# A central limit theorem for coefficients of L-functions in short intervals

and  Sun-Kai Leung  Département de mathématiques et de statistique
Université de Montréal
CP 6128 succ. Centre-Ville
Montréal, QC H3C 3J7
Canada<EMAIL_ADDRESS>
###### Abstract.

Assuming the generalized Lindelöf hypothesis (GLH), a weak generalized
Ramanujan conjecture and a Rankin–Selberg type partial sum estimate, we show
that the sum of coefficients of a general $L$-function in short intervals of
appropriate length has a Gaussian limiting distribution. The novelty lies in
the degree aspect under GLH. In particular, this generalizes Hughes–Rudnick’s
result on lattice point counts in thin annuli.

###### 2020 Mathematics Subject Classification:

11F30; 60F05

##  1\. Introduction

The distribution of arithmetic functions $a:\mathbb{N}\to\mathbb{C}$ is a
central topic in analytic number theory. In particular, it is natural to ask
how the partial sum $\sum_{n\leqslant x}a(n)$ is distributed as $x\in[1,X]$
varies. One might expect the partial sum to behave similarly to a sum of
independent and identically distributed random variables, and thus it should
exhibit a Gaussian law according to the central limit theorem. Although it may
come as a surprise that this is not typically the case, many arithmetic
functions actually follow the normal distribution in short intervals $[x,x+H]$
for some appropriate length $H=o(X)$ rather than in long intervals $[1,x].$

For instance, if the length of short intervals $H\to\infty$ but $\log H=o(\log
X)$ as $X\to\infty,$ i.e. very short, then Davenport and Erdős [DE52], and
more recently Lamzouri [Lam13], established central limit theorems for the
short character sum $\sum_{x<n\leqslant x+H}\chi(n),$ where $\chi\
(\operatorname{mod}q)$ is a Dirichlet character with $X=q.$ Similarly,
assuming a uniform Hardy–Littlewood prime $k$-tuple conjecture, Montgomery and
Soundararajan [MS04] also established a central limit theorem for the weighted
count of primes $\sum_{x<n\leqslant x+H}\Lambda(n)$ in the above range.

On the other hand, if the length of short intervals $H=o(X)$ but
$\log(X/H)=o(\log X)$ as $X\to\infty,$ i.e. shortish, Harper [Har22] proved
very recently that for almost all primes $q,$ the short character sum
$\sum_{x<n\leqslant x+H}\left(\frac{n}{q}\right)$ also follows the normal
distribution with $X=q.$ Similarly, assuming the Riemann hypothesis, the
linear independence hypothesis and its quantitative formulation, the author
has also proved very recently that the weighted count of primes
$\sum_{x<n\leqslant x+H}\Lambda(n)$ follows the normal distribution in the
above range, with respect to the logarithmic density (see [Leu24, Theorem
9.1]). In other words, as $u\in[1,U]$ varies, the weighted count of primes
$\sum_{e^{u}<n\leqslant e^{u+\delta}}\Lambda(n)$ has a Gaussian limiting
distribution, provided that $\delta=o(1)$ but $\log\frac{1}{\delta}=o(\log U)$
as $U\to\infty.$

Both results by Harper and the author rely heavily on the dual sum. In the
case of short (primitive) character sums, the dual sum is Pólya’s Fourier
expansion

$\displaystyle\sum_{x<n\leqslant
x+H}\chi(n)\approx\frac{2\epsilon_{\chi}}{\pi}\sum_{0<n\leqslant
q/H}\frac{\overline{\chi}(n)}{\sqrt{n}}\cdot\frac{\sin(\pi
nH/q)}{\sqrt{n/q}}\cdot\cos\left(\frac{\pi n}{q}(2x+H)-\frac{\pi}{2}\cdot
1_{\chi\text{ odd}}\right).$ (1.1)

In the case of primes in short intervals, the dual sum is von Mangoldt’s
explicit formula

$\displaystyle\sum_{e^{u}<n\leqslant
e^{u+\delta}}\Lambda(n)\approx(e^{u+\delta}-e^{u})-\frac{2}{\pi}\sum_{0<\widetilde{\gamma}\leqslant
1/\delta}\frac{\sin(\pi\widetilde{\gamma}\delta)}{\widetilde{\gamma}}\cdot\cos(\pi\widetilde{\gamma}(2u+\delta)),$
(1.2)

where $\widetilde{\gamma}:=\gamma/2\pi,$ i.e. the normalized positive ordinate
of a non-trivial Riemann zero.

Here, primitive Dirichlet characters $\chi(n)$ are considered as
“$GL_{1}$-arithmetic functions” as the Dirichlet L-function $L(s,\chi)$ is of
degree $1$ (see Section 2 for definition). Meanwhile, let
$\tau^{\prime}_{k}(n):=|\\{(d_{1},\ldots,d_{k})\in\mathbb{N}^{k}_{>1}\,:\,n=d_{1}\cdots
d_{k}\\}|.$ Then, Linnik’s identity (see [IK04, Chapter 13.3]) states that

$\displaystyle\frac{\Lambda(n)}{\log
n}=-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\tau^{\prime}_{k}(n).$

Also, for each $k\geqslant 1,$ since $\tau_{k}(n)$ is a “$GL_{k}$ arithmetic
function” by the same reasoning, one may regard the von-Mangoldt function
$\Lambda(n)$ as a “$GL_{\infty}$-arithmetic function”. Therefore, a natural
question arises: how about $GL_{m}$-arithmetic functions in general? Do they
follow the normal distribution in short intervals? In this paper, we provide
an affirmative answer conditionally, again by studying the dual sum, which is
the Voronoĭ summation formula (without twists).

##  2\. Definitions

We begin with introducing a general class of L-functions. Throughout the
paper, let

$\displaystyle L(f,s):=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{s}}$

be an L-function of degree $m\geqslant 2$ which is self-dual, i.e. the
coefficients $\lambda_{f}(n)$ are real, and is absolutely convergent for
$\mathrm{Re}(s)>1.$ We denote the gamma factor as

$\displaystyle
L_{\infty}(f,s):=\prod_{j=1}^{m}{\pi}^{-\frac{s+\mu_{j}}{2}}\Gamma\left(\frac{s+\mu_{j}}{2}\right),$

where $\mu_{j}$’s are the local parameters at $\infty.$ For convenience, we
shall call

$\displaystyle k:=\sum_{j=1}^{m}\mathrm{Re}(\mu_{j})$

the weight of $L(f,s).$ The complete $L$-function is defined as

$\displaystyle\Lambda(f,s):=D^{\frac{s}{2}}L_{\infty}(f,s)L(f,s)$

with conductor $D,$ which admits an analytic continuation to $\mathbb{C}$
except possibly poles of finite order at $s=1.$ Moreover, it satisfies the
function equation

$\displaystyle\Lambda(f,s)=w\Lambda(f,1-s)$

with root number $w=\pm 1.$

Let us denote $\Delta_{f}(x):=\sum_{n\leqslant
x}\lambda_{f}(n)-\mathop{\mathrm{Res}}_{s=1}L(f,s)x^{s}/s.$ Then in this
paper, we shall study the distribution of the remainder of the short
coefficient sum

$\displaystyle\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})=\sum_{x^{m}<n\leqslant(x+\delta)^{m}}\lambda_{f}(n)-\left(\mathop{\mathrm{Res}}_{s=1}L(f,s)\frac{(x+\delta)^{ms}-x^{ms}}{s}\right)$

conditioning on the following conjectures.

######  Conjecture 2.1 (Generalized Lindelöf hypothesis (GLH)).

Let $t\in\mathbb{R}.$ Then for any $\epsilon>0,$ we have
$L(f,\frac{1}{2}+it)\ll_{\epsilon}D^{\epsilon}\prod_{j=1}^{m}(1+|t+\mu_{j}|)^{\epsilon}.$

######  Conjecture 2.2 (Weak Generalized Ramanujan conjecture
($\text{GRC}^{\flat}$)).

There exists $\eta>0$ such that for any $n\geqslant 1,$ we have
$\lambda_{f}(n)\ll_{f}n^{\frac{1}{6}-\eta}.$

Similar to Weyl’s subconvexity bound, the exponent $1/6$ here is considered to
be critical due to its applications to many classical problems in analytic
number theory (see [BB13, Section 5.3] for instance), including shifted
convolution sums and hyperbolic circle problem (see [KS02, Section 8]).

######  Remark 2.1.

It is well-known that the normalized Fourier coefficients (Hecke eigenvalues)
of holomorphic or Hecke–Maass cusp forms satisfy
$\lambda_{f}(n)\leqslant\tau(n)$ by Deligne [Del74] or
$\lambda_{f}(n)\leqslant\tau(n)n^{7/64}$ by Kim–Sarnak [Kim03], respectively.
Therefore, the conjecture $\text{GRC}^{\flat}$ holds for such forms.

##  3\. Main results

With the definitions and conjectures in place, we can now state our main
result as follows.

######  Theorem 3.1.

Assume GLH and $\text{GRC}^{\flat}$ for a fixed L-function $L(f,s)$. Let
$X,\delta>0$ be real numbers for which $\delta=o_{X\to\infty}(1)$ but
$\log\left(1/\delta\right)=o_{X\to\infty}(\log X).$ Suppose

$\displaystyle\sum_{n\leqslant
y}\lambda_{f}(n)^{2}=(c_{f}+o_{y\to\infty}(1))y(\log 2y)^{r-1}$ (3.1)

for some constant $c_{f}>0$ and integer $r\geqslant 1.$ Then, if $x\in[X,2X]$
is chosen uniformly at random, we have the convergence in distribution to a
standard Gaussian

$\displaystyle\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}\sigma_{f}(\delta)}\xrightarrow[]{d}\mathcal{N}(0,1)$
(3.2)

as $X\to\infty,$ where

$\displaystyle{\sigma_{f}(\delta)}^{2}:=c_{f}m^{r}\delta\log^{r-1}\left(\frac{1}{\delta}\right),$

i.e. for any fixed real numbers $\alpha<\beta,$ we have

$\displaystyle\lim_{X\to\infty}\frac{1}{X}\mathop{\mathrm{meas}}\left\\{x\in[X,2X]\,:\,\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}\sigma_{f}(\delta)}\in(\alpha,\beta]\right\\}=\frac{1}{\sqrt{2\pi}}\int_{\alpha}^{\beta}e^{-\frac{t^{2}}{2}}dt.$

When $m=2,$ the assumption of GLH can be dropped.

######  Remark 3.1.

The Rankin–Selberg type partial sum estimate (3.1) is known for many
automorphic objects, especially the Hecke eigenvalues $A(n,1,\ldots,1)$ of
Hecke–Maass cusp forms for $SL_{m}(\mathbb{Z})$ (see [Jää20, Theorem 6]).

The theorem can be regarded as the short-interval counterpart of the work by
Kowalski and Ricotta [KR14, Corollary B] in the self-dual case, i.e.
coefficients are always real. Compared to their work, the source of normality
here stems from the oscillation of Bessel transforms instead of hyper-
Kloosterman sums. In this paper, the main difficulty lies in controlling the
errors incurred from approximating these Bessel transforms via the saddle
point method (see [FI05, p. 498] for instance). Therefore, unlike in their
work, the Voronoĭ summation formula with smooth weights offers little
advantage. Nonetheless, our proof is more elementary since it does not rely on
deep results regarding the equidistribution of hyper-Kloosterman sums.

It is certainly plausible to prove an analogous result for non self-dual
L-functions, i.e. coefficients are not always real. In such cases, the short
coefficient sum would follow a bivariate Gaussian distribution on the complex
plane. Besides, the proof should also work for $L$-functions belonging to the
Selberg class (see [KP99] for the definition) conditionally. To avoid
repetition, we refrain from pursuing these further here.

In the following, we provide several interesting examples.

######  Example 3.1.

Let $\lambda_{f}(n)$ be Hecke eigenvalues of a holomorphic Hecke cusp form or
a Hecke–Maass cusp form $f$ of level $1$ for $n\in\mathbb{N}.$ Then Theorem
3.1 holds unconditionally with $m=2$ and
${\sigma_{f}(\delta)}^{2}=\frac{12}{\pi^{2}}L(1,\mathop{\mathrm{Sym}}^{2}f)\delta.$
For $f$ holomorphic, this is essentially Lester–Yesha’s central limit theorem
[LY16, Theorem 1.4] with respect to a different scale (see Remark 3.2).

######  Example 3.2.

Given an integer $k\geqslant 2.$ Let $\lambda_{f}(n)=\tau_{k}(n)$ for
$n\in\mathbb{N}.$ Assume GLH for $k\geqslant 3$. Then Theorem 3.1 holds with
$m=k,D=1$ and
${\sigma_{f}(\delta)}^{2}=c_{f}k^{(k^{2})}\delta\log^{k^{2}-1}\left(\frac{1}{\delta}\right),$
where

$\displaystyle
c_{f}=\prod_{p}\left(1-\frac{1}{p}\right)^{k^{2}}\sum_{j=0}^{\infty}{j+k-1\choose
j}^{2}\frac{1}{p^{j}}.$

In particular, when $k=2,$ this is again essentially [LY16, Theorem 1.3] with
${\sigma_{f}(\delta)}^{2}=\frac{16}{\pi^{2}}\delta\log^{3}\left(\frac{1}{\delta}\right)$
with respect to a different scale.

######  Example 3.3.

Given a Galois extension $K/\mathbb{Q}$ of degree $d,$ let
$\lambda_{f}(n)=r_{K}(n)$ be the number of ideals of norm $n$ for
$n\in\mathbb{N}.$ Assume GLH for $k\geqslant 3$. Then Theorem 3.1 holds with
$m=d$ and
${\sigma_{f}(\delta)}^{2}=c_{f}d^{d}\delta\log^{d-1}\left(\frac{1}{\delta}\right).$
Here, the satisfaction of assumption (3.1) is guaranteed by [CN63, Theorem 3].
In particular, when $K=\mathbb{Q}(i),$ it becomes Hughes–Rudnick’s result on
lattice point counts in thin annuli [HR04, Theorem 1.3] with
${\sigma_{f}(\delta)}^{2}=\delta\log\left(\frac{1}{\delta}\right).$

######  Remark 3.2.

Suppose the conditions of Theorem 3.1 are satisfied. Let $H=H(x)=\delta
x^{1-\frac{1}{m}}.$ Then it is certainly plausible to establish a central
limit theorem for

$\displaystyle\frac{\Delta_{f}(x+H)-\Delta_{f}(x)}{x^{\frac{m-1}{2m}}\sigma_{f}(\delta)}$

as in [LY16] when $m=2$. However, our proof suggests that the left-hand side
of (3.2) is also a natural object to study (see [HR04, Theorem 1.3] for
instance).

Notation. Throughout the paper, we use the standard big $O$ and little $o$
notations as well as the Vinogradov notations $\ll,\gg,$ where from now on,
all implied constants depend on the fixed L-function $L(f,s)$ and subscripts
are omitted.

##  4\. Preliminaries

The Voronoĭ summation formula (without twists), as formulated by
Friedlander–Iwaniec lies at the heart of the matter; although to prove the
Theorem 3.1, we do not apply the formula directly as their error term is not
negligible when $m\geqslant 3.$

######  Proposition 4.1 (Voronoĭ summmation formula).

Assume (3.1). Let $x\geqslant D^{1/2m}.$ Then for any $1\leqslant N\leqslant
x^{m},$ we have

$\displaystyle\Delta_{f}(x^{m})=\frac{w}{\pi}\cdot
x^{\frac{m-1}{2}}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin\left(2\pi\breve{n}x+\varphi\right)}{\sqrt{\breve{n}}}+O_{\boldsymbol{\mu},\epsilon}((N/D)^{-\frac{1}{m}}x^{m-1+\epsilon}),$
(4.1)

where $\breve{n}:=m(n/D)^{1/m}$ for $n\in\mathbb{N}$ and
$\varphi:=\frac{\pi}{2}\left(\frac{m-1}{2}-k\right).$ The implied constant
here depends only on $\boldsymbol{\mu},\varepsilon.$

###### Proof.

This is essentially [FI05, Theorem 1.2], except instead of assuming the
generalized Ramanujan conjecture, i.e.
$\lambda_{f}(n)\ll_{\epsilon}n^{\epsilon}$ for any $n\geqslant 1,$ one can use
the Rankin–Selberg type partial sum estimate (3.1) to show that

$\displaystyle\frac{1}{2\pi
i}\int_{1+\epsilon+iT}^{1+\epsilon+i\infty}L(f,s)\frac{x^{ms}}{s}ds\ll_{\epsilon}$
$\displaystyle
x^{m\epsilon}\sum_{n=1}^{\infty}\frac{|\lambda_{f}(n)|}{n^{1+\epsilon}\max\\{1,T|\log(x^{m}/n)|\\}}$
$\displaystyle\ll_{\epsilon}$ $\displaystyle\frac{x^{(1+2\epsilon)m}}{T}$

(see [Kou19, p. 71] for instance). ∎

######  Remark 4.1.

A typographical error is present in the statement of [FI05, Theorem 1.2]:
$(\pi m)^{-\frac{1}{2}}$ should be corrected to $\pi m^{-\frac{1}{2}}.$

Instead of studying
$x^{-(m-1)/2}(\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m}))$ directly, we
define a similar quantity

$\displaystyle\Delta_{f}(x,\delta):=\frac{\Delta_{f}((x+\delta)^{m})}{(x+\delta)^{(m-1)/2}}-\frac{\Delta_{f}(x^{m})}{x^{(m-1)/2}},$

which is easier to handle, as the Voronoĭ summmation formula suggests that

$\displaystyle\Delta_{f}(x,\delta)\approx\frac{2w}{\pi}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin(\pi\breve{n}\delta)}{\sqrt{\breve{n}}}\cdot\cos\left(\pi\breve{n}(2x+\delta)+\varphi\right)$
(4.2)

as $m(N/D)^{1/m}\delta\to\infty$. This should be compared with (1.1), (1.2).
The pseudo-independence of $\cos\left(\pi\breve{n}(2x+\delta)+\varphi\right)$
for $x\in[X,2X]$ is provided by Lemma 4.1, so that heuristically we have

$\displaystyle\Delta_{f}(x,\delta)\approx\frac{2w}{\pi}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin(\pi\breve{n}\delta)}{\sqrt{\breve{n}}}\cdot\mathbb{X}_{n},$

where $\mathbb{X}_{n}\sim\mathrm{Re}(\mathbb{U}(S^{1}))$ are independent and
identically distributed random variables. Then, it follows from the classical
central limit theorem that $\Delta_{f}(x,\delta)$ is approximately normal with
mean $0$ and variance

$\displaystyle\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}.$
(4.3)

To argue rigorously, we compute the moments of (4.2) as in [HR04] and [LY16].

######  Lemma 4.1.

Let $q_{1},\ldots,q_{k}$ be distinct $m$-th power-free positive integers. Then
$q_{1}^{1/m},\ldots,q_{k}^{1/m}$ are linearly independent over $\mathbb{Q}.$

###### Proof.

Unlike in [HR04] and [LY16], the applicability of [Bes40, Theorem 2] is not
straightforward. Instead, we provide a different proof. Suppose
$\sum_{j=1}^{k}c_{j}q_{j}^{1/k}=0$ for some $c_{j}\in\mathbb{Q}.$ Fix
$j_{0}=1,\ldots,k.$ Then

$\displaystyle c_{j_{0}}=-\sum_{\begin{subarray}{c}j=1\\\ j\neq
j_{0}\end{subarray}}^{k}c_{j}\left(q_{j}/q_{j_{0}}\right)^{1/m}.$

Let $r_{j}=(q_{j}/q_{j_{0}})^{1/m}$ and $K$ be the finite extension of
$\mathbb{Q}$ by adjoining $r_{j}$ for $j\neq j_{0}.$ Taking the trace
$\operatorname{Tr}_{\mathbb{Q}}^{K}$ on both sides gives

$\displaystyle[K:\mathbb{Q}]c_{j_{0}}=-\sum_{\begin{subarray}{c}j=1\\\ j\neq
j_{0}\end{subarray}}^{k}c_{j}\operatorname{Tr}_{\mathbb{Q}}^{K}(r_{j}).$ (4.4)

Note that by assumption, there exists the smallest $1<m_{j}\leqslant m$ such
that $r_{j}^{m_{j}}\in\mathbb{Q},$ so that the minimal polynomial of $r_{j}$
is $x^{m_{j}}-r_{j}^{m_{j}}.$ Therefore, we have

$\displaystyle\operatorname{Tr}_{\mathbb{Q}}^{K}(r_{j})=[K:\mathbb{Q}(r_{j})]\sum_{i=1}^{m_{j}}\zeta_{m_{j}}^{i}r_{j}=0,$

so that the right-hand side of (4.4) vanishes and $c_{j_{0}}=0.$ Since $j_{0}$
is chosen arbitrarily, we have $c_{j}=0$ for any $j=1,\ldots,k$ and the lemma
follows. ∎

As we shall prove Theorem 3.1 via the method of moments, a quantitative lower
bound for nonvanishing alternating sums of $q_{j}^{1/m}$ is required.

######  Lemma 4.2.

For $j=1,\ldots,k,$ let $1\leqslant n_{j}\leqslant N$ and
$\epsilon_{j}\in\\{\pm 1\\}.$ Suppose

$\displaystyle\sum_{j=1}^{k}\epsilon_{j}n_{j}^{1/m}\neq 0.$

Then

$\displaystyle\left|\sum_{j=1}^{k}\epsilon_{j}n_{j}^{1/m}\right|\geqslant(kN^{1/m})^{-(m^{k}-1)}.$

###### Proof.

Let $G$ be the Galois group of the field extension $K/\mathbb{Q},$ where $K$
is the Galois closure of $\mathbb{Q}(n_{1}^{1/m},\ldots,n_{k}^{1/m})$. Then,
we consider the product

$\displaystyle P:=\prod_{\sigma\in
G}\sigma\left(\sum_{j=1}^{k}\epsilon_{j}n_{j}^{1/m}\right)=\prod_{\sigma\in
G}\left(\sum_{j=1}^{k}\epsilon_{j}\sigma(n_{j}^{1/m})\right).$

Since the set of algebraic integers forms a ring, the product $P$ is an
algebraic integer. Besides, as $P$ is fixed by $G,$ it is also a rational
number by Galois theory, so that $P$ is an integer. Since $\sigma\in G$ are
injective and $\sum_{j=1}^{k}\epsilon_{j}n_{j}^{1/m}\neq 0$ by assumption, no
terms in $P$ vanishes, and thus $|P|\geqslant 1.$ Then, we have

$\displaystyle\left|\sum_{j=1}^{k}\epsilon_{j}n_{j}^{1/m}\right|\geqslant\prod_{\begin{subarray}{c}\sigma\in
G\\\ \sigma\neq
e\end{subarray}}\left|\sum_{j=1}^{k}\epsilon_{j}\sigma(n_{j}^{1/m})\right|^{-1}\geqslant(kN^{1/m})^{-(m^{k}-1)},$

and the lemma follows. ∎

Finally, we provide an estimate for the variance (4.3).

######  Lemma 4.3.

Suppose the estimate (3.1) holds. Then as $\delta\to 0^{+},$ we have

$\displaystyle\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}=$
$\displaystyle(c_{f}+o(1))m^{r}\delta\log^{r-1}\left(\frac{1}{\delta}\right)$
$\displaystyle=$ $\displaystyle(1+o(1))\sigma_{f}(\delta)^{2}.$

###### Proof.

Let $S(y):=\sum_{n\leqslant y}\lambda_{f}(n)^{2}.$ Then by partial summation,
we have

$\displaystyle\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}=$
$\displaystyle\frac{2}{\pi^{2}}\int_{1}^{\infty}\frac{\sin^{2}(\pi
m(y/D)^{1/m}\delta)}{m(y/D)^{1/m}}\cdot\frac{dS(y)}{y}.$

We write $S(y)=M(y)+E(y),$ where the main term $M(y):=c_{f}y(\log y)^{r-1}$
and the error term $E(y)=o(y(\log 2y)^{r-1}).$ Then this becomes

$\displaystyle\frac{2}{\pi^{2}}$
$\displaystyle\int_{1}^{\infty}\frac{\sin^{2}(\pi
m(y/D)^{1/m}\delta)}{m(y/D)^{1/m}}\cdot\frac{d(c_{f}y(\log
y)^{r-1})}{y}+\frac{2}{\pi^{2}}\int_{1}^{\infty}\frac{\sin^{2}(\pi
m(y/D)^{1/m}\delta)}{m(y/D)^{1/m}}\cdot\frac{dE(y)}{y}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{2}}\cdot c_{f}\int_{1}^{\infty}\frac{\sin^{2}(\pi
m(y/D)^{1/m}\delta)}{m(y/D)^{1/m}y}\cdot(\log y)^{r-1}dy$
$\displaystyle+\frac{2}{\pi^{2}}\cdot c_{f}\int_{1}^{\infty}\frac{\sin^{2}(\pi
m(y/D)^{1/m}\delta)}{m(y/D)^{1/m}y}\cdot(r-2)(\log y)^{r-2}dy$
$\displaystyle+\frac{2}{\pi^{2}}\int_{1}^{\infty}E(y)d\left(\frac{\sin^{2}(\pi
m(y/D)^{1/m}\delta)}{m(y/D)^{1/m}y}\right)+O(\delta^{2})=:I_{1}+I_{2}+I_{3}+O(\delta^{2}).$
(4.5)

Let us estimate the integral $I_{1}$ first. Making the change of variables
$u=m(y/D)^{1/m}\delta,$ we have

$\displaystyle I_{1}=$ $\displaystyle\frac{2}{\pi^{2}}\cdot
c_{f}\int_{m\delta/D^{1/m}}^{\infty}\sin^{2}(\pi
u)\left(D\left(\frac{u}{m\delta}\right)^{m}\right)^{-(1+\frac{1}{m})}\log^{r-1}\left(D\left(\frac{u}{m\delta}\right)^{m}\right)\frac{D^{1+\frac{1}{m}}}{m\delta}\left(\frac{u}{m\delta}\right)^{m-1}du$
$\displaystyle=$ $\displaystyle
2c_{f}m\delta\int_{m\delta/D^{1/m}}^{\infty}\left(\frac{\sin(\pi u)}{\pi
u}\right)^{2}\log^{r-1}\left(D\left(\frac{u}{m\delta}\right)^{m}\right)du$
$\displaystyle=$ $\displaystyle
2c_{f}m\delta\left\\{\int_{0}^{\infty}-\int_{0}^{m\delta/D^{1/m}}\right\\}\left(\frac{\sin(\pi
u)}{\pi
u}\right)^{2}\log^{r-1}\left(m\log\frac{1}{\delta}+\log\left(D\left(\frac{u}{m}\right)^{m}\right)\right)^{r-1}du$
$\displaystyle=$ $\displaystyle
c_{f}m^{r}\delta\log^{r-1}\left(\frac{1}{\delta}\right)+O\left(\delta\log^{r-2}\left(\frac{1}{\delta}\right)\right).$

Arguing similarly, the integral $I_{2}$ is
$\ll\delta\log^{r-2}\left(\frac{1}{\delta}\right).$ It remains to estimate the
integral $I_{3}.$ Using the product rule, we have

$\displaystyle I_{3}=\int_{1}^{\infty}E(y)\left(\frac{\pi
m\delta}{D^{1/m}}\sin\left(2\pi
m\left(\frac{y}{D}\right)^{\frac{1}{m}}\delta\right)y^{\frac{2}{m}}-\frac{m+1}{D^{1/m}}\sin^{2}\left(\pi
m\left(\frac{y}{D}\right)^{\frac{1}{m}}\delta\right)y^{\frac{1}{m}}\right)$
$\displaystyle\cdot\left(m\left(\frac{y}{D}\right)^{\frac{1}{m}}y\right)^{-2}dy,$

which is

$\displaystyle o\left(\delta\int_{1}^{\infty}\sin\left(2\pi
m\left(\frac{y}{D}\right)^{\frac{1}{m}}\delta\right)\frac{(\log
2y)^{r-1}}{y}dy+\int_{1}^{\infty}\sin^{2}\left(\pi
m\left(\frac{y}{D}\right)^{\frac{1}{m}}\delta\right)\frac{(\log
2y)^{r-1}}{y^{1+\frac{1}{m}}}dy\right).$

Decomposing the range of integration $[1,\infty)$ into
$[1,\delta^{-m}]\cup[\delta^{-m},\infty),$ one can show that both integrals
here are $\ll\delta\log^{r-1}\left(\frac{1}{\delta}\right),$ so that
$I_{3}=o\left(\delta\log^{r-1}\left(\frac{1}{\delta}\right)\right).$ Combining
with the estimates for integrals $I_{1}$ and $I_{2}$, the lemma follows from
(4). ∎

##  5\. Mean and variance

As in [HR04] and [LY16], instead of the Lebesgue measure, we first integrate
with respect to a nonnegative Schwartz function $W$ of unit mass supported on
$(1/2,5/2).$ In particular, its Fourier transform $\widehat{W}$ decays rapidly
in the sense that for any $A>0,$ we have

$\displaystyle\widehat{W}(x)\ll_{A}\frac{1}{(1+|x|)^{A}},$

where the Fourier transform $\widehat{W}$ is defined as

$\displaystyle\widehat{W}(\xi):=\int_{-\infty}^{\infty}W(x)e^{-2\pi ix\xi}dx.$

For convenience, we write

$\displaystyle\mathbb{E}_{x\sim
X}^{W}\left(f(x)\right):=\frac{1}{X}\int_{-\infty}^{\infty}f(x)W\left(\frac{x}{X}\right)dx.$

In view of (4.2), we shall establish a central limit theorem for the finite
approximation

$\displaystyle\Delta_{f}(x,\delta;N):=\frac{2w}{\pi}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin(\pi\breve{n}\delta)}{\sqrt{\breve{n}}}\cdot\cos\left(\pi\breve{n}(2x+\delta)+\varphi\right).$

We also denote the truncated variance as

$\displaystyle\sigma_{f}(\delta;N)^{2}:=\frac{2}{\pi^{2}}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}.$

Let us begin with computing the mean and variance (or second moment) of
$\Delta_{f}(x,\delta;N).$

######  Lemma 5.1.

Let $X,\delta>0,N\geqslant 2.$ Then

$\displaystyle\mathbb{E}_{x\sim
X}^{W}\left(\Delta_{f}(x,\delta;N)\right)\ll_{A}\delta X^{-A}.$

In particular, if $\delta=o(1)$ as $X\to\infty,$ then $\mathbb{E}_{x\sim
X}^{W}\left(\Delta_{f}(x,\delta;N)\right)=o(\sigma_{f}(\delta;N)).$

###### Proof.

Let $1\leqslant n\leqslant N.$ Then

$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\cos\left(\pi\breve{n}(2x+\delta)+\varphi\right)W\left(\frac{x}{X}\right)dx$
$\displaystyle=\mathrm{Re}\left(\widehat{W}\left(-\breve{n}X\right)e^{i(\pi\breve{n}\delta+\varphi)}\right)$
$\displaystyle\ll_{A}(\breve{n}X)^{-A}.$

Recall that $\breve{n}:=m(n/D)^{1/m}.$ Therefore, we have

$\displaystyle\mathbb{E}_{x\sim X}^{W}\left(\Delta_{f}(x,\delta;N)\right)$
$\displaystyle=\frac{2w}{\pi}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin(\pi\breve{n}\delta)}{\sqrt{\breve{n}}}\cdot
O_{A}\left((\breve{n}X)^{-A}\right)$ $\displaystyle\ll_{A}\sum_{n\leqslant
N}\frac{|\lambda_{f}(n)|}{\sqrt{n}}\cdot\delta\sqrt{\breve{n}}\cdot(\breve{n}X)^{-A}$
$\displaystyle\ll_{A}\delta X^{-A}.$

Moreover, this is $\ll_{A}\sigma_{f}(\delta;N)X^{-A}$ since
$\sigma_{f}(\delta;N)^{2}\gg\delta^{2}$ when considering only the first term
with $n=1$. Therefore, the lemma follows. ∎

######  Lemma 5.2.

Let $X,\delta>0,N\geqslant 2.$ Suppose $N\leqslant
X^{\frac{m(1-\epsilon)}{m-1}}$ for some fixed small $\epsilon>0.$ Then

$\displaystyle\mathbb{E}_{x\sim
X}^{W}(\Delta_{f}(x,\delta;N)^{2})=\sigma_{f}(\delta;N)^{2}+O_{A}\left(\delta^{2}X^{-A}\right).$

In particular, if $\delta=o(1)$ as $X\to\infty,$ then $\mathbb{E}_{x\sim
X}^{W}(\Delta_{f}(x,\delta;N)^{2})=(1+o(1))\sigma_{f}(\delta;N)^{2}.$

###### Proof.

Let $1\leqslant n_{1},n_{2}\leqslant N.$ Then

$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\cos\left(\pi\breve{n_{1}}(2x+\delta)+\varphi\right)\cos\left(\pi\breve{n_{2}}(2x+\delta)+\varphi\right)W\left(\frac{x}{X}\right)dx$
$\displaystyle=\frac{1}{2}\cdot\mathrm{Re}\left(\widehat{W}((\breve{n_{1}}+\breve{n_{2}})X)e^{-i(\pi(\breve{n_{1}}+\breve{n_{2}})\delta+2\varphi)}+\widehat{W}((\breve{n_{1}}-\breve{n_{2}})X)e^{-i\pi(\breve{n_{1}}-\breve{n_{2}})\delta}\right)$
$\displaystyle=\begin{cases}1/2+O_{A}\left(((\breve{n_{1}}+\breve{n_{2}})X)^{-A}\right)&\mbox{{if
$n_{1}=n_{2},$ } }\\\ \hfil
O_{A}\left(((\breve{n_{1}}-\breve{n_{2}})X)^{-A}\right)&\mbox{{if $n_{1}\neq
n_{2}.$ } }\end{cases}$ (5.1)

Since by definition

$\displaystyle\breve{n_{1}}-\breve{n_{2}}=\frac{m}{D^{1/m}}(n_{1}^{1/m}-n_{2}^{1/m})\gg
N^{\frac{1}{m}-1}$

if $n_{1}\neq n_{2},$ the expression (5.1) is

$\displaystyle\begin{cases}1/2+O_{A}\left(((\breve{n_{1}}+\breve{n_{2}})X)^{-A}\right)&\mbox{{if
$n_{1}=n_{2},$ } }\\\ \hfil O_{A}((N^{\frac{1}{m}-1}X)^{-A})&\mbox{{if
$n_{1}\neq n_{2}.$ } }\end{cases}$

Therefore, we have

$\displaystyle\mathbb{E}_{x\sim
X}^{W}(\Delta_{f}(x,\delta;N)^{2})=\frac{4}{\pi^{2}}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}\cdot\left(\frac{1}{2}+O_{A}\left((2\breve{n}X)^{-A}\right)\right)$
$\displaystyle+\sum_{n_{1}\neq n_{2}\leqslant
N}\frac{\lambda_{f}(n_{1})\lambda_{f}(n_{2})}{\sqrt{n_{1}n_{2}}}\cdot\frac{\sin(\pi\breve{n_{1}}\delta)\sin(\pi\breve{n_{2}}\delta)}{\sqrt{\breve{n_{1}}\breve{n_{2}}}}\cdot
O_{A}((XN^{\frac{1}{m}-1})^{-A})$
$\displaystyle=\frac{2}{\pi^{2}}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}+O_{A}(\delta^{2}X^{-A})+O_{A}(\delta^{2}N^{2}(XN^{\frac{1}{m}-1})^{-A}).$

Note that the first error term is absorbed by the second. Also, since by
assumption $N\leqslant X^{\frac{m(1-\epsilon)}{m-1}},$ this is

$\displaystyle\sigma_{f}(\delta;N)^{2}+O_{B}\left(\delta^{2}X^{-B}\right).$

Moreover, this is $(1+O_{B}(X^{-B}))\sigma_{f}(\delta;N)^{2}$ since
$\sigma_{f}(\delta;N)^{2}\gg\delta^{2}.$ Therefore, the lemma follows. ∎

In fact, the truncated variance $\sigma_{f}(\delta;N)^{2}$ is a good
approximant.

######  Lemma 5.3.

Let $\delta>0,N\geqslant 2.$ Then

$\displaystyle\sigma_{f}(\delta;N)^{2}=\sigma_{f}(\delta)^{2}+O(N^{-\frac{1}{m}}\log^{r-1}N).$

In particular, if $\delta^{m}N\to\infty$ as $\delta\to 0^{+},$ then

$\displaystyle\sigma_{f}(\delta;N)^{2}=$
$\displaystyle(1+o(1))\sigma_{f}(\delta)^{2}$ $\displaystyle=$
$\displaystyle(c_{f}+o(1))m^{r}\delta\log^{r-1}\left(\frac{1}{\delta}\right).$

###### Proof.

By definition, we have

$\displaystyle\sigma_{f}(\delta)^{2}=\sigma_{f}(\delta;N)^{2}+\frac{2}{\pi^{2}}\sum_{n>N}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{\sin^{2}(\pi\breve{n}\delta)}{\breve{n}}.$

Assuming (3.1), the last sum is

$\displaystyle\ll\sum_{n>N}\frac{\lambda_{f}(n)^{2}}{n}\cdot\frac{1}{n^{1/m}}\ll\frac{\log^{r-1}N}{N^{1/m}}.$

Then, the lemma follows from Lemma 4.3. ∎

##  6\. Higher moments

Given $k\geqslant 3,\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k},\boldsymbol{r}\in\mathbb{N}^{k}$ and a subset
$S\subseteq[k]:=\\{1,2,\ldots,k\\},$ we define

$\displaystyle\langle\boldsymbol{\epsilon},\boldsymbol{r}\rangle|_{S}:=\sum_{j\in
S}\epsilon_{j}r_{j}.$

Also, let us denote

$\displaystyle D_{q}(S):=$
$\displaystyle\left(\frac{w}{\pi}\right)^{|S|}\underset{\begin{subarray}{c}\epsilon_{j},r_{j},\,j\in
S\\\
\langle\boldsymbol{\epsilon},\boldsymbol{r}\rangle|_{S}=0\end{subarray}}{\sum\sum}\prod_{j\in
S}\frac{\lambda_{f}(n_{j})}{\sqrt{n_{j}}}\cdot\frac{\sin\left(\pi\breve{n_{j}}\delta\right)}{\sqrt{\breve{n_{j}}}}\cdot
e^{i\epsilon_{j}\varphi},$

where $\epsilon_{j}=\pm 1,1\leqslant r_{j}\leqslant(N/q)^{1/m}$ for each $j\in
S,$ and $n_{j}=qr_{j}^{m}$ with $q$ being $m$-th power-free.

To compute higher moments, we need an estimate for the $L^{1}$-norm of
$D_{q}(S)$ over $m$-th power-free numbers.

######  Lemma 6.1.

Assume $\text{GRC}^{\flat}.$ Let $\delta>0,N\geqslant 2.$ Suppose
$\delta^{m}N\to\infty$ as $\delta\to 0^{+}.$ Then

$\displaystyle\frac{1}{\sigma_{f}(\delta;N)^{|S|}}\sum_{\begin{subarray}{c}q\leqslant
N\\\ \text{$m$-th power-
free}\end{subarray}}\left|D_{q}(S)\right|=\begin{cases}\hfil 0&\mbox{{if
$|S|=1,$ }}\\\ \hfil 1&\mbox{{if $|S|=2,$ }}\\\
O(\delta^{m((\frac{1}{3}+\eta)|S|-1)})&\mbox{{if $|S|\geqslant 3$.
}}\end{cases}$

###### Proof.

We adapt the proof of [HR04, Lemma 9] with $\text{GRC}^{\flat}.$ By
definition, when $|S|=1,$ the sum $D_{q}(S)$ is empty. Also, if $|S|=2,$ then
the sum is simply $\sigma_{f}(\delta;N)^{2}.$ Therefore, it remains to deal
with the case $|S|\geqslant 3.$ Using the assumption $\text{GRC}^{\flat},$ we
have

$\displaystyle\sum_{\begin{subarray}{c}q\leqslant N\\\ \text{$m$-th power-
free}\end{subarray}}\left|D_{q}(S)\right|\ll_{k}\sum_{q=1}^{\infty}\frac{E_{q}(S)}{q^{(\frac{1}{3}+\frac{1}{2m}+\eta)|S|}},$
(6.1)

where

$\displaystyle
E_{q}(S):=\underset{\begin{subarray}{c}\epsilon_{j},r_{j},\,j\in S\\\
\langle\boldsymbol{\epsilon},\boldsymbol{r}\rangle|_{S}=0\end{subarray}}{\sum\sum}\prod_{j\in
S}\frac{|\sin\left(\pi\breve{n_{j}}\delta\right)|}{r_{j}^{\frac{m}{3}+\frac{1}{2}+m\eta}}.$
(6.2)

Note that $E_{q}(S)\ll_{k}1$ for any $q.$ Moreover, we show that when
$q\leqslant\delta^{-m},$ a more careful treatment of the product yields a
sharper upper bound. In order for
$\langle\boldsymbol{\epsilon},\boldsymbol{r}\rangle|_{S}=0$ to hold, at least
of two of the $\epsilon$’s must have opposite signs. For convenience, we
assume without loss of generality that $S=\\{1,2,\ldots,|S|\\}$ and
$\epsilon_{|S|}=-1,\epsilon_{|S|-1}=+1$ so that

$\displaystyle r_{|S|}=r_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}r_{j}.$

In order for $r_{|S|},r_{|S|-1}\geqslant 1$ to hold, we have
$r_{|S|-1}\geqslant 1+\max\\{0,-\sum_{j=1}^{|S|-2}\epsilon_{j}r_{j}\\}.$
Therefore, the expression (6.2) is

$\displaystyle E_{q}(S)\leqslant 2$
$\displaystyle\sum_{\begin{subarray}{c}\epsilon_{j}=\pm 1\\\
j=1,\ldots,|S|-2\end{subarray}}\sum_{\begin{subarray}{c}r_{j}\geqslant 1\\\
j=1,\ldots,|S|-2\end{subarray}}\sum_{r_{|S|-1}\geqslant
1+\max\\{0,-\sum_{j=1}^{|S|-2}\epsilon_{j}r_{j}\\}}$
$\displaystyle\left(\prod_{j=1}^{|S|-1}\frac{|\sin\left(\pi
mr_{j}(q/D)^{1/m}\delta\right)|}{r_{j}^{\frac{m}{3}+\frac{1}{2}+m\eta}}\right)\frac{|\sin\left(\pi
m\left(r_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}r_{j}\right)(q/D)^{1/m}\delta\right)|}{\left(r_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}r_{j}\right)^{\frac{m}{3}+\frac{1}{2}+m\eta}},$

which is

$\displaystyle\ll\sum_{\begin{subarray}{c}\epsilon_{j}=\pm 1\\\
j=1,\ldots,|S|-2\end{subarray}}$
$\displaystyle\underset{\begin{subarray}{c}x_{1},\ldots,x_{|S|-2}\geqslant
1\\\ x_{|S|-1}\geqslant
1+\max\\{0,-\sum_{j=1}^{|S|-2}\epsilon_{j}x_{j}\\}\end{subarray}}{\int\cdots\int}\left(\prod_{j=1}^{|S|-1}\frac{|\sin\left(\pi
mx_{j}(q/D)^{1/m}\delta\right)|}{x_{j}^{\frac{m}{3}+\frac{1}{2}+m\eta}}\right)$
$\displaystyle\cdot\frac{|\sin\left(\pi
m\left(x_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}x_{j}\right)(q/D)^{1/m}\delta\right)|}{\left(x_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}x_{j}\right)^{\frac{m}{3}+\frac{1}{2}+m\eta}}\,dx_{1}\cdots
dx_{|S|-1}.$

Making the change of variables $y_{j}=mx_{j}(q/D)^{1/m}\delta$ for
$j=1,\ldots,|S|-1,$ this becomes

$\displaystyle\left(m\left(\frac{q}{D}\right)^{1/m}\delta\right)^{1+\left(\frac{m}{3}-\frac{1}{2}+m\eta\right)|S|}\sum_{\begin{subarray}{c}\epsilon_{j}=\pm
1\\\
j=1,\ldots,|S|-2\end{subarray}}\underset{\begin{subarray}{c}y_{1},\ldots,y_{|S|-2}\geqslant
m(q/D)^{1/m}\delta\\\ y_{|S|-1}\geqslant
m(q/D)^{1/m}\delta+\max\\{0,-\sum_{j=1}^{|S|-2}\epsilon_{j}y_{j}\\}\end{subarray}}{\int\cdots\int}$
$\displaystyle\left(\prod_{j=1}^{|S|-1}\frac{|\sin\left(\pi
y_{j}\right)|}{y_{j}^{\frac{m}{3}+\frac{1}{2}+m\eta}}\right)\frac{\left|\sin\left(\pi\left(y_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}y_{j}\right)\right)\right|}{\left(y_{|S|-1}+\sum_{j=1}^{|S|-2}\epsilon_{j}y_{j}\right)^{\frac{m}{3}+\frac{1}{2}+m\eta}}\,dy_{1}\cdots
dy_{|S|-1}.$

Since the multiple integral is uniformly bounded, we conclude that

$\displaystyle
E_{q}(S)\ll_{k}\begin{cases}(q^{1/m}\delta)^{1+\left(\frac{m}{3}-\frac{1}{2}+m\eta\right)|S|}&\mbox{{if
$q\leqslant\delta^{-m},$ }}\\\ \hfil 1&\mbox{{if $q>\delta^{-m}.$
}}\end{cases}$ (6.3)

Then, substituting this into (6.1) gives

$\displaystyle\sum_{\begin{subarray}{c}q\leqslant N\\\ \text{$m$-th power-
free}\end{subarray}}\left|D_{q}(S)\right|\ll_{k}$
$\displaystyle\sum_{q\leqslant\delta^{-m}}q^{-\left(\frac{1}{3}+\frac{1}{2m}+\eta\right)|S|}\left(q^{1/m}\delta\right)^{1+\left(\frac{m}{3}-\frac{1}{2}+m\eta\right)|S|}$
$\displaystyle+$
$\displaystyle\sum_{q>\delta^{-m}}q^{-\left(\frac{1}{3}+\frac{1}{2m}+\eta\right)|S|}.$
(6.4)

Appealing to Lemma 5.3, the second sum here is

$\displaystyle\ll\delta^{-m+\left(\frac{m}{3}+\frac{1}{2}+m\eta\right)|S|}\ll\delta^{m\left(\left(\frac{1}{3}+\eta\right)|S|-1\right)}\sigma_{f}(\delta;N)^{|S|}.$
(6.5)

On the other hand, since

$\displaystyle\sum_{q\leqslant\delta^{-m}}q^{-\left(\frac{1}{3}+\frac{1}{2m}+\eta\right)|S|}\cdot
q^{\frac{1}{m}+\left(\frac{1}{3}-\frac{1}{2m}+\eta\right)|S|}=\sum_{q\leqslant\delta^{-m}}q^{-\frac{1}{m}(|S|-1)}$
$\displaystyle\ll_{k}\begin{cases}\hfil\delta^{|S|-1-m}&\mbox{{if
$|S|\leqslant m,$ }}\\\ \hfil\log\frac{1}{\delta}&\mbox{{if $|S|=m+1,$ }}\\\
\hfil 1&\mbox{{if $|S|>m+1,$ }}\end{cases}$

the first sum of (6.1) is

$\displaystyle\ll_{k}\begin{cases}\hfil\delta^{-m+\left(\frac{m}{3}+\frac{1}{2}+m\eta\right)|S|}&\mbox{{if
$|S|\leqslant m,$ }}\\\
\hfil\delta^{1+\left(\frac{m}{3}+\frac{1}{2}+m\eta\right)|S|}\log\frac{1}{\delta}&\mbox{{if
$|S|=m+1,$ }}\\\
\hfil\delta^{1+\left(\frac{m}{3}+\frac{1}{2}+m\eta\right)|S|}&\mbox{{if
$|S|>m+1.$ }}\end{cases}$

Combining with (6.5), the lemma follows. ∎

With the lemma in hand, we are able to compute higher moments.

######  Proposition 6.1.

Given $X,\delta>0$ and $k\geqslant 3,$ let $2\leqslant N\leqslant
X^{\frac{m(1-\eta)}{m^{k}-1}}$ for some fixed small $\eta>0.$ Suppose
$\delta=o(1)$ but $\delta^{m}N\to\infty$ as $X\to\infty.$ Then

$\displaystyle\mathbb{E}_{x\sim
X}^{W}(\Delta_{f}(x,\delta;N)^{k})=(\mu_{k}+o_{k}(1))\sigma_{f}(\delta;N)^{k},$

where

$\displaystyle\mu_{k}:=\begin{cases}\frac{k!}{2^{k/2}(k/2)!}&\mbox{{if $k$ is
even,} }\\\ \hfil 0&\mbox{{otherwise. }}\end{cases}$

###### Proof.

Let $[N]:=\\{1,2,\ldots,N\\}.$ Then similar to Lemma 5.1 and 5.2, for each
$\boldsymbol{n}\in[N]^{k}$, we have

$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\prod_{j=1}^{k}\cos\left(\pi\breve{n_{j}}(2x+\delta)+\varphi\right)W\left(\frac{x}{X}\right)dx$
$\displaystyle=$
$\displaystyle\frac{1}{2^{k}}\sum_{\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k}}\prod_{j=1}^{k}e^{i\epsilon_{j}(\pi\breve{n_{j}}\delta+\varphi)}\cdot\widehat{W}\left(-\left(\sum_{j=1}^{k}\epsilon_{j}\breve{n_{j}}\right)X\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2^{k}}\sum_{\begin{subarray}{c}\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k}\\\
\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle=0\end{subarray}}\prod_{j=1}^{k}e^{i\epsilon_{j}\varphi}+O_{A}\left(\frac{1}{2^{k}}\sum_{\begin{subarray}{c}\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k}\\\ \langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle\neq
0\end{subarray}}\left(\left|\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle\right|X\right)^{-A}\right),$

where
$\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle:=\sum_{j=1}^{k}\epsilon_{j}\breve{n_{j}}.$
Since by Lemma 4.2

$\displaystyle\left|\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle\right|\geqslant\frac{m/D^{1/m}}{\left(kN^{1/m}\right)^{m^{k}-1}}$

whenever $\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle\neq 0,$
the error term here is

$\displaystyle\ll_{A}\left(\frac{X}{\left(kN^{1/m}\right)^{m^{k}-1}}\right)^{-A}$
$\displaystyle\ll_{A,k}N^{A(m^{k}-1)/m}X^{-A}$ $\displaystyle\ll_{A,k}X^{-\eta
A}.$

Therefore, we have

$\displaystyle\left(\frac{2w}{\pi}\right)^{k}\underset{\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle\neq
0}{\sum_{\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k}}\sum_{\boldsymbol{n}\in[N]^{k}}}\prod_{j=1}^{k}\frac{\lambda_{f}(n_{j})}{\sqrt{n_{j}}}\cdot\frac{\sin\left(\pi\breve{n_{j}}\delta\right)}{\sqrt{\breve{n_{j}}}}\cdot
O_{A,k}\left(X^{-\eta A}\right)\ll_{A,k}$
$\displaystyle\delta^{k}N^{k}X^{-\eta A}$ $\displaystyle\ll_{B,k}$
$\displaystyle\delta^{k}X^{-B},$

so that

$\displaystyle\mathbb{E}_{x\sim X}^{W}(\Delta_{f}(x,\delta;N)^{k})=$
$\displaystyle\left(\frac{w}{\pi}\right)^{k}\underset{\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle=0}{\sum_{\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k}}\sum_{\boldsymbol{n}\in[N]^{k}}}\prod_{j=1}^{k}\frac{\lambda_{f}(n_{j})}{\sqrt{n_{j}}}\cdot\frac{\sin\left(\pi\breve{n_{j}}\delta\right)}{\sqrt{\breve{n_{j}}}}\cdot
e^{i\epsilon_{j}\varphi}+O_{B,k}(\delta^{k}X^{-B}).$

It remains to deal with the diagonal sum. Appealing to Lemma 4.1, if
$\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle=0,$ then there
must exist a decomposition

$\displaystyle[k]:=\\{1,2,\ldots,k\\}=\bigsqcup_{i=1}^{l}S_{i}$

such that for $i=1,\ldots,l$ and $j\in S_{i},$ we have $n_{j}=q_{i}r_{j}^{m}$
with $q_{i}$’s being distinct $m$-th power-free and with $r_{j}$’s satisfying

$\displaystyle\langle\boldsymbol{\epsilon},\boldsymbol{r}\rangle|_{S_{i}}:=\sum_{j\in
S_{i}}\epsilon_{j}r_{j}=0.$

Therefore, the diagonal sum is

$\displaystyle\left(\frac{w}{\pi}\right)^{k}\underset{\langle\boldsymbol{\epsilon},\boldsymbol{\breve{n}}\rangle=0}{\sum_{\boldsymbol{\epsilon}\in\\{\pm
1\\}^{k}}\sum_{\boldsymbol{n}\in[N]^{k}}}\prod_{j=1}^{k}\frac{\lambda_{f}(n_{j})}{\sqrt{n_{j}}}\cdot\frac{\sin\left(\pi\breve{n_{j}}\delta\right)}{\sqrt{\breve{n_{j}}}}\cdot
e^{i\epsilon_{j}\varphi}$
$\displaystyle=\sum_{l=1}^{k}\sum_{[k]=\bigsqcup_{i=1}^{l}S_{i}}\sum_{\begin{subarray}{c}\boldsymbol{q}\in[N]^{l}\\\
\text{$q_{i}$ distinct $m$-th power-
free}\end{subarray}}\prod_{i=1}^{l}D_{q_{i}}(S_{i}),$

which is

$\displaystyle\sum_{\begin{subarray}{c}[k]=\bigsqcup_{i=1}^{k/2}S_{i}\\\
|S_{i}|=2\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{q}\in[N]^{l}\\\
\text{$q_{i}$ distinct $m$-th power-
free}\end{subarray}}\prod_{i=1}^{l}D_{q_{i}}(S_{i})+\sum_{l=1}^{k}\sum_{\begin{subarray}{c}[k]=\bigsqcup_{i=1}^{l}S_{i}\\\
\exists\,i\,:\,|S_{i}|\neq
2\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{q}\in[N]^{l}\\\
\text{$q_{i}$ distinct $m$-th power-
free}\end{subarray}}\prod_{i=1}^{l}D_{q_{i}}(S_{i})$
$\displaystyle=:\Sigma_{1}+\Sigma_{2}.$

Let us deal with the second sum $\Sigma_{2}$ first. Since $D_{q}(S)=0$ if
$|S|=1$, we can assume that $|S_{i}|\geqslant 2$ for all $i=1,\ldots,l.$
Therefore, Lemma 6.1 implies that

$\displaystyle\Sigma_{2}\ll_{k}\sigma(\delta;N)^{k}\delta^{3m\eta.}$ (6.6)

On the other hand, note that the first sum $\Sigma_{1}$ vanishes if $k$ is
odd. Otherwise, if $k$ is even, then we have

$\displaystyle\Sigma_{1}$
$\displaystyle=\sum_{\begin{subarray}{c}[k]=\bigsqcup_{i=1}^{k/2}S_{i}\\\
|S_{i}|=2\end{subarray}}\prod_{i=1}^{k/2}\sum_{\begin{subarray}{c}q\leqslant
N\\\ \text{$m$-th power-
free}\end{subarray}}D_{q}(S_{i})-\sum_{\begin{subarray}{c}[k]=\bigsqcup_{i=1}^{k/2}S_{i}\\\
|S_{i}|=2\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{q}\in[N]^{k/2}\\\
\text{$q_{i}$ non-distinct $m$-th power-
free}\end{subarray}}\prod_{i=1}^{k/2}D_{q_{i}}(S_{i}),$

where the first sum here is simply $\mu_{k}\cdot\sigma_{f}(\delta;N)^{k}$ by
Lemma 6.1.

On the other hand, since

$\displaystyle
D_{q}(S)\ll_{k}\frac{E_{q}(S)}{q^{\left(\frac{1}{3}+\frac{1}{2m}+\eta\right)|S|}},$

where $E_{q}(S)$ is defined in (6.2), it follows from (6.3) that
$D_{q}(S)\ll_{k}\delta^{\left(\frac{2}{3}+2\eta\right)m}.$ Therefore, the
second sum is
$\ll_{k}\sigma(\delta;N)^{k-2}\delta^{\left(\frac{2}{3}+2\eta\right)m}$ again
by Lemma 6.1. Then, combining with (6.6), the proposition follows from Lemma
5.3. ∎

##  7\. $L^{2}$-deviation of the approximants

When the degree $m\geqslant 3,$ the (pointwise) error term given by
Proposition 4.1 is not sufficient to prove Theorem 3.1. Inspired by the work
of Lester [Les16] and Jääsaari [Jää20], we establish bounds for the
$L^{2}$-deviation of the approximants $\Delta_{f}(x,\delta;N)$ and
$(x+\delta)^{-\frac{m-1}{2}}\Delta_{f}((x+\delta)^{m}),$ except that in order
to prove Theorem 3.1, our truncation is at $N=X^{o(1)}$ instead of a fixed
power of $X$ as $X\to\infty.$

######  Lemma 7.1.

Assume GLH. Let $X,\delta>0$ be real numbers satisfying the conditions in
Theorem 3.1. Then there exists $N\geqslant 2$ for which $\delta^{m}N\to\infty$
as $X\to\infty$ such that

$\displaystyle\mathbb{E}^{W}_{x\sim
X}\left(\left(\Delta_{f}(x,\delta)-\Delta_{f}(x,\delta;N)\right)^{2}\right)=o(\sigma_{f}(\delta)^{2}).$

When $m=2,$ the assumption of GLH can be dropped.

###### Proof.

Let us denote

$\displaystyle\Delta_{f}(x^{m};N):=\frac{w}{\pi}\cdot
x^{\frac{m-1}{2}}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin\left(2\pi\breve{n}x+\varphi\right)}{\sqrt{\breve{n}}}.$

Then by definition, we have

$\displaystyle\left(\Delta_{f}(x,\delta)-\Delta_{f}(x,\delta;N)\right)^{2}$
$\displaystyle=\left(\left(\frac{\Delta_{f}((x+\delta)^{m})}{(x+\delta)^{\frac{m-1}{2}}}-\frac{\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}}\right)-\left(\frac{\Delta_{f}((x+\delta)^{m};N)}{(x+\delta)^{\frac{m-1}{2}}}-\frac{\Delta_{f}(x^{m};N)}{x^{\frac{m-1}{2}}}\right)\right)^{2}$
$\displaystyle\leqslant
2\left(\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}((x+\delta)^{m};N)}{(x+\delta)^{\frac{m-1}{2}}}\right)^{2}+2\left(\frac{\Delta_{f}(x^{m})-\Delta_{f}(x^{m};N)}{x^{\frac{m-1}{2}}}\right)^{2}.$

Therefore, it suffices to bound

$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\left(\frac{\Delta_{f}(x^{m})-\Delta_{f}(x^{m};N)}{x^{\frac{m-1}{2}}}\right)^{2}W\left(\frac{x}{X}\right)dx.$
(7.1)

When $m=2,$ Proposition 4.1 implies that

$\displaystyle\frac{\Delta_{f}(x^{m})-\Delta_{f}(x^{m};N)}{x^{\frac{m-1}{2}}}=\frac{w}{\pi}\sum_{N<n\leqslant
x^{m}}\frac{\lambda_{f}(n)}{\sqrt{n}}\cdot\frac{\sin\left(2\pi\breve{n}x+\varphi\right)}{\sqrt{\breve{n}}}+O_{\epsilon}(x^{-\frac{1}{2}+\epsilon}).$

Arguing similarly as in Lemma 5.3, one can show that (7.1) is

$\displaystyle\ll_{\epsilon}\sum_{N<n\leqslant(5X/2)^{m}}\frac{\lambda_{f}(n)^{2}}{n}+X^{-1+2\epsilon}.$

Assuming (3.1), the sum here is

$\displaystyle\ll\frac{\log^{r-1}N}{N^{1/m}}=o(\sigma_{f}(\delta)^{2}).$

Also, since $\log\frac{1}{\delta}=o(\log X)$ as $X\to\infty,$ we have
$X^{-1+2\epsilon}=o(\sigma_{f}(\delta)^{2})$ as well.

It remains to bound (7.1) for $m\geqslant 3$ under GLH. Similar to [Les16,
Proposition 2.2] and [Jää20, Proposition 9], by truncating and shifting the
contour of the integral

$\displaystyle\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}L(f,s)\frac{x^{ms}}{s}ds$

appropriately, one can show that

$\displaystyle\Delta_{f}(x^{m})=\sum_{n\leqslant
x^{m}}\lambda_{f}(n)-\mathop{\mathrm{Res}}_{s=1}L(f,s)\frac{x^{ms}}{s}$
$\displaystyle=\mathrm{Re}\left(\frac{1}{\pi
i}\left\\{\int_{-\epsilon}^{-\epsilon+iT_{0}}+\int_{\frac{1}{2}+iT_{0}}^{\frac{1}{2}+iT}+\int_{-\epsilon+iT_{0}}^{\frac{1}{2}+iT_{0}}+\int_{\frac{1}{2}+iT}^{1+\epsilon+iT}+\int_{1+\epsilon+iT}^{1+\epsilon+i\infty}\right\\}L(f,s)x^{ms}\frac{ds}{s}\right)+L(f,0)$

for $1\leqslant T_{0}<T\leqslant x^{m}$ to be determined later. Using an
estimate due to Friedlander–Iwaniec (see [FI05, Section 2]), there exists a
positive increasing function $\psi(x)\ll_{\epsilon}x^{\epsilon}$ for which
$\psi(x)\to\infty$ as $x\to\infty$ such that

$\displaystyle\mathrm{Re}\left(\frac{1}{\pi
i}\int_{-\epsilon}^{-\epsilon+iT_{0}}L(f,s)x^{ms}\frac{ds}{s}\right)=\Delta_{f}(x^{m};N)+O\left(x^{m}\psi(x)^{m}{T_{0}}^{-\frac{m+1}{2}}+\psi(x)^{m}{T_{0}}^{\frac{m}{2}-1}\right),$

where we set $T_{0}=2\pi(N/D)^{\frac{1}{m}}x.$ Using the assumption (3.1), one
can show that

$\displaystyle\int_{1+\epsilon+iT}^{1+\epsilon+i\infty}L(f,s)x^{ms}\frac{ds}{s}\ll_{\epsilon}$
$\displaystyle
x^{m\epsilon}\sum_{n=1}^{\infty}\frac{|\lambda_{f}(n)|}{n^{1+\epsilon}\max\\{1,T|\log(x^{m}/n)|\\}}$
$\displaystyle\ll_{\epsilon}$ $\displaystyle x^{(1+2\epsilon)m}T^{-1}$

(see [Kou19, p. 71] for instance). Also, assuming GLH, we have

$\displaystyle\int_{\frac{1}{2}+iT}^{1+\epsilon+iT}L(f,s)x^{ms}\frac{ds}{s}\ll_{\epsilon}x^{\frac{m}{2}}T^{-1+m\epsilon}$

and

$\displaystyle\int_{-\epsilon+iT_{0}}^{\frac{1}{2}+iT_{0}}L(f,s)x^{ms}\frac{ds}{s}\ll_{\epsilon}x^{-m\epsilon}{T_{0}}^{\frac{m}{2}-1+2m\epsilon},$

so that (7.1) is

$\displaystyle\ll_{\epsilon}$
$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\left|\int_{\frac{1}{2}+iT_{0}}^{\frac{1}{2}+iT}L(f,s)x^{ms}\frac{ds}{s}\right|^{2}W\left(\frac{x}{X}\right)\frac{dx}{x^{m-1}}+\frac{T_{0}^{m-2+4m\epsilon}}{X^{m-1-2m\epsilon}}+\frac{X^{m+1}\psi(X)^{2m}}{T_{0}^{m+1}}+\frac{X^{m+1+4m\epsilon}}{T^{2-2m\epsilon}}$
$\displaystyle=$
$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\left|\int_{\frac{1}{2}+iT_{0}}^{\frac{1}{2}+iT}L(f,s)x^{ms}\frac{ds}{s}\right|^{2}W\left(\frac{x}{X}\right)\frac{dx}{x^{m-1}}+\frac{N^{1-\frac{2}{m}+4\epsilon}}{X^{1-6m\epsilon}}+\frac{\psi(X)^{2m}}{N^{1+\frac{1}{m}}}+\frac{X^{m+1+4m\epsilon}}{T^{2-2m\epsilon}}.$
(7.2)

It remains to bound the first integral, which is

$\displaystyle\iint_{(u,v)\in[T_{0},T]^{2}}\frac{L(f,\frac{1}{2}+iu)L(f,\frac{1}{2}-iv)}{(\frac{1}{2}+iu)(\frac{1}{2}-iv)}\left(\frac{1}{X}\int_{-\infty}^{\infty}x^{1+im(u-v)}W\left(\frac{x}{X}\right)dx\right)dudv.$
(7.3)

Since $\widehat{W}$ decays rapidly, by making the change of variables
$y=\frac{x}{X},$ the inner integral is

$\displaystyle
X^{1+im(u-v)}\int_{-\infty}^{\infty}y^{1+im(u-v)}W(y)dy\leqslant
g(A)X\min\\{1,|u-v|^{-A}\\}$

for some positive increasing function $g(A).$ Let $\theta(X)$ be a positive
(slowly) decreasing function for which $\theta(X)=o_{X\to\infty}(1)$ and
$g(\theta(X)^{-1})\leqslant\log X$. Then the expression (7.3) is

$\displaystyle\leqslant
Xg(A)\iint_{\begin{subarray}{c}(u,v)\in[T_{0},T]^{2}\\\ |u-v|\leqslant
X^{2\theta(X)}\end{subarray}}\left|\frac{L(f,\frac{1}{2}+iu)L(f,\frac{1}{2}-iv)}{(\frac{1}{2}+iu)(\frac{1}{2}-iv)}\right|dudv$
$\displaystyle+X^{1-2A\theta(X)}g(A)\iint_{\begin{subarray}{c}(u,v)\in[T_{0},T]^{2}\\\
|u-v|>X^{2\theta(X)}\end{subarray}}\left|\frac{L(f,\frac{1}{2}+iu)L(f,\frac{1}{2}-iv)}{(\frac{1}{2}+iu)(\frac{1}{2}-iv)}\right|dudv.$

Assuming GLH, there exists a positive increasing function
$\phi(x)\ll_{\epsilon}x^{\epsilon}$ satisfying $\phi(x)\to\infty$ as
$x\to\infty$ for which the integrands here are
$\leqslant(\phi(u)\phi(v))^{m}(uv)^{-1}.$ Then this is

$\displaystyle\ll$ $\displaystyle
X^{1+2\theta(X)}g(A)\phi(2T_{0})^{2m}T_{0}^{-1}+X^{1-2A\theta(X)}g(A)\phi(T)^{2m}\log^{2}T$
$\displaystyle=$ $\displaystyle
X^{2\theta(X)}g(A)\phi(4\pi(N/D)^{1/m}X)^{2m}N^{-\frac{1}{m}}+X^{1-2A\theta(X)}g(A)\phi(T)^{2m}\log^{2}T.$

Finally, combining with (7), the lemma follows from taking
$T=(X/2)^{m},A=\theta(X)^{-1},N=\delta^{-m}g(\theta(X)^{-1})^{m}X^{2m\theta(X)}\psi(X)^{2m}\phi(X^{2})^{2m^{2}}$
and letting $\epsilon\to 0^{+}.$ ∎

######  Lemma 7.2.

Let $X,\delta>0$ be real numbers satisfying the conditions in Theorem 3.1.
Then as $X\to\infty,$ we have

$\displaystyle\mathbb{E}^{W}_{x\sim
X}\left(\left(\frac{\Delta_{f}((x+\delta)^{m})}{(x+\delta)^{\frac{m-1}{2}}}-\frac{\Delta_{f}((x+\delta)^{m})}{x^{\frac{m-1}{2}}}\right)^{2}\right)=o(\sigma_{f}(\delta)^{2}).$

###### Proof.

Since

$\displaystyle\frac{\Delta_{f}((x+\delta)^{m})}{(x+\delta)^{\frac{m-1}{2}}}=\frac{\Delta_{f}((x+\delta)^{m})}{x^{\frac{m-1}{2}}}+O\left(\delta\cdot\frac{\Delta_{f}((x+\delta)^{m})}{x^{\frac{m+1}{2}}}\right),$

it suffices to bound

$\displaystyle\frac{1}{X}\int_{-\infty}^{\infty}\left(\frac{\Delta_{f}(x^{m})}{x^{\frac{m+1}{2}}}\right)^{2}W\left(\frac{x}{X}\right)dx,$

which is

$\displaystyle\ll\frac{1}{X}\int_{-\infty}^{\infty}\left(\frac{\Delta_{f}(x^{m})-\Delta_{f}(x^{m};N)}{x^{\frac{m+1}{2}}}\right)^{2}W\left(\frac{x}{X}\right)dx+\frac{1}{X}\int_{-\infty}^{\infty}\left(\frac{\Delta_{f}(x^{m};N)}{x^{\frac{m+1}{2}}}\right)^{2}W\left(\frac{x}{X}\right)dx$

with
$N=\delta^{-m}g(\theta(X)^{-1})^{m}X^{2m\theta(X)}\psi(X)^{2m}\phi(X^{2})^{2m^{2}}$
as defined in the last line of the proof of Lemma 7.1. Appealing to Lemma 7.1
and (7.1), the first integral is $o(\sigma_{f}(\delta)^{2}X^{-2}).$

On the other hand, arguing similarly as in Lemma 5.3, the second integral is

$\displaystyle\ll\frac{1}{X^{2}}\sum_{n\leqslant
N}\frac{\lambda_{f}(n)^{2}}{n^{1+\frac{1}{m}}}\ll\frac{1}{X^{2}}\cdot\frac{\log^{r-1}N}{N^{1/m}},$

which is again $o(\sigma_{f}(\delta)^{2}X^{-2}).$ Therefore, the proof is
completed. ∎

##  8\. Proof of Theorem 3.1

We are now set to prove Theorem 3.1. Given a Borel subset
$B\subseteq\mathbb{R},$ we denote

$\displaystyle\mathbb{P}_{x\sim
X}^{W}\left(B\right):=\frac{1}{X}\int_{B}W\left(\frac{x}{X}\right)dx.$

Let
$N=\delta^{-m}g(\theta(X)^{-1})^{m}X^{2m\theta(X)}\psi(X)^{2m}\phi(X^{2})^{2m^{2}}$
as defined in the last line of the proof of Lemma 7.1. In particular, it
satisfies $\delta^{m}N\to\infty$ but $\log N=o\left(\log X\right)$ as
$X\to\infty,$ so that Lemma 5.1, Lemma 5.2 and Proposition 6.1 are applicable.
Then, using the method of moments (see [Bil12, Theorem 30.1] for instance), we
have

$\displaystyle\mathbb{P}_{x\sim
X}^{W}\left(\frac{\Delta_{f}(x,\delta;N)}{\sigma_{f}(\delta;N)}\in(\alpha,\beta]\right)=\frac{1}{\sqrt{2\pi}}\int_{\alpha}^{\beta}e^{-\frac{t^{2}}{2}}dt+o(1)$

for any fixed real numbers $\alpha<\beta.$ Since
$\sigma_{f}(\delta;N)=(1+o(1))\sigma_{f}(\delta)$ by Lemma 5.3, it follows
that

$\displaystyle\mathbb{P}_{x\sim
X}^{W}\left(\frac{\Delta_{f}(x,\delta;N)}{\sigma_{f}(\delta)}\in(\alpha,\beta]\right)=\frac{1}{\sqrt{2\pi}}\int_{\alpha}^{\beta}e^{-\frac{t^{2}}{2}}dt+o(1).$

Appealing to Lemma 7.1, we have

$\displaystyle\mathbb{E}^{W}_{x\sim
X}\left(\left(\Delta_{f}(x,\delta)-\Delta_{f}(x,\delta;N)\right)^{2}\right)=o(\sigma_{f}(\delta)^{2}),$

so that Chebyshev’s inequality gives

$\displaystyle\mathbb{P}_{x\sim
X}^{W}\left(\frac{\Delta_{f}(x,\delta)}{\sigma_{f}(\delta)}\in(\alpha,\beta]\right)=\frac{1}{\sqrt{2\pi}}\int_{\alpha}^{\beta}e^{-\frac{t^{2}}{2}}dt+o(1).$

Similarly, since

$\displaystyle\mathbb{E}^{W}_{x\sim
X}\left(\left(\frac{\Delta_{f}((x+\delta)^{m})}{(x+\delta)^{\frac{m-1}{2}}}-\frac{\Delta_{f}((x+\delta)^{m})}{x^{\frac{m-1}{2}}}\right)^{2}\right)=o(\sigma_{f}(\delta)^{2})$

by Lemma 7.2, we arrive at

$\displaystyle\mathbb{P}_{x\sim
X}^{W}\left(\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}\sigma_{f}(\delta)}\in(\alpha,\beta]\right)=\frac{1}{\sqrt{2\pi}}\int_{\alpha}^{\beta}e^{-\frac{t^{2}}{2}}dt+o(1).$

Finally, let $0<\epsilon<1.$ We choose $W$ such that $W\equiv 1$ on
$(1+\epsilon,2-\epsilon).$ Since $W$ is nonnegative and of unit mass, we have

$\displaystyle\frac{1}{X}\mathop{\mathrm{meas}}\left\\{x\in[X,2X]\,:\,\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}\sigma_{f}(\delta)}\in(\alpha,\beta]\right\\}$
$\displaystyle-\mathbb{P}_{x\sim
X}^{W}\left(\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}\sigma_{f}(\delta)}\in(\alpha,\beta]\right)$
$\displaystyle=\frac{1}{X}\int_{-\infty}^{\infty}1_{(\alpha,\beta]}\left(\frac{\Delta_{f}((x+\delta)^{m})-\Delta_{f}(x^{m})}{x^{\frac{m-1}{2}}\sigma_{f}(\delta)}\right)\left(1_{[1,2]}\left(\frac{x}{X}\right)-W\left(\frac{x}{X}\right)\right)dx$
$\displaystyle\leqslant\int_{\mathbb{R}\setminus(1+\epsilon,2-\epsilon)}\left(1_{[1,2]}(x)+W(x)\right)dx=4\epsilon.$

Letting $\epsilon\to 0^{+},$ the proof is completed.

##  9\. Uniformity in L-functions

Since it is unclear how the Rankin–Selberg type partial sum estimate (3.1)
depends on $L(f,s)$ in general, our main result, Theorem 3.1, lacks uniformity
in L-functions. Nevertheless, in certain special cases, such as Example 3.2
where $\lambda_{f}(n)=\tau_{k}(n),$ we claim that by carefully tracking all
dependencies on $k,$ if $\delta=o\left(\frac{1}{k}\right)$ but
$\log\left(\frac{1}{k\delta}\right)=o\left(\frac{1}{k}\log x\right)$ as
$X\to\infty,$ then the theorem holds uniformly in $k=k(X).$ A detailed proof
will be provided in the author’s doctoral thesis.

## Acknowledgements

The author is grateful to Andrew Granville for his advice and encouragement.
He would also like to thank Bingrong Huang and Steve Lester for helpful
discussion. Part of the work was supported by the Swedish Research Council
under grant no. 2016-06596 while the author was in residence at Institut
Mittag-Leffler in Djursholm, Sweden during the semester of Winter 2024.

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