# Sharp upper bounds for moments of quadratic Dirichlet $L$-functions

Peng Gao  School of Mathematical Sciences, Beihang University, Beijing 100191,
P. R. China<EMAIL_ADDRESS>
###### Abstract.

We establish unconditional sharp upper bounds of the $k$-th moments of the
family of quadratic Dirichlet $L$-functions at the central point for $0\leq
k\leq 2$.

Mathematics Subject Classification (2010): 11M06

Keywords: moments, quadratic Dirichlet $L$-functions, upper bounds

##  1\. Introduction

Moments of central values of families of $L$-functions have attracted much
attention in research as they can be applied to address the non-vanishing
issue of these values, which in turn carry significant arithmetic
implications. Much progress has been made in late years that largely enhances
our understanding of these moments. A conjecture concerning asymptotic
expressions for the moments of various families of $L$-functions has been made
by J. P. Keating and N. C. Snaith in [Keating-Snaith02], in connection with
random matrix theory. Another conjecture of J. B. Conrey, D. W. Farmer, J. P.
Keating, M. O. Rubinstein and N. C. Snaith in [CFKRS] gives more precise
predictions by including lower order terms on the asymptotic behaviors of the
moments for certain families of $L$-functions.

In [R&Sound] and [R&Sound1], Z. Rudnick and K. Soundararajan developed a
simple and powerful method towards establishing lower bounds for rational
moments of families of $L$-functions of the conjectured order of magnitude.
The method is further extended by M. Radziwiłł and K. Soundararajan in
[Radziwill&Sound1] to all moments larger than the first. On the other hand, an
approach due to K. Soundararajan in [Sound01] enables one to obtain the
corresponding upper bounds under the generalized Riemann hypothesis (GRH).
This approach was further sharpened by A. J. Harper [Harper] to give upper
bounds of desired order of magnitude for moments of $L$-functions in many
families conditionally.

In [Radziwill&Sound], M. Radziwiłł and K. Soundararajan developed a new
principle that allows one to seek upper bounds of all smaller moments with the
knowledge of an upper bound for a particular moment. This principle is further
implemented by W. Heap and K. Soundararajan in [H&Sound] to treat the case of
lower bounds.

In this paper, we are interested in the family of quadratic Dirichlet
$L$-functions. Asymptotic formulas for the first two moments of this family
was obtained by M. Jutila in [Jutila] with the error terms being subsequently
improved in [ViTa, DoHo, Young1, sound1, Sono]. For the purpose of this paper,
we are interested in the family given by $\\{L(s,\chi_{8d})\\}$ for $d$ being
odd and square-free integers. Here $\chi_{8d}=\left(\frac{8d}{\cdot}\right)$
is the Kronecker symbol. The study of the above family was initiated by K.
Soundararajan in [sound1], who obtained mollified first two moments of the
family to show that at least $87.5\%$ of the members of this family have non-
vanishing central values. The third moment of the above family is also
obtained for the first time in [sound1]. The error term was improved by M. P.
Young in [Young2] for a smoothed version. In [Shen], Q. Shen obtained the
fourth moment of the above family under GRH.

Asymptotic formulas for all positive real moments of
$\\{L(\tfrac{1}{2},\chi_{8d})\\}$ are conjectured by J. C. Andrade and
Keating, J. P. in [Andrade-Keating01]. Combining the above mentioned work of
[Harper, Sound01, R&Sound, R&Sound1, Radziwill&Sound1], we now know that

$\displaystyle X(\log
X)^{\frac{k(k+1)}{2}}\ll_{k}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}0<d<X\\\
(d,2)=1\end{subarray}}|L(\tfrac{1}{2},\chi_{8d})|^{k}\ll_{k}X(\log
X)^{\frac{k(k+1)}{2}},$

where the lower bound above holds for all real $k\geq 1$ unconditionally and
the upper bound above holds for all $k\geq 0$ under GRH. Here and throughout
the paper, we denote $\sideset{}{{}^{*}}{\sum}$ for the sum over square-free
integers.

It is our goal in this paper to apply the principal of obtaining upper bounds
given in [Radziwill&Sound] by M. Radziwiłł and K. Soundararajan in our
setting. Before we state our result, we would like to recall the principle.
Although the principle works for general $L$-functions, we only adapt it for
the family $\\{L(s,\chi_{8d})\\}$ for simplicity. For this, we recall that a
result of B. Hough [Hough] shows that under GRH, the quantity
$\log|L(\tfrac{1}{2},\chi_{8d})|$ with $X<d\leq 2X$ for large $X$ is normally
distributed with mean $\tfrac{1}{2}\log\log d$ and variance $\log\log d$. On
the other hand, it is shown in [Hough] that the sum

(1.1) $\displaystyle\sum_{n\leq
z}\frac{\Lambda(n)\chi_{8d}(n)}{n^{\frac{1}{2}}}$

is similarly distributed as $\log|L(\tfrac{1}{2},\chi_{8d})|$ for
$z=X^{1/(\log\log X)^{2}}$, say. Here $\Lambda(n)$ is the von Mangoldt
function. As the contribution of prime powers $\geq 3$ is negligible in the
above sum and the contribution of prime squares is about $\log\log X$, we see
that the difference between the expression given in (1.1) and $\log\log X$ is
mainly determined by

$\displaystyle{\mathcal{P}}(d)=\sum_{p\leq z}\frac{1}{\sqrt{p}}\chi_{8d}(p).$

Taking exponentials, this implies that the quantity
$|L(\tfrac{1}{2},\chi_{8d})|(\log|d|)^{1/2}\exp(-{\mathcal{P}}(d))$ is usually
small. Thus, for any given real numbers $n>0,0<k<1$, we may estimate the
quantity $(|L(\tfrac{1}{2},\chi_{8d})|(\log|d|^{1/2})^{nk}$ by writing it as

$\displaystyle\Big{(}|L(\tfrac{1}{2},\chi_{8d})|(\log|d|)^{1/2}\Big{)}^{nk}=|L(\tfrac{1}{2},\chi_{8d})|^{nk}(\log|d|)^{nk/2}\exp(-nk(1-k){\mathcal{P}}(d))\cdot\exp(nk(1-k){\mathcal{P}}(d)).$

We now recall that Young’s inequality asserts that we have $ab\leq
a^{p}/p+b^{q}/q$ for real numbers $a,b\geq 0$ and real numbers $p,q\geq 1$
satisfying $1/p+1/q=1$. Applying this with
$a=|L(\tfrac{1}{2},\chi_{8d})|^{nk}(\log|d|)^{nk/2}\exp(-nk(1-k){\mathcal{P}}(d)),b=\exp(nk(1-k){\mathcal{P}}(d))$
and $p=1/k,q=1/(1-k)$, we see that

(1.2)
$\displaystyle\Big{(}|L(\tfrac{1}{2},\chi_{8d})|(\log|d|)^{1/2}\Big{)}^{nk}\leq
k|L(\tfrac{1}{2},\chi_{8d})|^{n}(\log|d|)^{n/2}\exp(-n(1-k){\mathcal{P}}(d))+(1-k)\exp(kn{\mathcal{P}}(d)).$

As we expect
$|L(\tfrac{1}{2},\chi_{8d})|^{n}(\log|d|)^{n/2}\exp(-n{\mathcal{P}}(d))$ to be
small most of the time, the right side of (1.2) should be bounded above by
$\exp(kn{\mathcal{P}}(d))$ on average. As expanding $\exp(kn{\mathcal{P}}(d))$
into Euler products leads to a too long Dirichlet polynomial to estimate, we
approximate it by taking a suitably long Taylor expansion to achieve our goal
since ${\mathcal{P}}(d)$ is often small in size.

We note that the bound given in (1.2) may be modified to establish similar
bounds for general $L$-functions. If we further know that the corresponding
$L$-function satisfies $L(\tfrac{1}{2})\geq 0$, then we may replace
$|L(\tfrac{1}{2})|^{n}$ on the right side of (1.2) by $L(\tfrac{1}{2})^{n}$ to
see that if we have a good understanding of the $n$-th moment (twisted by
another character) of the corresponding family of $L$-functions at the central
value, then we shall be able to obtain sharper upper bounds for every $m$-th
moment of the same family of $L$-functions with $0\leq m\leq n$. This is
precisely what has been carried out by Radziwiłł and Soundararajan in
[Radziwill&Sound] to treat the moments of quadratic twists of $L$-functions
attached to elliptic curves, since in this case it is known that the
corresponding $L$-functions have non-negative values at $1/2$ and the first
moment of the family can be evaluated.

For the family of quadratic twists of Dirichlet $L$-functions, although our
current stage of knowledge is short of determining whether
$L(\tfrac{1}{2},\chi_{8d})\geq 0$ is always valid, we do know that these
values are real so that $L(\tfrac{1}{2},\chi_{8d})^{2}\geq 0$ is always true.
Thanks to the work of K. Soundararajan in [sound1], we also have a good
knowledge of the twisted second moment of the same family. Combining these
with the above mentioned principle of Radziwiłł and Soundararajan, we are
therefore able to establish unconditionally the correct order of magnitude of
every moment less than the second of the family of quadratic Dirichlet
$L$-functions. This is given in the following result.

######  Theorem 1.1.

Unconditionally, for every $0\leq k\leq 2$, we have

(1.3) $\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}0<d<X\\\
(d,2)=1\end{subarray}}|L(\tfrac{1}{2},\chi_{8d})|^{k}\ll_{k}X(\log
X)^{\frac{k(k+1)}{2}}.$

We notice that the twisted third moment of the quadratic family of Dirichlet
$L$-functions has been evaluated by M. P. Young in [Young2]. Thus, if one
assumes that $L(\tfrac{1}{2},\chi_{8d})\geq 0$ for all $d$ under consideration
(which follows from GRH), then we are able to obtain the following result.

######  Theorem 1.2.

Assume that $L(\tfrac{1}{2},\chi_{8d})\geq 0$ for all odd, square-free $d$.
Then the bound given in (1.3) is valid for every $0\leq k\leq 3$. In
particular, this is true under GRH.

We omit the proof of Theorem 1.2 in the paper as it is similar to that of
Theorem 1.1. We also notice that using the approach of Radziwiłł and
Soundararajan in [Radziwill&Sound], we may be able to give an alternative
proof of a result of B. Hough [Hough, Corollary 1.1] that says the
distribution of logarithms of central values of $L(\tfrac{1}{2},\chi_{8d})$ is
bounded above by the Gaussian distribution.

We would like to point out here that in order for the principle of Radziwiłł
and Soundararajan on obtaining upper bounds for moments of $L$-functions to
work, one in general needs to evaluate a given moment of $L$-functions twisted
by certain character instead of just evaluating the moment itself. Previously,
it is known that these twisted moments play vital roles when using mollifiers
to study the non-vanishing issues of central values of $L$-functions. They are
also necessary in the work of M. P. Young [Young1, Young2] to use a recursive
method to reduce the sizes of error terms in the moments of $L$-functions at
the central point. Now, the principle of Radziwiłł and Soundararajan provides
a further evidence on the importance of studying these twisted moments.

In the last section of this paper, we propose a variant of the above mentioned
principle of Radziwiłł and Soundararajan for obtaining upper bounds of moments
of $L$-functions. This may potentially lead to a slightly simpler treatment
(at least in some cases) when acquiring such upper bounds.

##  2\. Preliminaries

We include here some tools needed in our proof of Theorem 1.1 together with an
initial treatment of the proof.

###  2.1. Tools

From now on, we reserve the letter $p$ for a prime number and we recall the
following well-known Mertens’ formula (see [MVa1, Theorem 2.7]) and a
consequence of it (via partial summation).

######  Lemma 2.2.

Let $x\geq 2$. We have, for some constant $b$,

$\sum_{p\leq x}\frac{1}{p}=\log\log x+b+O\Big{(}\frac{1}{\log x}\Big{)}.$

Also, for any integer $j\geq 1$, we have

$\sum_{p\leq x}\frac{(\log p)^{j}}{p}=\frac{(\log x)^{j}}{j}+O((\log
x)^{j-1}).$

Now, we denote $\Phi$ for a smooth, non-negative function compactly supported
on $[1/2,5/2]$ with $\Phi(x)=1$ for $x\in[1,2]$, and define, for any complex
number $s$,

${\widehat{\Phi}}(s)=\int_{0}^{\infty}\Phi(x)x^{s}\frac{dx}{x}.$

We define $\delta_{n=\square}$ to be $1$ when $n=\square$ and $0$ otherwise,
where we write $\square$ for a perfect square. Similar to the proof of
[Radziwill&Sound, Proposition 1], we have the following result concerning a
smoothed sum of quadratic characters.

######  Lemma 2.3.

For large $X$ and any odd positive integer $n$, we have

$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}(d,2)=1\end{subarray}}\chi_{8d}(n)\Phi\Big{(}\frac{d}{X}\Big{)}=\displaystyle\delta_{n=\square}{\widehat{\Phi}}(1)\frac{2X}{3\zeta(2)}\prod_{p|n}\Big{(}\frac{p}{p+1}\Big{)}+O(X^{\frac{1}{2}+\epsilon}\sqrt{n}).$

We denote further $d(n)$ for the divisor function and $\sigma(n)$ for the sum
of the positive divisors of $n$. Also define $\Lambda_{j}(n)$ for all integers
$j\geq 0$ to be the coefficient of $n^{-s}$ in the Dirichlet series expansion
of $(-1)^{j}\zeta^{(j)}(s)/\zeta(s)$. Note that this implies that
$\Lambda_{1}(n)=\Lambda(n)$ and that $\Lambda_{j}(n)$ is supported on integers
having at most $j$ distinct prime factors such that
$\Lambda_{j}(n)\ll_{j}n^{j}$.

Combining Proposition 1.1 and Proposition 1.3 in [sound1] and setting
$Y=X^{1/4},M=1$ there, we readily deduce the following asymptotic result
concerning the twisted second moment of quadratic Dirichlet $L$-functions.

######  Lemma 2.4.

Writing any odd $l$ as $l=l_{1}l^{2}_{2}$ with $l_{1}$ square-free, we have
for any $\varepsilon>0$,

$\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{(d,2)=1}L(\tfrac{1}{2},\chi_{8d})^{2}\chi_{8d}(l)\Phi(\frac{d}{X})=&\frac{D\widehat{\Phi}(1)}{36\zeta(2)}\frac{d(l_{1})}{\sqrt{l_{1}}}\frac{l_{1}}{\sigma(l_{1})h(l)}X\Big{(}\log^{3}\Big{(}\frac{X}{l_{1}}\Big{)}-3\sum_{\begin{subarray}{c}p|l_{1}\end{subarray}}\log^{2}p\log\Big{(}\frac{X}{l_{1}}\Big{)}+O(l)\Big{)}+O\left(X^{\frac{3}{4}+\varepsilon}l^{\tfrac{1}{2}+\varepsilon}_{1}\right),\end{split}$

where $D=\frac{1}{8}\displaystyle\prod_{\begin{subarray}{c}p\geq
3\end{subarray}}\left(1-\frac{1}{p}\right)h(p)$ and $h$ is the multiplicative
function defined on prime powers by

$\displaystyle h(p^{k})=1+\frac{1}{p}+\frac{1}{p^{2}}-\frac{4}{p(p+1)},\quad
k\geq 1.$

Also,

$\displaystyle O(l)=$
$\displaystyle\sum^{3}_{j,k=0}\sum_{\begin{subarray}{c}m|l_{1}\end{subarray}}\sum_{\begin{subarray}{c}n|l_{1}\end{subarray}}\frac{\Lambda_{j}(m)}{m}\frac{\Lambda_{k}(n)}{n}D(m,n)Q_{j,k}\Big{(}\log\frac{X}{l_{1}}\Big{)}-3\Big{(}A+B\frac{\widehat{\Phi}^{\prime}(1)}{\widehat{\Phi}(1)}\Big{)}\sum_{\begin{subarray}{c}p|l\end{subarray}}\log^{2}p,$

where $A$ and $B$ are absolute constants and $D(m,n)\ll 1$ uniformly for all
$m$ and $n$. The $Q_{j,k}$ are polynomials of degree $\leq 2$ whose
coefficients involve only absolute constants and linear combinations of
$\frac{\widehat{\Phi}^{(j)}(1)}{\widehat{\Phi}(1)}$ for $1\leq j\leq 3$.

Lastly, we define for any non-negative integer $\ell$ and any real number $x$,

(2.1) $E_{\ell}(x)=\sum_{j=0}^{\ell}\frac{x^{j}}{j!}.$

We recall the following two key inequalities from [Radziwill&Sound, Lemma 1,
Lemma 2].

######  Lemma 2.5.

Let $\ell\geq 0$ be an even integer. The function $E_{\ell}(x)$ is positive,
convex and satisfies $E_{\ell}(x)\geq e^{x}$ for $x\leq 0$. Moreover, for
$x\leq\ell/e^{2}$, we have

$e^{x}\leq\Big{(}1+\frac{e^{-\ell}}{16}\Big{)}E_{\ell}(x).$

######  Lemma 2.6.

Let $x_{1}$, $\ldots$, $x_{R}$ be real numbers and let
$C=\exp((e^{-\ell_{1}}+\ldots+e^{-\ell_{R}})/16)$, where $\ell_{1}$,
$\ldots,\ell_{R}$ are positive even integers. Then we have for any $y\geq 0$
and $0\leq k\leq 1$,

$\displaystyle y^{k}\leq$ $\displaystyle
Cky\prod_{j=1}^{R}E_{\ell_{j}}((k-1)x_{j})+C(1-k)\prod_{j=1}^{R}E_{\ell_{j}}(kx_{j})$
$\displaystyle+\sum_{r=0}^{R-1}\Big{(}Cky\prod_{j=1}^{r}E_{\ell_{j}}((k-1)x_{j})+C(1-k)\prod_{j=1}^{r}E_{\ell_{j}}(kx_{j})\Big{)}\Big{(}\frac{e^{2}x_{r+1}}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}.$

###  2.7. Initial treatment of the proof of Theorem 1.1

Let $X$ be a large number and $\\{\ell_{j}\\}_{1\leq j\leq R}$ be a sequence
of even natural numbers defined by $\ell_{1}=2\lceil 100\log\log X\rceil$ and
$\ell_{j+1}=2\lceil 100\log\ell_{j}\rceil$ for $j\geq 1$. Here we choose $R$
to be the largest natural number such that $\ell_{R}>10^{4}$ and observe that
we have $\ell_{j}>\ell_{j+1}^{2}$ for all $1\leq j\leq R-1$. Let ${P}_{1}$ be
the set of odd primes below $X^{1/\ell_{1}^{2}}$. For $2\leq j\leq R$, we
define ${P_{j}}$ to be the set of primes lying in the interval
$(X^{1/\ell_{j-1}^{2}},X^{1/\ell_{j}^{2}}]$. We also define

${\mathcal{P}}_{j}(d)=\sum_{p\in P_{j}}\frac{1}{\sqrt{p}}\chi_{8d}(p).$

Given a real number $\alpha$, we denote

(2.2)
$\displaystyle{\mathcal{N}}_{j}(d,\alpha)=E_{\ell_{j}}(\alpha{\mathcal{P}}_{j}(d)),\quad\mathcal{N}(d,\alpha)=\prod_{j=1}^{R}{\mathcal{N}}_{j}(d,\alpha).$

We further set for two real numbers $n,k$ satisfying $0\leq k\leq 1$,

${\mathcal{A}}_{j}(d)={\mathcal{N}}_{j}(d,(k-1)n),\quad{\mathcal{B}}_{j}(d)={\mathcal{N}}_{j}(d,nk).$

We apply Lemma 2.6 with $y=|L(\frac{1}{2},\chi_{8d})|^{n}(\log
d)^{-\frac{n}{2}}$ and $x_{j}=n{\mathcal{P}}_{j}(d)$ to obtain the following
bounds for the moments of $L(\frac{1}{2},\chi_{8d})$.

######  Proposition 2.8.

With notations as above, we have

(2.3) $\displaystyle\begin{split}\Big{(}|L(\frac{1}{2},\chi_{8d})|^{n}(\log
d)^{-\frac{n}{2}}\Big{)}^{k}\leq&Ck|L(\frac{1}{2},\chi_{8d})|^{n}(\log
d)^{-\frac{n}{2}}\Big{(}\prod_{j=1}^{R}{\mathcal{A}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}\\\
&+C(1-k)\Big{(}\prod_{j=1}^{R}{\mathcal{B}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{B}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}.\end{split}$

Arguing similar to [Radziwill&Sound, Section 3.4], we see that in order to
prove Theorem 1.1, it suffices to show that the right side of (2.3) averaged
over $d$ is $\ll X(\log X)^{\frac{(nk)^{2}}{2}}$. We close this section by
giving such an estimation for the terms involving with ${\mathcal{B}}_{j}(d)$.

######  Proposition 2.9.

With notations as above, we have

$\sideset{}{{}^{*}}{\sum}_{(d,2)=1}\Big{(}\prod_{j=1}^{R}{\mathcal{B}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{B}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}\Phi\Big{(}\frac{d}{X}\Big{)}\ll
X(\log X)^{\frac{(nk)^{2}}{2}}.$

###### Proof.

Let $w(n)$ be the multiplicative function defined by $w(p^{\alpha})=\alpha!$
for prime powers $p^{\alpha}$ and let $\Omega(n)$ denote the number of
distinct prime powers dividing $n$. We also define functions
$b_{j}(n),p_{j}(n)$ for $1\leq j\leq R$ such that $b_{j}(n),p_{j}(n)=0$ or
$1$, and we have $b_{j}(n)=1$ ($p_{j}(n)=1$) if and only if $n$ is composed of
at most (exactly, counted with multiplicity) $\ell_{j}$ primes, all from the
interval $P_{j}$ . Using these notations, we see that

(2.4)
${\mathcal{B}}_{j}(d)=\sum_{n_{j}}\frac{1}{\sqrt{n_{j}}}\frac{(nk)^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\chi_{8d}(n_{j}),\quad\frac{1}{\ell_{j}!}{\mathcal{P}}_{j}(d)^{\ell_{j}}=\sum_{n_{j}}\frac{1}{w(n_{j})\sqrt{n_{j}}}p_{j}(n_{j})\chi_{8d}(n_{j}),\quad
1\leq j\leq R.$

We note here that both ${\mathcal{B}}_{j}(d)$ and
${\mathcal{P}}_{j}(d)^{\ell_{j}}$ are short Dirichlet polynomials since
$b_{j}(n_{j}),p_{j}(n_{j})=0$ unless
$n_{j}\leq(X^{1/\ell_{j}^{2}})^{\ell_{j}}=X^{1/\ell_{j}}$. It follows that the
expressions
$\prod_{j=1}^{R}{\mathcal{B}}_{j}(d),\prod_{j=1}^{r}{\mathcal{B}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}(d)$
are all short Dirichlet polynomials of length at most
$X^{1/\ell_{1}+\ldots+1/\ell_{R}}<X^{1/1000}$.

We expand the term
$\prod_{j=1}^{r}{\mathcal{B}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}(d)$ for
some $0\leq r\leq R-1$ using (2.4) and apply Lemma 2.3 to estimate it. By
doing so, we may ignore the error term in Lemma 2.3 as both
${\mathcal{B}}_{j}$ and ${\mathcal{P}_{j}}^{\ell_{j}}(d)$ are short Dirichlet
polynomials. Considering the main term contributions from Lemma 2.3, we see
that

$\displaystyle\prod_{j=1}^{r}{\mathcal{B}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}\ll
X$
$\displaystyle\prod_{j=1}^{r}\Big{(}\sum_{n_{j}=\square}\frac{1}{\sqrt{n_{j}}}\frac{(nk)^{\Omega(n_{j})}}{w(n_{j})}\prod_{p|n_{j}}\Big{(}\frac{p}{p+1}\Big{)}b_{j}(n_{j})\Big{)}$
$\displaystyle\times\Big{(}\ell_{r+1}!\sum_{n_{r+1}=\square}\frac{1}{w(n_{r+1})\sqrt{n_{r+1}}}\prod_{p|n_{r+1}}\Big{(}\frac{p}{p+1}\Big{)}p_{r+1}(n_{r+1})\Big{)}.$

The proof of the proposition now follows by arguing in the same way as in the
proof of Proposition 4 in [Radziwill&Sound]. ∎

##  3\. Proof of Theorem 1.1

In view of our discussions in the previous section, it remains to show that
the right side of (2.3) averaged over $d$ for the terms involving with
${\mathcal{A}}_{j}(d)$ is also $\ll X(\log X)^{\frac{(nk)^{2}}{2}}$ for $n=2$.
As our approach here may be applied to treat other values of $n$, we shall
retain the symbol $n$ in most of the places in the rest of the section instead
of specitying it to be $2$. Thus, to conclude the proof of Theorem 1.1, it
suffices to establish the following result.

######  Proposition 3.1.

With notations as above, we have for $n=2$,

(3.1)
$\displaystyle\sideset{}{{}^{*}}{\sum}_{(d,2)=1}|L(\frac{1}{2},\chi_{8d})|^{n}\Big{(}\prod_{j=1}^{R}{\mathcal{A}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}\Phi\Big{(}\frac{d}{X}\Big{)}\ll
X(\log X)^{\frac{(nk)^{2}+n}{2}}.$

In the remaining of this section, we give a proof of Proposition 3.1. First
note that we may replace $|L(\frac{1}{2},\chi_{8d})|^{n}$ by
$L(\frac{1}{2},\chi_{8d})^{n}$ when $n=2$ since $L(\frac{1}{2},\chi_{8d})$ is
real. As the arugments are similar, it suffices to show that

(3.2) $\displaystyle\sideset{}{{}^{*}}{\sum}_{(d,2)=1}$ $\displaystyle
L(\tfrac{1}{2},\chi_{8d})^{n}\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Phi\Big{(}\frac{d}{X}\Big{)}\ll
X(\log X)^{\frac{(nk)^{2}+n}{2}}.$

Note first that we have

${\mathcal{A}}_{j}(d)=\sum_{n_{j}}\frac{1}{\sqrt{n_{j}}}\frac{(n(k-1))^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\chi_{8d}(n_{j}),\quad
1\leq j\leq R.$

Analogue to our discussions above, the product
$\prod_{j=1}^{r}{\mathcal{A}}_{j}(d){\mathcal{P}}_{{r+1}}^{\ell_{r+1}}(d)$ for
all $0\leq r\leq R-1$ are short Dirichlet polynomials of length at most
$X^{1/1000}$.

We now apply Lemma 2.4 to evaluate
$\prod_{j=1}^{r}{\mathcal{A}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}$(d) for
some $0\leq r\leq R-1$ by expanding it into Dirichlet series. Once again we
may focus only on the main term to see that, upon writing
$n_{j}=(n_{j})_{1}(n_{j})_{2}^{2}$ with $(n_{j})_{1}$ being square-free,

(3.3)
$\displaystyle\begin{split}&\sideset{}{{}^{*}}{\sum}_{(d,2)=1}L(\tfrac{1}{2},\chi_{8d})^{n}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}(d)\Phi\Big{(}\frac{d}{X}\Big{)}\\\
\ll&X\sum_{n_{1},\cdots,n_{r+1}}\Big{(}\prod_{j=1}^{r}\frac{1}{\sqrt{n_{j}(n_{j})_{1}}}\frac{(n(k-1))^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\frac{d((n_{j})_{1})(n_{j})_{1}}{\sigma((n_{j})_{1})h(n_{j})}\Big{)}\Big{(}\frac{\ell_{r+1}!}{\sqrt{n_{r+1}(n_{r+1})_{1}}}\frac{p_{r+1}(n_{r+1})}{w(n_{r+1})}\frac{d((n_{r+1})_{1})(n_{r+1})_{1}}{\sigma((n_{r+1})_{1})h(n_{r+1})}\Big{)}\\\
&\times\Big{(}\log^{3}\Big{(}\frac{X}{(n_{1})_{1}\cdots(n_{r+1})_{1}}\Big{)}-3\sum_{\begin{subarray}{c}p|(n_{1})_{1}\cdots(n_{r+1})_{1}\end{subarray}}\log^{2}p\log\Big{(}\frac{X}{(n_{1})_{1}\cdots(n_{r+1})_{1}}\Big{)}+O(n_{1}\cdots
n_{r+1})\Big{)}.\end{split}$

As the estimations are similar, we may consider the above sums involving with
the terms $\log^{3}(X/((n_{1})_{1}\cdots(n_{r+1})_{1}))=(\log
X-\log((n_{1})_{1}\cdots(n_{r+1})_{1}))^{3}$ only. Upon expanding, we observe
that we can write $\log^{3}(X/((n_{1})_{1}\cdots(n_{r+1})_{1}))$ as linear
combinations of the sum:

(3.4) $\displaystyle C(m_{0},\ldots,m_{r+1})(\log
X)^{m_{0}}\sum_{\begin{subarray}{c}p_{i}|(n_{i})_{1}\\\ 1\leq i\leq
r+1\end{subarray}}\prod_{1\leq i\leq r+1}(\log p_{i})^{m_{i}},$

where $m_{j},0\leq j\leq r+1$ are non-negative integers satisfying
$\sum_{0\leq j\leq r+1}m_{j}=3$ and where $C(m_{0},\ldots,m_{r+1})$ are
bounded constants. Without loss of generality, we may group terms to consider
the total contribution to (3.3) from all terms of the above form corresponding
to $m_{i_{1}}=m_{i_{2}}=m_{i_{3}}=1$ for some $1\leq i_{1}<i_{2}<i_{3}\leq
r+1$. For example, when the corresponding $p_{i_{1}}\in P_{1},p_{i_{2}}\in
P_{2},p_{i_{3}}\in P_{r+1}$, the contribution is

(3.5) $\displaystyle\begin{split}\ll&\sum_{l_{1},l_{2},l_{3}\geq
0}\prod^{3}_{s=1}\Big{(}\frac{\log
p_{i_{s}}}{p^{l_{s}+1}_{i_{s}}}\frac{|(n(k-1))|^{2l_{s}+1}}{(2l_{s}+1)!}\frac{np_{i_{s}}}{(p_{i_{s}}+1)h(p^{2l_{s}+1}_{i_{s}})}\Big{)}\\\
&\times\prod_{j=1}^{r}\Big{(}\sum_{(n_{j},p_{i_{1}}p_{i_{2}})=1}\frac{1}{\sqrt{n_{j}(n_{j})_{1}}}\frac{(n(k-1))^{\Omega(n_{j})}}{w(n_{j})}\widetilde{b}_{j,l_{1},l_{2}}(n_{j})\frac{d((n_{j})_{1})(n_{j})_{1}}{\sigma((n_{j})_{1})h(n_{j})}\Big{)}\\\
&\times\Big{(}\ell_{r+1}!\sum_{(n_{r+1},p_{i_{3}})=1}\frac{1}{\sqrt{n_{r+1}(n_{r+1})_{1}}}\frac{p_{r+1}(n_{r+1}p^{2l_{3}+1}_{i_{3}})}{w(n_{r+1})}\frac{d((n_{r+1})_{1})(n_{r+1})_{1}}{\sigma((n_{r+1})_{1})h(n_{r+1})}\Big{)},\end{split}$

where we define
$\widetilde{b}_{j,l_{1},l_{2}}(n_{j})=b_{j}(n_{j}p^{l_{j}}_{i_{j}})$ for
$j=1,2$ and $\widetilde{b}_{j,l_{1},l_{2}}(n_{j})=b_{j}(n_{j})$ otherwise.

Let us consider the sum over $n_{1}$ in (3.5). If we replace the factor
$\widetilde{b}_{l,l_{1},l_{2}}(n_{1})$ by $1$, then the sum becomes

(3.6) $\displaystyle\begin{split}&\prod_{\begin{subarray}{c}p\in P_{1}\\\
(p,p_{i_{1}})=1\end{subarray}}\Big{(}\sum_{j=0}^{\infty}\frac{1}{p^{j}}\frac{(n(k-1))^{2j}}{(2j)!h(p^{2j})}+\sum_{j=0}^{\infty}\frac{1}{p^{j+1}}\frac{(n(k-1))^{2j+1}}{(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}\\\
\ll&\Big{(}\prod_{\begin{subarray}{c}p\in P_{1}\\\
(p,p_{i_{1}})=1\end{subarray}}C(p)\Big{)}\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in
P_{1}\end{subarray}}\frac{1}{p}\Big{)},\end{split}$

where for some constant $A$ independent of $p$,

$\displaystyle\begin{split}C(p)=&\exp(\frac{A}{p^{2}})\Big{(}1-\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\frac{1}{p}\Big{)}\Big{(}\sum_{j=0}^{\infty}\frac{1}{p^{j}}\frac{(n(k-1))^{2j}}{(2j)!h(p^{2j})}+\sum_{j=0}^{\infty}\frac{1}{p^{j+1}}\frac{(n(k-1))^{2j+1}}{(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}.\end{split}$

Note here that $C(p)$ is well-defined as one checks readily checked that each
factor in the above product is positive for $n=2,0\leq k\leq 1$ and $p\geq 3$.
Meanwhile, we note that the left side of (3.6) is also

(3.7)
$\displaystyle\begin{split}\gg\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in
P_{1}\end{subarray}}\frac{1}{p}\Big{)}.\end{split}$

On the other hand, using Rankin’s trick by noticing that
$2^{\Omega(n_{1})-\ell_{1}}\geq 1$ if $\Omega(n_{1})>\ell_{1}$, we see that
the error introduced by replacing $\widetilde{b}_{l,l_{1},l_{2}}(n_{1})$ with
$1$ does not exceed

(3.8)
$\displaystyle\begin{split}&\sum_{n_{1}}\frac{1}{\sqrt{n_{1}(n_{1})_{1}}}\frac{|n(k-1)|^{\Omega(n_{1})}}{w(n_{1})}2^{\Omega(n_{1})-\ell_{1}}\frac{d((n_{1})_{1})(n_{1})_{1}}{\sigma((n_{1})_{1})h(n_{1})}\\\
\leq&2^{-\ell_{1}}\prod_{\begin{subarray}{c}p\in P_{1}\\\
(p,p_{i_{1}})=1\end{subarray}}\Big{(}1+\sum_{j=1}^{\infty}\frac{1}{p^{j}}\frac{(n(k-1))^{2j}2^{2j}}{(2j)!}\frac{1}{h(p^{2j})}+\sum_{j=0}^{\infty}\frac{1}{p^{j+1}}\frac{|n(k-1)|^{2j+1}2^{2j+1}}{(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}\\\
\ll&2^{-\ell_{1}}\exp\Big{(}\big{(}2(n(k-1))^{2}+2n^{2}(1-k)\big{)}\sum_{p\in
P_{1}}\frac{1}{p}\Big{)}.\end{split}$

We deduce from (3.6), (3.7) and (3.8) that the error term is

(3.9) $\displaystyle\ll$ $\displaystyle
2^{-\ell_{1}}\exp\Big{(}\big{(}\frac{3}{2}(n(k-1))^{2}+3n^{2}(1-k)\big{)}\sum_{p\in
P_{1}}\frac{1}{p}\Big{)}\Big{(}\prod_{\begin{subarray}{c}p\in P_{1}\\\
(p,p_{i_{1}})=1\end{subarray}}C(p)\Big{)}\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in
P_{1}\end{subarray}}\frac{1}{p}\Big{)}.$

Note that Lemma 2.2 implies that for all $1\leq j\leq R$, we have $\sum_{p\in
P_{j}}1/p\leq 2\log\ell_{j-1}\leq\ell_{j}/36$ from our definition on
$\ell_{j}$. We obtain from this and (3.9) that for $n=2$, we have

(3.10)
$\displaystyle\begin{split}&\sum_{(n_{1},p_{i_{1}})=1}\frac{1}{\sqrt{n_{1}(n_{1})_{1}}}\frac{(n(k-1))^{\Omega(n_{1})}}{w(n_{1})}\widetilde{b}_{1,l_{1},l_{2}}(n_{1})\frac{d((n_{1})_{1})(n_{1})_{1}}{\sigma((n_{1})_{1})h(n_{1})}\\\
\ll&(1+O(2^{-\ell_{1}/2}))\Big{(}\prod_{\begin{subarray}{c}p\in P_{1}\\\
(p,p_{i_{1}})=1\end{subarray}}C(p)\Big{)}\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in
P_{1}\end{subarray}}\frac{1}{p}\Big{)}.\end{split}$

We may also establish similar estimations for sums over $n_{j},2\leq j\leq r$.

Next, we apply Rankin’s trick again to see that the contribution of the
$n_{r+1}$ terms in (3.5) is

$\leq\ell_{r+1}!10^{-\ell_{r+1}}\prod_{p\in
P_{r+1}}\Big{(}\sum_{j=0}^{\infty}\frac{10^{2j}}{p^{j}(2j)!h(p^{2j})}+\sum_{j=1}^{\infty}\frac{10^{2j+1}}{p^{j+1}(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}.$

We apply Lemma 2.2, the estimation
$\ell_{r+1}!\leq\ell_{r+1}(\ell_{r+1}/e)^{\ell_{r+1}}$ and the definition of
$\ell_{r+1}$ to see that the above is

$\ll\ell_{r+1}\Big{(}\frac{\ell_{r+1}}{10e}\Big{)}^{\ell_{r+1}}\exp\Big{(}70\sum_{p\in
P_{r+1}}\frac{1}{p}\Big{)}\ll\ell_{r+1}\Big{(}\frac{\ell_{r+1}}{10e}\Big{)}^{\ell_{r+1}}\exp(\tfrac{5}{7}\ell_{r+1}).$

Combining the this with (3.5) and (3.10), we conclude that the contribution
from all terms of the forms given in (3.4) corresponding to
$m_{i_{1}}=m_{i_{2}}=m_{i_{3}}=1$ for some $1\leq i_{1}<i_{2}<i_{3}\leq r+1$
to the left side of (3.2) is

$\displaystyle\ll$ $\displaystyle{X}e^{-\ell_{r+1}/7}\prod_{1\leq j\leq
r}\Big{(}1+O(2^{-\ell_{j}/2})\Big{)}\prod_{\begin{subarray}{c}p\in\bigcup_{j=1}^{r}P_{j}\end{subarray}}C(p)\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in\bigcup_{j=1}^{r}P_{j}\end{subarray}}\frac{1}{p}\Big{)}$
$\displaystyle\times\Big{(}\sum_{p\in\bigcup_{j=1}^{r}P_{j}}\sum_{l\geq
0}\frac{\log
p}{p^{l+1}}\frac{|(n(k-1))|^{2l+1}}{(2l+1)!}\frac{np}{(p+1)h(p^{2l+1})}\Big{)}^{3}$
$\displaystyle\ll$ $\displaystyle e^{-\ell_{r+1}/7}X(\log
X)^{\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)+3},$

by using Lemma 2.2 while noting that $\displaystyle\prod_{p}C(p)\ll 1$ and
$\displaystyle\prod_{1\leq j\leq r}\Big{(}1+O(2^{-\ell_{j}/2})\Big{)}\ll 1$
since $\ell_{j}>\ell_{j+1}^{2}$.

Summing over $r$, we deduce that the estimation in (3.2) is valid and this
completes the proof of the proposition.

##  4\. A further remark

In this section we describe a variant on the principle of Radziwiłł and
Soundararajan mentioned in Section 1 for obtaining upper bounds of moments of
$L$-functions. This is based on the following simple observation.

######  Lemma 4.1.

Let $\ell$ be a non-negative even integer. For all real number $x$, let
$E_{\ell}(x)$ be defined in (2.1). We have

$E_{\ell}(x)E_{\ell}(-x)\geq 1.$

###### Proof.

We expand out the product $f_{\ell}(x):=E_{\ell}(x)E_{\ell}(-x)-1$ as a
polynomial to see that it suffices to show that the coefficients of
$x^{i},0\leq j\leq 2\ell$ are all non-negative. Note that the function
$f_{\ell}(x)$ is even so that only even powers of $x$ appear in the expansion.
Furthermore, it follows from the binomial theorem that the only non-zero even
powers of $x$ involved are those of the form $x^{2j}$ for $2j>l$. We then set
$\ell=2k$ to see that the coefficient of $x^{2(k+j)}$ for some integer $j>0$
is given by

$\frac{1}{(2(k+j))!}\sum^{2k}_{i=2j}\binom{2(k+j)}{i}(-1)^{i}=-\frac{2}{(2(k+j))!}\sum^{2j-1}_{i=0}\binom{2(k+j)}{i}(-1)^{i}\geq
0,$

where the last inequality above follows from [FI10, (6.6)] and this completes
the proof. ∎

Now, for two real numbers $n,k$ with $0<k<1$, we apply Lemma 4.1 to see that

$\displaystyle\sideset{}{{}^{*}}{\sum}_{(d,2)=1}|L(\tfrac{1}{2},\chi_{8d})|^{nk}\Phi(\frac{d}{X})\leq\sideset{}{{}^{*}}{\sum}_{(d,2)=1}|L(\tfrac{1}{2},\chi_{8d})|^{nk}{\mathcal{N}}(d,n(k-1))^{k}{\mathcal{N}}(d,n(1-k))^{k}\Phi(\frac{d}{X}),$

where we recall that the definition of ${\mathcal{N}}(d,\alpha)$ is given in
(2.2). We further apply Hölder’s inequality to the right side expression above
to deduce that

$\displaystyle\sideset{}{{}^{*}}{\sum}_{(d,2)=1}|L(\tfrac{1}{2},\chi_{8d})|^{nk}\Phi(\frac{d}{X})\leq\Big{(}\sideset{}{{}^{*}}{\sum}_{(d,2)=1}|L(\tfrac{1}{2},\chi_{8d})|^{n}{\mathcal{N}}(d,n(k-1))\Phi(\frac{d}{X})\Big{)}^{k}\Big{(}\sideset{}{{}^{*}}{\sum}_{(d,2)=1}{\mathcal{N}}(d,n(1-k))^{k/(1-k)}\Phi(\frac{d}{X})\Big{)}^{1-k}.$

This approach can be applied to obtain upper bounds for the $k$-th moment of
$L$-functions. In particalur, it is convenient to study the $\tfrac{1}{2}$-th
moment as it suffices to evaluate the average over $d$ of
$|L(\tfrac{1}{2},\chi_{8d})|{\mathcal{N}}(d,-\tfrac{1}{2})$ and
${\mathcal{N}}(d,\tfrac{1}{2})$ by taking $n=1,k=1/2$ above. If we assume that
$L(\tfrac{1}{2},\chi_{8d})\geq 0$ (which follows from GRH), then we just need
to consider $L(\tfrac{1}{2},\chi_{8d}){\mathcal{N}}(d,-\tfrac{1}{2})$, which
is relatively simpler compared to our work above. The same method applies
equally to the study on the moments of quadratic twists of $L$-functions
attached to elliptic curves (as we know in this case the corresponding
$L$-functions have non-negative values at the central point).

More generally, when $k=1/m$ with $m>2$ a positive integer, we denote
$\\{L(s,f)\\}_{f\in\mathcal{F}}$ for a general family of $L$-functions, we may
apply Lemma 4.1 (with a suitable adjustment on the defintion of
${\mathcal{N}}$) to see that

$\displaystyle\sum_{f\in\mathcal{F}}|L(\tfrac{1}{2},f)|^{k}\leq\sum_{f\in\mathcal{F}}|L(\tfrac{1}{2},f)|^{k}{\mathcal{N}}(f,k-1)^{k}\Big{(}\prod^{m-2}_{i=1}\big{(}{\mathcal{N}}(f,1-ik){\mathcal{N}}(f,(i+1)k-1)\big{)}^{k}\Big{)}{\mathcal{N}}(f,1-(m-1)k)^{k}.$

Applying Hölder’s inequality $m-1$ times to the right side expression above,
we deduce that

$\displaystyle\sum_{f\in\mathcal{F}}|L(\tfrac{1}{2},f)|^{k}\leq$
$\displaystyle\Big{(}\sum_{f\in\mathcal{F}}|L(\tfrac{1}{2},f)|{\mathcal{N}}(f,k-1)\Big{)}^{k}\prod^{m-2}_{i=1}\Big{(}\sum_{f\in\mathcal{F}}{\mathcal{N}}(f,1-ik){\mathcal{N}}(f,(i+1)k-1)\Big{)}^{k}\Big{(}\sum_{f\in\mathcal{F}}{\mathcal{N}}(f,1-(m-1)k)\Big{)}^{k}.$

It follows that in order to obtain upper bounds for the $1/m$-th moment of the
corresponding family, we only need to be able to evaluate the average over
$f\in\mathcal{F}$ of $|L(\tfrac{1}{2},f)|{\mathcal{N}}(f,k-1)$ and quantities
involving with products of at most two copies of ${\mathcal{N}}$. The above
approach can also be adapted to treat moments of the Riemann zeta function on
the critical line. We shall however not go any further in this direction here.

Acknowledgments. P. G. is supported in part by NSFC grant 11871082.

## References