in research
pertaining to TTI models. Despite this scarcity, advancements have been made
to amplify cultural diversity within these models. liu2023cultural and
liu2021visually have pioneered new benchmarks to address this concern: the
former emphasizes the incorporation of Chinese culture, while the latter
broadens the scope to Western European/American cultures, forging a
multilingual resource. Concurrently, diverse efforts have been channeled to
unearth cultural biases, with studies like cho2022dalleval unveiling skin tone
biases and bird2023typology probing the accommodated risks. Furthermore,
naik2023social have undertaken an endeavor closely aligned with ours,
executing a comprehensive analysis of social bias representation in TTI models
across varied dimensions such as gender, race, age, and geographical location.
This is facilitated by designing template-based prompts in English that
meticulously isolate the variable under examination, be it gender, skin tone,
or geographic location.

Our work, however, differs from naik2023social in a few key areas. Firstly,
our emphasis is predominantly on the cultural dimensions of TTI models rather
than social ones. Secondly, we employ multilingual prompts to delve deeper
into the cultural nuances, acknowledging the pivotal role of text language in
channeling the model to represent diverse cultures. Finally, we establish a
foundational framework of cultural characterization, drawing insights from
seminal cultural research studies. This framework elucidates cultural concepts
and dimensions (see §LABEL:sec:culture), providing a comprehensive lens
through which the representation of culture in TTI models can be understood
and analyzed.

Table 1: Prompt Templates