# Local Fourier uniformity of higher divisor functions on average

Mengdi Wang  Department of Mathematics and Statistics, University of Turku,
Turku 20014, Finland<EMAIL_ADDRESS>
###### Abstract.

Let $\tau_{k}$ be the $k$-fold divisor function. By constructing an
approximant of $\tau_{k}$, denoted as $\tau_{k}^{*}$, which is a normalized
truncation of the $k$-fold divisor function, we prove that when
$\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leqslant H\leqslant X$ and
$C>0$ is sufficiently large, the following estimate holds for almost all
$x\in[X,2X]$:

$\Big{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau_{k}^{*}(n))e(\alpha_{d}n^{d}+\cdots+\alpha_{1}n)\Big{|}=o(H\log^{k-1}X),$

where $\alpha_{1},\dots,\alpha_{d}\in\mathbb{R}$ are arbitrary frequencies.

##  1\. Introduction

Denote $\tau_{k}(n)$ as the $k$-fold divisor function, signifying the number
of representations of $n$ as the product $n=n_{1}\cdots n_{k}$ of $k$ natural
numbers, where $k\geqslant 2$ is a fixed integer. The Dirichlet hyperbola
method leads to the following asymptotic expression

$\sum_{X<n\leqslant 2X}\tau_{k}(n)=XP_{k}(\log X)+O(X^{1-1/k+\varepsilon}),$

where $P_{k}$ is a polynomial of degree $k-1$ and $\varepsilon>0$ is
arbitrarily small. We expect a similar mean value result for $\tau_{k}$ within
a relatively short interval $[X,X+H]$. For instance, according to [10, Theorem
1.3], when $X^{0.55+\varepsilon}\leqslant H\leqslant X$, one has

$\sum_{X<n\leqslant X+H}\tau_{k}(n)=HP_{k}(\log
X)+O(H\log^{(2/3+\varepsilon)k-1}X).$

Meanwhile, in a recent paper [13], it is demonstrated that, assuming
$\log^{k\log k-k+1+\varepsilon}X\leqslant H\leqslant X$, for almost all
integers $x\in(X,2X]$, the following holds

$\sum_{x<n\leqslant x+H}\tau_{k}(n)=C_{k}H\log^{k-1}X+o(H\log^{k-1}X),$

where $C_{k}>0$ is a constant. Building on these observations, we say a
1-bounded function $g:\mathbb{N}\to\mathbb{C}$ is orthogonal to $\tau_{k}$ in
the interval $[X,X+H]$ if

$\sum_{X<n\leqslant X+H}\tau_{k}(n)g(n)=o(H\log^{k-1}X);$

otherwise, we say $\tau_{k}$ and $g$ are correlated in the interval $[X,X+H]$.

Assume $e(n\alpha)$ denotes a major-arc phase. It is known that $\tau_{k}$ and
$e(\cdot\alpha)$ are correlated in $[X,2X]$. To address this and establish the
orthogonality of the divisor function with all phases, we can either transfer
our focus to consider the $W$-tricked divisor functions, as demonstrated in
[11, Theorem 9.1] (for $k=2$); or, as in [5, 9], subtract off a suitable
approximant from $\tau_{k}$.

In the series of papers [5, 9], Matomäki, Radziwiłł, Shao, Tao and Teräväinen
construct an approximant $\tau_{k}^{\\#}$ (defined in [9, (1.2)]) and show
that $\tau_{k}-\tau_{k}^{\\#}$ is orthogonal to degree $d$ nilsequences in all
short intervals and almost all very short intervals. If we focus solely on
polynomial phase functions, [9, Theorem 1.1] demonstrates that when
$X^{3/5+\varepsilon}\leqslant H\leqslant X$, we have

$\sup_{\alpha_{1},...,\alpha_{d}\in\mathbb{T}}\Bigl{|}\sum_{X<n\leqslant
X+H}(\tau_{k}(n)-\tau_{k}^{\\#}(n))e(\alpha_{d}n^{d}+\cdots+\alpha_{1}n)\Bigr{|}\ll
H\log^{3k/4-1}X,$

and when $k=2$ the above $H$ can be as small as $X^{1/3+\varepsilon}$; and [5,
Theorem 1.1] shows that if $X^{\varepsilon}\leqslant H\leqslant X$ then

$\displaystyle\sup_{\alpha_{1},...,\alpha_{d}\in\mathbb{T}}\Bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau_{k}^{\\#}(n))e(\alpha_{d}n^{d}+\cdots+\alpha_{1}n)\Bigr{|}=o(H\log^{k-1}X)$
(1.1)

holds for all but $O(X\log^{-A}X)$ integers $x\in[X,2X]$.

Suppose that $n\in[X,2X]$ and $0<\gamma<\frac{1}{5k}$ is a constant. Let’s
define

$\tau_{k}^{*}(n)=\gamma^{1-k}\sum_{\begin{subarray}{c}m|n\\\ m\leqslant
X^{\gamma}\end{subarray}}\tau_{k-1}(m).$

The approximant111We would refer the readers to [6, Lemma 6.1] and [11,
Section 4] for the construction of a majorant rather than approximant, and
also refer to [9, Section 3.1] for the discussion of distinct approximants.
$\tau_{k}^{*}$ can be viewed as a normalized truncation of $\tau_{k}$, and
this construction is inspired by the minorant in [1] but with slight
modifications. Our main result is as follows.

######  Theorem 1.1.

Suppose that $0<\eta<1$ is a parameter and $X\geqslant 1$ is sufficiently
large, there is a large constant $C=C(\eta)>0$ such that when
$\exp\bigl{(}C\log^{1/2}X(\log\log X)^{1/2}\bigr{)}\leqslant H\leqslant X$ we
have

$\int_{X}^{2X}\sup_{\alpha_{1},\dots,\alpha_{d}\in\mathbb{T}}\Bigl{|}\sum_{x\leqslant
n\leqslant
x+H}\bigl{(}\tau_{k}(n)-\tau_{k}^{*}(n)\bigr{)}e(\alpha_{d}n^{d}+\cdots+\alpha_{1}n)\Bigr{|}\,\,\mathrm{d}x\leqslant\eta
XH\log^{k-1}X.$

Theorem 1.1 suggests that, by replacing the approximant with ours, one can
extend the estimate (1.1) to the range $\exp\bigl{(}C\log^{1/2}X(\log\log
X)^{1/2}\bigr{)}\leqslant H\leqslant X$ for all but $o(X)$ integers
$x\in[X,2X]$.

### Outline of the proof

Our proof draws inspiration from the exploration of the local Fourier
uniformity conjecture. Suppose that $\lambda$ is the Liouville function. [7,
Conjecture 1] predicts that if $H\to_{X}\infty$, then

$\int_{X}^{2X}\sup_{\alpha\in\mathbb{T}}\bigl{|}\sum_{x<n\leqslant
x+H}\lambda(n)e(n\alpha)\bigr{|}\,\,\mathrm{d}x=o(XH).$

This conjecture has been investigated in papers [7, 8, 15, 16, 17], and the
current best-known result confirms its validity for
$\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leqslant H\leqslant X$ by
Walsh [16]. Moreover, under the assumption of the Generalized Riemann
Hypothesis, Walsh [17] verifies its effectiveness when $H\geqslant(\log
X)^{\psi(X)}$ where $\psi(X)$ tends to infinity arbitrarily slowly.

In the following, we will present a brief outline of our proof, omitting
several challenges we encountered. One notable challenge is that in the proof,
for distinct purposes, we need to shift and shrink the supported intervals.
However, especially when dealing with very short intervals, it becomes
difficult to shift and shrink the supported intervals while maintaining local
correlation information.

Following the ideas in studying the local Fourier uniformity conjecture, let’s
assume

$\int_{X}^{2X}\sup_{\alpha\in\mathbb{T}}\bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau_{k}^{*}(n))e(n\alpha)\bigr{|}\,\,\mathrm{d}x\gg
XH\log^{k-1}X.$

The discretization process allows us to identify $\gg X/H$ of $H$-separated
integer $x\in(X,2X]$ such that, for each of these $x$, there exists a
corresponding frequency $\alpha_{x}$ where the local correlation holds,

$\displaystyle\bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau_{k}^{*}(n))e(n\alpha_{x})\bigr{|}\gg H\log^{k-1}X.$
(1.2)

We then apply Ramaré’s identity, serving a similar purpose as Elliott’s
inequality in the works [7, 8, 15, 16, 17], to extract a small prime factor
$p\in[P/2,P]$ with $P\approx H^{\varepsilon}$ from almost all $n\in(x,x+H]$.
More or less, we can assume that

$\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{\tau_{k}(n)-\tau_{k}^{*}(n)}{\omega_{(P^{\varepsilon},P]}(n)+1}e(n(p\alpha_{x}))\Bigr{|}\gg\frac{H\log^{k-1}X}{p},$

where $\omega_{(P^{\varepsilon},P]}(n)$ counts the number of prime factors of
$n$ in the interval $(P^{\varepsilon},P]$. Because $x$ lies in the dyadic
interval $[X,2X]$ and $p$ is within the dyadic interval $[P/2,P]$, varying
both $p$ and $x$ allows us to predict substantial overlaps in the intervals
$[\frac{x}{p},\frac{x+H}{p}]$. Let’s say, $I\subseteq[X/P,(4X)/P]$ is an
interval of length $|I|\asymp H/P$ and there exist numerous pairs
$(p,(x,\alpha_{x}))$ such that

$\displaystyle\Bigl{|}\sum_{n\in
I}\frac{\tau_{k}(n)-\tau_{k}^{*}(n)}{\omega_{(P^{\varepsilon},P]}(n)+1}e(n(p\alpha_{x}))\Bigr{|}\gg\frac{H\log^{k-1}X}{P}.$
(1.3)

We expect that there should only have a few frequencies dominates the above
exponential sum, which means that there should be many pairs
$(p,(x,\alpha_{x}))$ and $(q,(y,\alpha_{y}))$ such that

$\displaystyle p\alpha_{x}\approx q\alpha_{y}\,(\mathrm{mod}\,1)\quad\text{
and }\quad x/p\approx y/q.$ (1.4)

To make the information we obtained about $\alpha_{x}$ valuable, it is
essential to upgrade the congruence relation held
$\,(\mathrm{mod}\,p^{\prime})$ for large primes
$p^{\prime}\in[P^{\varepsilon^{2}},P^{\varepsilon}]$ rather than
$\,(\mathrm{mod}\,1)$. Following the argument in [7, Proposition 3.2], we
successfully deduce that

$\displaystyle p\gamma_{z_{1}}\approx
q\gamma_{z_{2}}\,(\mathrm{mod}\,p^{\prime})\quad\text{ and }\quad
z_{1}/p\approx z_{2}/q,$ (1.5)

for many pairs
$(z_{1},\gamma_{z_{1}}),(z_{2},\gamma_{z_{2}})\in[X/(4P^{1+\varepsilon}),4X/P^{1+\varepsilon}]\times\mathbb{T}$,
$p,q\in[P/2,P]$, and many primes
$p^{\prime}\in[P^{\varepsilon^{2}},P^{\varepsilon}]$. Besides, we also have
the local correlation below

$\Bigl{|}\sum_{z<n\leqslant
z+H/P^{1+\varepsilon}}\frac{\tau_{k}(n)-\tau_{k}^{*}(n)}{(\omega_{(P^{\varepsilon},P]}(n)+1)(\omega_{(P^{\varepsilon^{2}},P^{\varepsilon}]}(n)+1)}e(n\gamma_{z})\Bigr{|}\gg(H/P^{1+\varepsilon})\log^{k-1}X.$

Building on the idea of Walsh [16], we can use the information (1.5) to build
relatively long paths. This approach effectively establishes a global
frequency $\gamma$ with $e(n\gamma)\approx e(an/q)n^{iT}$ for some small $q$
and such that $\gamma\approx\gamma_{z}$ for many local frequencies
$\gamma_{z}$. To derive the contradiction, it is important to demonstrate that
for almost all $z\in[X/P^{1+\varepsilon},4X/P^{1+\varepsilon}]$, we have

$\Bigl{|}\sum_{z<n\leqslant
z+H/P^{1+\varepsilon}}\frac{\tau_{k}(n)-\tau_{k}^{*}(n)}{(\omega_{(P^{\varepsilon},P]}(n)+1)(\omega_{(P^{\varepsilon^{2}},P^{\varepsilon}]}(n)+1)}e(an/q)n^{iT}\Bigr{|}\ll(H/P^{1+\varepsilon})\log^{k-1}X.$

However, the function $\omega_{(P^{\varepsilon},P]}(n)$ introduced through the
use of Ramaré’s identity is not regular, posing challenges in establishing the
above inequality. Therefore, instead of following the approach outlined in [7,
Proposition 3.2], we upgrade the congruent in (1.4) to
$\,(\mathrm{mod}\,p^{\prime})$ and analyze configurations
$(x,\alpha_{x})\in[X,2X]\times\mathbb{T}^{d}$. By demonstrating the existence
of numerous $\alpha_{x}$ satisfying $e(n\alpha_{x})\approx e(an/q)n^{iT}$ for
a small $q$ and extending [4, Theorem 9.2] to the divisor function, we can
derive a contradiction with (1.2).

Finally, let’s redirect to (1.3). Observing that within such a short interval
direct handling of the exponential in terms of $\tau_{k}$ becomes challenging,
we intend to eliminate the weight $\tau_{k}$ at a notable cost. However noting
that the second moment of divisor function causes a logarithmic loss, i.e.

$X^{-1}\sum_{n\sim X}\tau_{k}^{2}(n)\gg\log^{2k-1}X\gg\log
X\bigl{(}X^{-1}\sum_{X<n\leqslant 2X}\tau_{k}(n)\bigr{)}^{2},$

the approach outlined in [7, Lemma 2.4] cannot be applied here. To navigate
this obstacle, we address the local divisor correlations $\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)$ with the assistance of [6].

### Acknowledgement

The author would like to thank Kaisa Matomäki, Yu-Chen Sun, and Joni
Teräväinen for their valuable discussions and suggestions. The author also
thanks Yang Cao for helping create the diagrams. This work was supported by
the Academy of Finland grant no. 333707.

##  2\. Notations and preliminaries

This section serves as introducing fundamental notations and collecting
essential results concerning $k$-fold divisor function and frequencies that we
required in the paper. Suppose that $2Q_{0}<Q$ are two numbers, denote
$\mathcal{P}(Q_{0},Q)$ as the product of the primes in the interval
$(Q_{0},Q]$. Let $c$ and $\varepsilon$ be small constants, both independent of
$\eta$, and we allow them to vary in distinct occurrence. Besides, we use the
vector $\vec{\alpha}=(\alpha^{(d)},\dots,\alpha^{(1)})\in\mathbb{T}^{d}$ to
represent a $d$-tuple frequency.

If $A$ is a set, we use $1_{A}$ to denote the indicator of $A$; and if $A$ is
a statement, we let $1_{A}$ denote the number 1 when $A$ is true and $0$ when
$A$ is false.

###  2.1. Properties of divisor functions

######  Lemma 2.1 (Shiu’s bound).

Let $A\geqslant 1$ and $\varepsilon>0$ be fixed. Let $X^{\varepsilon}\leqslant
H\leqslant X$ and $q\leqslant H^{1-\varepsilon}$. Let $f$ be a non-negative
multiplicative function such that $f(p^{l})\leqslant A^{l}$ for every prime
power $p^{l}$ and $f(n)\ll_{c}n^{c}$ for every $c>0$. Then

$\sum_{X<n\leqslant X+H\atop n\equiv
a\,(\mathrm{mod}\,q)}f(n)\ll\frac{H}{\phi(q)\log
X}\exp\Bigl{(}\sum_{p\leqslant 2X,p\nmid q}\frac{f(p)}{p}\Bigr{)}.$

###### Proof.

[12, Theorem 1]. ∎

######  Lemma 2.2 (Local uniformity of divisor function).

Suppose that $\log^{k\log k}X\leqslant H\leqslant X$, then we have

$\int_{X}^{2X}\Bigl{|}H^{-1}\sum_{x\leqslant n\leqslant
x+H}\tau_{k}(n)-x^{-1}\sum_{x<n\leqslant
2x}\tau_{k}(n)\Bigr{|}\,\,\mathrm{d}x=o(X\log^{k-1}X).$

###### Proof.

[13, Theorem 1.1]. ∎

We’d like to introduce a corollary to this lemma to make the exceptional set
result clear.

######  Corollary 2.3 (Exceptional set and pointwise upper bound).

Let $0<\delta<1$ be a parameter and $\log^{k\log k}X\leqslant H\leqslant X$.
There is an exceptional set $\mathcal{E}\subseteq[X,2X]$ with cardinality
$\\#\mathcal{E}\ll\delta X$ and satisfing

  1. (1)

$\sum_{x\in\mathcal{E}}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\ll\delta XH\log^{k-1}X$ and,

  2. (2)

when $x\in[X,2X]\backslash\mathcal{E}$ we have the pointwise bound

$\sum_{x<n\leqslant x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\ll H\log^{k-1}X.$

###### Proof.

Suppose that $\mathcal{E}$ is the set of integers $x\in(X,2X]$ such that the
pointwise bound $\sum_{x<n\leqslant x+H}\tau_{k}(n)\ll H\log^{k-1}X$ doesn’t
hold, it can be deduced from Lemma 2.2 that $\\#\mathcal{E}\ll\delta X$ and
$\sum_{x\in\mathcal{E}}\sum_{x<n\leqslant x+H}\tau_{k}(n)\ll\delta
XH\log^{k-1}X$. Meanwhile, as it can be seen from the notion of the
approximant that $\tau_{k}^{*}\leqslant\gamma^{1-k}\tau_{k}$ pointwise, one
thus has

$\sum_{x\in\mathcal{E}}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\leqslant\sum_{x\in\mathcal{E}}\sum_{x<n\leqslant
x+H}(1+\gamma^{1-k})\tau_{k}(n)\ll\delta XH\log^{k-1}X,$

and when $x\in[X,2X]\backslash\mathcal{E}$ one has

$\sum_{x<n\leqslant
x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\leqslant\sum_{x<n\leqslant
x+H}(1+\gamma^{1-k})\tau_{k}(n)\ll H\log^{k-1}X.$

∎

######  Lemma 2.4 (Local divisor correlations on average).

Let $0<\delta<1$ be a parameter and $\log^{10000k\log k}X\leqslant H\leqslant
X^{1/2-\varepsilon}$. There is an exceptional set $\mathcal{E}\subseteq[-H,H]$
with cardinality $\\#\mathcal{E}\ll\delta^{2}H$ such that the following
statement holds for all but $O(\delta X)$ integers $x\in[X,2X]$:

  1. (1)

(pointwise divisor correlation) If $h\in[-H,H]\backslash\mathcal{E}$ we have

$\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h)\ll\delta^{-1}H\log^{2k-2}X;$

  2. (2)

(negligible contribution from the exceptional set)

$\sum_{h\in\mathcal{E}}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\ll\delta^{2}H^{2}\log^{2k-2}X.$

###### Proof.

Let’s fix an integer $0\neq|h|\leqslant H$ for a moment. When $X+H\leqslant
n\leqslant 2X-H$, each term $\tau_{k}(n)\tau_{k}(n+h)$ occurs $H$ times in the
expression

$\sum_{X+H<x\leqslant 2X-H}\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h).$

Simultaneously, by considering the estimate $\sum_{X<x\leqslant
X+H}\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h)\ll H^{2}X^{\varepsilon}$,
one can deduce from [6, Theorem 1.1] that for each $0\neq|h|\leqslant H$ there
exists a constant $c_{k,h}>0$ such that

$\sum_{0<|h|\leqslant H}\Bigl{|}\sum_{x\sim X}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)-c_{k,h}XH\log^{2k-2}X\Bigr{|}\leqslant\delta^{4}XH^{2}\log^{2k-2}X.$

If we set $\mathcal{E}$ as the set of $|h|\leqslant H$ satisfying

$\sum_{x\sim X}\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h)\gg
XH\log^{2k-2}X,$

it is immediately clear that $\\#\mathcal{E}\ll\delta^{4}H$. Moreover,
considering the estimate $\sum_{x\sim X}\tau_{k}^{2}(x)\ll X\log^{2k-1}X$,
derived from Lemma 2.1, we can also obtain that

$\sum_{x\sim X}\sum_{h\in\mathcal{E}}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\\\ \leqslant\sum_{x\sim X}\sum_{x<n\leqslant
x+H}\tau_{k}^{2}(n)+\sum_{h\in\mathcal{E}\backslash\left\\{0\right\\}}\sum_{x\sim
X}\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h)\\\ \ll
XH\log^{2k-1}X+\delta^{4}XH^{2}\log^{2k-2}X\ll\delta^{4}XH^{2}\log^{2k-2}X.$

Therefore, it follows from the pigeonhole principle that for all but
$O(\delta^{2}X)$ integers $x\in[X,2X]$ we have

$\displaystyle\sum_{h\in\mathcal{E}}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\ll\delta^{2}H^{2}\log^{2k-2}X.$ (2.1)

Now we may assume that $h\not\in\mathcal{E}$ which means that

$\sum_{x\sim X}\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h)\ll
XH\log^{2k-2}X.$

Similarly, pigeonhole principle indicates that when $x\in[X,2X]$ is outside a
set with cardinality $O(\delta X)$ the inequality

$\displaystyle\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\ll\delta^{-1}H\log^{2k-2}X$ (2.2)

holds. The lemma is established by recognizing that there are, in total, at
most $O(\delta X)$ integers $x$ in the interval $(X,2X]$ for which either
(2.1) or (2.2) does not hold. ∎

###  2.2. Frequencies and pyramids

The following lemma is an extension of [16, Lemma 2.1].

######  Lemma 2.5 (Pair of frequencies).

Let $\varepsilon,\varepsilon^{\prime}>0$ be small parameters and
$Q\in\mathbb{N}$. Let $\vec{\alpha}_{1},\vec{\alpha}_{2}\in\mathbb{T}^{d}$ be
frequencies and $p\neq q$ be primes not dividing $Q$. Suppose that

$\displaystyle p^{j}\alpha_{1}^{(j)}\equiv
q^{j}\alpha_{2}^{(j)}+O(\varepsilon^{j}+\varepsilon^{\prime
j})\,(\mathrm{mod}\,Q)\text{ for all }1\leqslant j\leqslant d.$ (2.3)

Then there is a frequency $\vec{\alpha}\in\mathbb{T}^{d}$ such that

$p^{j}\alpha^{(j)}\equiv\alpha_{2}^{(j)}+O\bigl{(}\bigl{(}\frac{\varepsilon^{\prime}}{q}\bigr{)}^{j}\bigr{)}\,(\mathrm{mod}\,Q)\text{
and
}q^{j}\alpha^{(j)}\equiv\alpha_{1}^{(j)}+O\Bigl{(}\bigl{(}\frac{\varepsilon}{p}\bigr{)}^{j}\Bigr{)}\,(\mathrm{mod}\,Q).$

###### Proof.

We are going to fix the degree $1\leqslant j\leqslant d$ in the proof. If we
take $p^{j}\beta_{1}^{(j)}\equiv\alpha_{2}^{(j)}\,(\mathrm{mod}\,Q)$ and
$q^{j}\beta_{2}^{(j)}\equiv\alpha_{1}^{(j)}\,(\mathrm{mod}\,Q)$, the
assumption (2.3) implies that

$(pq)^{j}\beta_{1}^{(j)}\equiv(pq)^{j}\beta_{2}^{(j)}+O\bigl{(}\varepsilon^{j}+\varepsilon^{\prime
j}\bigr{)}\,(\mathrm{mod}\,Q).$

On the other hand, if it is necessary one can add an integer multiple of
$\frac{Q}{p^{j}}$ to $\beta_{1}^{(j)}$ and an integer multiple of
$\frac{Q}{q^{j}}$ to $\beta_{2}^{(j)}$ to assume that

$\beta_{1}^{(j)}\equiv\beta_{2}^{(j)}+O(\frac{Q}{2p^{j}q^{j}})\,(\mathrm{mod}\,Q).$

Combining these two estimates we have

$\beta_{1}^{(j)}\equiv\beta_{2}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}+\varepsilon^{\prime
j}}{p^{j}q^{j}}\Bigr{)}\,(\mathrm{mod}\,Q).$

Taking
$\alpha^{(j)}=\frac{\varepsilon^{j}}{\varepsilon^{j}+\varepsilon^{\prime
j}}\beta_{1}^{(j)}+\frac{\varepsilon^{\prime
j}}{\varepsilon^{j}+\varepsilon^{\prime j}}\beta_{2}^{(j)}$, it is easy to see
that

$p^{j}\alpha^{(j)}-\alpha^{(j)}_{2}\equiv\frac{\varepsilon^{\prime
j}p^{j}}{\varepsilon^{j}+\varepsilon^{\prime
j}}\bigl{(}\beta_{2}^{(j)}-\beta_{1}^{(j)}\bigr{)}\equiv
O\Bigl{(}\frac{\varepsilon^{\prime j}}{q^{j}}\Bigr{)}\,(\mathrm{mod}\,Q),$

and similarly,
$q^{j}\alpha^{(j)}\equiv\alpha_{1}^{(j)}+O\bigl{(}\frac{\varepsilon^{j}}{p^{j}}\bigr{)}\,(\mathrm{mod}\,Q)$.
Therefore, the frequency $\vec{\alpha}=(\alpha^{(d)},\dots,\alpha^{(1)})$
satisfies the conditions of this lemma. ∎

We now explore the situation where there exists an ordered sequence of
frequencies, and any two adjacent frequencies in the sequence satisfy
condition (2.3). Let’s say,
$\vec{\alpha}_{1},\dots,\vec{\alpha}_{k+1}\in\mathbb{T}^{d}$ are frequencies,
$p_{1},\dots,p_{k},q_{1},\dots,q_{k}$ are distinct primes not dividing $Q$.
Suppose that for every $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant d$
we have

$\displaystyle p_{i}^{j}\alpha_{i}^{(j)}\equiv
q_{i}^{j}\alpha_{i+1}^{(j)}+O(\varepsilon^{j})\,(\mathrm{mod}\,Q).$ (2.4)

The first application of Lemma 2.5 with $\varepsilon=\varepsilon^{\prime}$
gives an ordered sequence
$\left\\{\vec{\alpha}_{2,1},\dots,\vec{\alpha}_{2,k}\right\\}$ such that

$\displaystyle
p_{i}^{j}\alpha_{2,i}^{(j)}\equiv\alpha_{i+1}^{(j)}+O\Bigl{(}(\frac{\varepsilon}{q_{i}})^{j}\Bigr{)}\,(\mathrm{mod}\,Q)\text{
and
}q_{i}^{j}\alpha_{2,i}^{(j)}\equiv\alpha_{i}^{(j)}+O\Bigl{(}(\frac{\varepsilon}{p_{i}})^{j}\Bigr{)}\,(\mathrm{mod}\,Q).$
(2.5)

Taking into account the relations between both $\vec{\alpha}_{2,i}$ and
$\vec{\alpha}_{2,i+1}$ with $\vec{\alpha}_{i+1}$ through the aforementioned
congruence condition, it can be demonstrated, using the triangle inequality,
that $\vec{\alpha}_{2,i}$ is also related to $\vec{\alpha}_{2,i+1}$ by the
congruence equation below

$p_{i}^{j}\alpha_{2,i}^{(j)}\equiv
q_{i+1}^{j}\alpha_{2,i+1}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{q_{i}^{j}}+\frac{\varepsilon^{j}}{p_{i+1}^{j}}\Bigr{)}\,(\mathrm{mod}\,Q),$

and Lemma 2.5 can be utilized once more. One can repeatedly apply Lemma 2.5 to
build a pyramid.

######  Definition 2.6 (Pre-paths and pyramids).

Suppose that $Q\in\mathbb{N}$. We say that a parameter $0<\varepsilon<1$, a
$2k$-tuple distinct primes $p_{1},\dots,p_{k},q_{1},\dots,q_{k}$ not dividing
$Q$ and an ordered $(k+1)$-tuple $(\vec{\alpha}_{1},\dots,\vec{\alpha}_{k+1})$
together form a pre-path $\,(\mathrm{mod}\,Q)$ of length $k$ if the tuples
satisfy (2.4). We also say that the initial sequence
$\\{\vec{\alpha}_{1},\dots,\vec{\alpha}_{k+1}\\}$ and the corresponding
sequence $\left\\{\vec{\alpha}_{t,i}\right\\}_{2\leqslant t\leqslant
k+1,1\leqslant i\leqslant k+2-t}$ obtained in the previous paragraph
collectively form a pyramid, with $\vec{\alpha}_{k+1,1}$ referred to as the
top element.

It’s worth noting that this definition slightly differs from the one in [16,
Definition 2.3], where it was initially introduced.

---
$\textstyle{\vec{\alpha}_{4,1}}$$\textstyle{\vec{\alpha}_{3,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{q_{3}}$$\scriptstyle{q_{3}}$$\textstyle{\vec{\alpha}_{3,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{\vec{\alpha}_{2,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{2}}$$\textstyle{\vec{\alpha}_{2,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{q_{3}}$$\scriptstyle{q_{3}}$$\scriptstyle{p_{1}}$$\textstyle{\vec{\alpha}_{2,3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\vec{\alpha}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{1}}$$\textstyle{\vec{\alpha}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{2}}$$\scriptstyle{p_{1}}$$\textstyle{\vec{\alpha}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{3}}$$\scriptstyle{q_{3}}$$\scriptstyle{q_{3}}$$\scriptstyle{p_{2}}$$\textstyle{\vec{\alpha}_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{3}}$

The diagram above illustrates a pyramid of length 3.

For a pre-path $\,(\mathrm{mod}\,Q)$ of length $k$, we can determine the
congruence relations among any two elements in the pyramid. Nevertheless, we
are particularly interested in the congruence relations between the top
element and elements in the initial sequence.

######  Lemma 2.7.

Let $Q\in\mathbb{N}$,
$\left\\{\varepsilon,p_{1},\dots,q_{k},(\vec{\alpha}_{1},\dots,\vec{\alpha}_{k+1})\right\\}$
form a pre-path $\,(\mathrm{mod}\,Q)$ of length $k$, and $\vec{\alpha}$ be the
top element of the pyramid. Suppose that $\prod_{i_{1}\leqslant i\leqslant
i_{2}}p_{i}\asymp\prod_{i_{1}\leqslant i\leqslant i_{2}}q_{i}$ whenever
$1\leqslant i_{1}<i_{2}\leqslant k$, then for all $0\leqslant
i^{\prime}\leqslant k$ and $1\leqslant j\leqslant d$ we have

$\prod_{i\leqslant i^{\prime}}p_{i}^{j}\prod_{i^{\prime}<i\leqslant
k}q_{i}^{j}\alpha^{(j)}\equiv\alpha_{i^{\prime}+1}^{(j)}+O(k\varepsilon^{j})\,(\mathrm{mod}\,Q).$

###### Proof.

For simplicity, we may represent $\vec{\alpha}_{i}$ as $\vec{\alpha}_{1,i}$,
and denote the top element $\vec{\alpha}$ as $\vec{\alpha}_{k+1,1}$ in the
proof. Our approach follows the principle of the left lower triangular,
signifying that we exclusively utilize relations in the lower directions first
and then employ relations in the left-lower directions. Assuming $2\leqslant
t\leqslant k+1$ and $1\leqslant i\leqslant k+2-t$, we assert that for all
$1\leqslant j\leqslant d$, the following two congruence equations hold:

$\displaystyle
q^{j}_{i+t-2}\alpha_{t,i}^{(j)}\equiv\alpha_{t-1,i}^{(j)}+\Bigl{(}\frac{\varepsilon^{j}}{(p_{i}\dots
p_{i+t-2})^{j}}\Bigr{)}\,(\mathrm{mod}\,Q)$ (2.6)

and

$\displaystyle
p^{j}_{i}\alpha_{t,i}^{(j)}\equiv\alpha_{t-1,i+1}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{(q_{i}\dots
q_{i+t-2})^{j}}\Bigr{)}\,(\mathrm{mod}\,Q).$ (2.7)

This claim can be easily confirmed through induction on $t$. When $t=2$ it
follows from (2.5). Now we suppose the claim holds for $2\leqslant t\leqslant
k$ and examine the case for $t+1$. Utilizing (2.6) with $i$ replaced by $i+1$
and keeping (2.7) in mind, the triangle inequality demonstrates that

$q^{j}_{i+t-1}\alpha_{t,i+1}^{(j)}\equiv
p_{i}^{j}\alpha_{t,i}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{(p_{i+1}\dots
p_{i+t-1})^{j}}+\frac{\varepsilon^{j}}{(q_{i}\dots
q_{i+t-2})^{j}}\Bigr{)}\,(\mathrm{mod}\,Q).$

The application of Lemma 2.5 with $\varepsilon=\frac{\varepsilon}{p_{i+1}\dots
p_{i+t-1}}$ and $\varepsilon^{\prime}=\frac{\varepsilon}{q_{i}\dots
q_{i+t-2}}$ then gives a frequency $\alpha_{t+1,i}^{(j)}$ such that

$q^{j}_{i+t-1}\alpha_{t+1,i}^{(j)}\equiv\alpha_{t,i}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{(p_{i}\dots
p_{i+t-1})^{j}}\Bigr{)}\,(\mathrm{mod}\,Q)$

and

$p_{i}^{j}\alpha_{t+1,i}^{(j)}\equiv\alpha_{t,i+1}^{(j)}+O\bigl{(}\frac{\varepsilon^{j}}{(q_{i}\dots
q_{i+t-1})^{j}}\bigr{)}\,(\mathrm{mod}\,Q).$

Let’s redirect our focus to the lemma itself. First of all, we aim to
establish the relations between $\vec{\alpha}$ and the elements that share the
same vertical line with $\vec{\alpha}$. Assuming $0\leqslant
i^{\prime}\leqslant k$, iteratively applying (2.6) with $i=1$ and considering
the given assumption $\prod_{i_{1}\leqslant i\leqslant
i_{2}}p_{i}\asymp\prod_{i_{1}\leqslant i\leqslant i_{2}}q_{i}$ one can get
that

$\prod_{i^{\prime}<t\leqslant
k}q^{j}_{t}\alpha^{(j)}\equiv\prod_{i^{\prime}<t\leqslant
k-1}q_{t}^{j}\Bigl{(}\alpha_{k,1}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{(p_{1}\dots
p_{k})^{j}}\Bigr{)}\Bigr{)}\\\ \equiv\prod_{i^{\prime}<t\leqslant
k-1}q_{t}^{j}\alpha_{k,1}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{\prod_{t\leqslant
i^{\prime}}p_{t}^{j}}\Bigr{)}\\\
\equiv\cdots\equiv\alpha_{i^{\prime}+1,1}^{(j)}+O\Bigl{(}(k-i^{\prime})\frac{\varepsilon^{j}}{\prod_{t\leqslant
i^{\prime}}p_{t}^{j}}\Bigr{)}\,(\mathrm{mod}\,Q).$

Next, we consider the situation involving elements in the horizontal line,
namely, $\vec{\alpha}_{1,1},\dots,\vec{\alpha}_{1,k+1}$. Suppose that
$0\leqslant i^{\prime}\leqslant k$, by employing (2.7), we can establish the
relation between $\vec{\alpha}_{1,i^{\prime}+1}$ and
$\vec{\alpha}_{i^{\prime}+1,1}$.

$\prod_{i\leqslant
i^{\prime}}p_{i}^{j}\alpha_{i^{\prime}+1,1}^{(j)}\equiv\prod_{i\leqslant
i^{\prime}-1}p_{i}^{j}\Bigl{(}\alpha_{i^{\prime},2}^{(j)}+O\Bigl{(}\frac{\varepsilon^{j}}{(q_{1}\dots
q_{i^{\prime}})^{j}}\Bigr{)}\Bigr{)}\\\ \equiv\prod_{i\leqslant
i^{\prime}-1}p_{i}^{j}\alpha_{i^{\prime},2}^{(j)}+O(\varepsilon^{j})\equiv\cdots\equiv\alpha_{1,i^{\prime}+1}^{(j)}+O(i^{\prime}\varepsilon^{j})\,(\mathrm{mod}\,Q).$

Therefore, combining the above two estimates together, one has

$\prod_{i\leqslant i^{\prime}}p_{i}^{j}\prod_{i^{\prime}<i\leqslant
k}q_{i}^{j}\alpha^{(j)}\equiv\prod_{i\leqslant
i^{\prime}}p_{i}^{j}\Bigl{(}\alpha_{i^{\prime}+1,1}^{(j)}+O\Bigl{(}(k-i^{\prime})\frac{\varepsilon^{j}}{\prod_{t\leqslant
i^{\prime}}p_{t}^{j}}\Bigr{)}\Bigr{)}\equiv\alpha_{1,i^{\prime}+1}^{(j)}+O(k\varepsilon^{j})\,(\mathrm{mod}\,Q),$

this concludes the lemma.

∎

##  3\. Major arc estimates

We are going to concentrate on integers having both small prime factors and
also large prime factors in this section. In practice, suppose that
$2\leqslant h\leqslant X$ and $\delta=\eta^{A}$ for some large constant $A>0$.
Let $P_{1}=h^{2\delta^{3}}$, $Q_{1}=h^{\delta^{2}}$, $P_{2}=h^{100\delta^{2}}$
and $Q_{2}=h^{\delta}$. Additionally, following the notation used in [8, page
90], set $v_{1}=\delta^{2}/4000$, $v_{2}=\frac{1}{10}$, $P_{3}=X^{v_{1}}$,
$Q_{3}=P_{4}=X^{\sqrt{v_{1}v_{2}}}$, and $Q_{4}=X^{v_{2}}$. Now, let
$S\subseteq(X,2X]$ denote the set of integers with a prime factor in each
interval $(P_{i},Q_{i}]$ for $1\leqslant i\leqslant 4$. That is,

$\displaystyle S=\left\\{n\in(X,2X]:(n,\mathcal{P}(P_{i},Q_{i}))>1\text{ for
all }1\leqslant i\leqslant 4\right\\}.$ (3.1)

######  Proposition 3.1 (Major-arc approximation).

Let $0<\delta<\eta<1$. Suppose that $d\in\mathbb{N}$ is an integer,
$\exp(\log^{1/2}X)\leqslant h\leqslant X^{1/2-\varepsilon}$, $|T^{*}|\leqslant
X^{d+1}$, $S\subseteq(X,2X]$ is as in (3.1), and $1\leqslant a\leqslant q\ll
1$ are integers, then we have

$\Biggl{|}\sum_{\begin{subarray}{c}x<n\leqslant x+h\\\ n\equiv
a\,(\mathrm{mod}\,q)\end{subarray}}1_{S}(n)\tau_{k}(n)n^{iT^{*}}-\sum_{\begin{subarray}{c}x<n\leqslant
x+h\\\ n\equiv
a\,(\mathrm{mod}\,q)\end{subarray}}\tau_{k}^{*}(n)n^{iT^{*}}\Biggr{|}\ll\delta
h\log^{k-1}X$

for all but $O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$
integers $x\in(X,2X]$.

As in [5, Section 3] and [9, Section 3], we are going to introduce long
averages into the above left-hand side expression so that Proposition 3.1 can
be simplified into proving the following three assertions.

######  Proposition 3.2 (Long and short averages of $\tau_{k}$).

Let $\delta,T^{*},S,q$ be as in Proposition 3.1 and $\chi\,(\mathrm{mod}\,q)$
be a Dirichlet character. Then when $\exp(\log^{1/2}X)\leqslant h\leqslant
X^{1/2-\varepsilon}\leqslant h_{1}\leqslant X^{1-\frac{1}{100k}}$ one of the
following statements holds for all but
$O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$ integers
$x\in(X,2X]$:

  1. (1)

if $\chi\neq\chi_{0}$ or $|T^{*}|>X$ then

$\bigl{|}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi(n)n^{iT^{*}}\bigr{|}\ll\delta h\log^{k-1}X;$

  2. (2)

otherwise we have

$\biggl{|}h^{-1}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi_{0}(n)n^{iT^{*}}-h^{-1}\int_{x}^{x+h}u^{iT^{*}}\,\,\mathrm{d}u\cdot
h_{1}^{-1}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\chi_{0}(n)\biggr{|}\ll\delta\log^{k-1}X.$

###### Proof.

Let $|t_{0}|\leqslant X$ be the number $t\in[-X,X]$ that minimizes the
distance

$\sum_{p\leqslant
2X}\frac{\tau_{k}(p)-\Re\tau_{k}(p)\chi(p)p^{iT^{*}-it}}{p}.$

First of all, we aim to prove that for all but at most
$O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$ integers
$x\in(X,2X]$ the following inequality holds

$\displaystyle\biggl{|}h^{-1}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi(n)n^{iT^{*}}-\frac{\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u}{h}\cdot
h_{1}^{-1}\sum_{x<n\leqslant x+h_{1}\atop n\in
S}\tau_{k}(n)\chi(n)n^{i(T^{*}-t_{0})}\biggr{|}\leqslant\delta\log^{k-1}X.$
(3.2)

The proof essentially follows from [4, Theorem 9.2], but it is somewhat
simpler since our $h_{1}$ still maintains some distance from $X$. We will
provide a brief proof with a particular focus on adapting the argument from
[4, Theorem 9.2] to our cases with the assistance of [13]. Taking
$T_{0}=h_{1}^{\frac{1}{200k}},P^{\prime}=P_{3},Q^{\prime}=Q_{3},P=P_{4}$ and
$Q=Q_{4}$ in [4, Proposition 8.3], considering the definition of $R_{C}(1+it)$
in [4, (44)], the application of [4, Proposition 8.3] shows that

$h^{-1}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi(n)n^{iT^{*}}-\frac{\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u}{h}\cdot
h_{1}^{-1}\sum_{x<n\leqslant x+h_{1}\atop n\in
S}\tau_{k}(n)\chi(n)n^{i(T^{*}-t_{0})}\\\
=A(x,h,h_{1},t_{0},h_{1}^{\frac{1}{100k}},\mathcal{U})+O_{\varepsilon}\Bigl{(}\log^{k-1}X\Bigl{(}\frac{h_{1}^{1+1/100k}}{X}+\frac{1}{h}+\frac{(1+C\log\delta^{-1})^{2}}{h_{1}^{1/20k}}\Bigr{)}\Bigr{)}\\\
+O\biggl{(}(1+C\log\delta^{-1})^{2}\sup_{|t|\leqslant X/2,|t-t_{0}|\geqslant
h_{1}^{\frac{1}{200k}}\atop\frac{X}{16Q_{3}Q_{4}}\leqslant
C\leqslant\frac{16X}{P_{3}P_{4}}}\biggl{|}\sum_{m\sim
C}\frac{f(m)}{m^{1+it}(\omega_{(P_{3},Q_{3}]}(m)+1)(\omega_{(P_{4},Q_{4}]}(m)+1)}\biggr{|}\biggr{)},$

where $f(m)=\tau_{k}(m)\chi(m)m^{iT^{*}}$ and $\omega_{(P_{i},Q_{i}]}(m)$
counts the prime factors of $m$ in the interval $(P_{i},Q_{i}]$. The second
term is obviously bounded by $O(h^{-\frac{1}{(101k)^{2}}})$ from the
assumptions of $h$ and $h_{1}$. Meanwhile, taking note that [5, Lemma 3.2]
holds whenever $f$ is a $\tau_{k}$-bounded function, we can estimate the above
third term as

$\ll\frac{(1+C\log\delta^{-1})^{2}\log^{O(1)}X}{h_{1}^{\frac{1}{400k}}}\ll
h_{1}^{-\frac{1}{500k}}.$

In light of the pigeonhole principle, it suffices to prove that

$\frac{1}{X}\int_{X}^{2X}|A(x,h,h_{1},t_{0},h_{1}^{\frac{1}{200k}},\mathcal{U})|^{2}\,\,\mathrm{d}x\ll
h^{-\frac{\delta^{3}}{50k}}+X^{-\frac{\delta^{6}}{10^{16}}}.$

It follows from [4, (46)] that

$\frac{1}{X}\int_{X}^{2X}|A(x,h,h_{1},t_{0},h_{1}^{\frac{1}{200k}},\mathcal{U})|^{2}\,\,\mathrm{d}x\ll\max_{X/h\leqslant
T\leqslant
X}\frac{X/h}{T}\int_{[-T,T]\backslash\mathcal{U}}\Bigl{|}\sum_{n\sim X\atop
n\in
S}\frac{\tau_{k}(n)\chi(n)n^{iT^{*}}}{n^{1+it}}\Bigr{|}^{2}\,\,\mathrm{d}t\\\
+\frac{\log^{2k}X}{h}+\frac{\log^{5}X}{X^{(\delta/20)^{6}/160}}\max_{X/h\leqslant
T\leqslant
X}\Bigl{(}\frac{\bigl{|}\bigl{(}\mathcal{U}+[-X^{\varepsilon},X^{\varepsilon}]\bigr{)}\cap[-T,T]\bigr{|}\cdot
Q_{3}}{hT^{1/2}}+1\Bigr{)},$

since the second term is obviously negligible, it can be simplified to prove
the following two events:

  1. (1)

$\mathcal{U}\subseteq(X,2X]$ is a set such that for any $T\in[X/h,X]$ we have

$\bigl{|}\bigl{(}\mathcal{U}+[-X^{\varepsilon},X^{\varepsilon}]\bigr{)}\cap[-T,T]\bigr{|}\ll\frac{hT^{1/2}}{Q_{3}};$

  2. (2)

The following estimate holds uniformly for $X/h\leqslant T\leqslant X$

$\int_{[-T,T]\backslash\mathcal{U}}\Bigl{|}\sum_{n\sim X\atop n\in
S}\frac{\tau_{k}(n)\chi(n)n^{iT^{*}}}{n^{1+it}}\Bigr{|}^{2}\,\,\mathrm{d}t\ll
P_{1}^{-1/100}\frac{T\log^{2k-2}X}{X/h}.$

Given a number $X/h\leqslant T\leqslant X$, let’s assume that
$\mathcal{T}_{1}$, $\mathcal{T}_{2}$, and $\mathcal{U}$ are disjoint sets with
a union of $[-T,T]$. The precise definitions are in accordance with the proof
of [13, Proposition 4.4], with parameters $P_{1}$, $P_{2}$, $Q_{1}$, and
$Q_{2}$ substituted as in our case. It follows from the assumption of
$\mathcal{U}$ that if $t\in\mathcal{U}$, there exists some
$v\in\mathcal{I}_{2}$ such that
$|Q_{v,2}(1+it_{0}+it)|>e^{-\alpha_{2}v/P_{1}^{1/6}}$ with
$\alpha_{2}=\frac{1}{4}-\frac{1}{100}$, then [13, Lemma 4.5 (ii)] reveals that

$|\mathcal{U}|\ll|\mathcal{I}_{2}|T^{2\alpha_{2}}Q_{2}^{2\alpha_{2}}\exp\bigl{(}2k\frac{\log
X\log_{2}X}{\log P_{2}}\bigr{)}\ll
hT^{1/2}(\frac{h}{X})^{1/50}h^{-1+\frac{\delta}{2}}P_{1}^{1/6}.$

The first claim follows by noting that $h\leqslant X^{1/2-\varepsilon}$ and
$Q_{3}=X^{\delta/200}$. And the second claim follows from the estimates of
$E_{1}$ and $E_{2}$ on [13, pages 25-26]. Thus, we have finished the proof of
(3.2) so far.

We can actually remove the restriction $n\in S$ from the long averages in
(3.2) by observing, from Lemma 2.1, that

$\Bigl{|}\sum_{x<n\leqslant x+h_{1}\atop n\in S}\tau_{k}(n)-\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\Bigr{|}\ll\sum_{P_{i}<p_{i}\leqslant Q_{i}\atop
i=1,2,3,4}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)1_{(n,\mathcal{P}(P_{i},Q_{i}))=1}\ll\delta^{k}h_{1}\log^{k-1}X.$

Besides, [5, Lemma 3.6] also informs us that when $|T^{*}-t_{0}|\geqslant
X/h_{1}^{1-\varepsilon}$, we have

$\bigl{|}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\chi(n)n^{i(T^{*}-t_{0})}\bigr{|}\ll\delta h_{1}.$

Furthermore, when $|T^{*}-t_{0}|\leqslant X/h_{1}^{1-\varepsilon}$ we can
reduce to the case that $T^{*}-t_{0}=0$ by decomposing the long intervals into
short intervals of length $h_{1}^{1-2\varepsilon}$. In practice,

$\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\chi(n)n^{i(T^{*}-t_{0})}=\sum_{x_{0}}\sum_{x_{0}<n\leqslant
x_{0}+h_{1}^{1-2\varepsilon}}\tau_{k}(n)\chi(n)n^{i(T^{*}-t_{0})},$

where $x_{0}\in[x,x+h_{1})$ takes $h_{1}^{1-2\varepsilon}$-separated values.
Since Taylor expansion denominates that
$n^{i(T^{*}-t_{0})}=x_{0}^{i(T^{*}-t_{0})}+O(|T^{*}-t_{0}|h_{1}^{1-2\varepsilon}/x)$
whenever $|n-x_{0}|\leqslant h_{1}^{1-2\varepsilon}$, one thus has

$\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\chi(n)n^{i(T^{*}-t_{0})}=\sum_{x_{0}}x_{0}^{i(T^{*}-t_{0})}\sum_{x_{0}<n\leqslant
x_{0}+h_{1}^{1-2\varepsilon}}\tau_{k}(n)\chi(n)+O(h_{1}^{1-\varepsilon}).$

Combining the above analysis, we can therefore conclude that

$\biggl{|}h^{-1}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi(n)n^{iT^{*}}-\frac{\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u}{h}\cdot
h_{1}^{-1}\sum_{x_{0}}x_{0}^{i(T^{*}-t_{0})}\sum_{x_{0}<n\leqslant
x_{0}+h_{1}^{1-2\varepsilon}}\tau_{k}(n)\chi(n)\biggr{|}\\\
\ll\delta\log^{k-1}X$

for all but at most $O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$
integers $x\in(X,2X]$.

Our task now reduces to prove that when $\chi\neq\chi_{0}$ or $\chi=\chi_{0}$
but $|T^{*}|>X$ the long average in the left-hand side expression above is
always bounded by $O(\delta\log^{k-1}X)$, because when $\chi=\chi_{0}$ and
$|T^{*}|\leqslant X$ it is clear from the notion of $t_{0}$ that
$t_{0}=T^{*}$. To achieve this, let $h_{2}=h_{1}^{1-2\varepsilon}$ and assume
$\chi\neq\chi_{0}$. Perron’s formula leads us to

$\sum_{x_{0}<n\leqslant x_{0}+h_{2}}\tau_{k}(n)\chi(n)=\frac{1}{2\pi
i}\int_{-T}^{T}L(1+it,\chi)^{k}\frac{(x_{0}+h_{2})^{1+it}-x_{0}^{1+it}}{1+it}\,\,\mathrm{d}t+O(\frac{x_{0}^{\varepsilon}}{T}).$

By shifting the contour of the integration to the region $\sigma=1/2$ we get
that

$\sum_{x_{0}<n\leqslant x_{0}+h_{2}}\tau_{k}(n)\chi(n)=\frac{1}{2\pi
i}\int_{-T}^{T}L(1/2+it,\chi)^{k}\frac{(x_{0}+h_{2})^{1/2+it}-x_{0}^{1/2+it}}{1/2+it}\,\,\mathrm{d}t\\\
\pm\frac{1}{2\pi i}\int_{1}^{1/2}L(u\mp
iT,\chi)^{k}\frac{(x_{0}+h_{2})^{u+iT}-x_{0}^{u+iT}}{u+iT}\,\,\mathrm{d}u+O(\frac{x_{0}^{\varepsilon}}{T}).$

We may apply [3, Lemmas 10.9-10.10] and Taylor expansion to bound the vertical
integral as follows

$\ll\frac{h_{2}}{x_{0}^{1/2}}\sup_{|t|\leqslant
T}|L(\frac{1}{2}+it,\chi)|^{\max\left\\{0,k-4\right\\}}\int_{-T}^{T}|L(\frac{1}{2}+it,\chi)|^{4}\,\,\mathrm{d}t\ll\frac{h_{2}}{x_{0}^{1/2}}T^{k/2-1+\varepsilon}\ll
h_{2}^{1-\varepsilon}$

if we take $T=X^{1/k}$. Besides, when $k\leqslant 4$ similar result can be
obtained by an application of Cauchy-Schwarz inequality to derive the fourth
moment of the $L$-function. Simultaneously, the horizontal line integrals can
be estimated as follows:

$\ll T^{-1}\int_{1/2}^{1}T^{1-u+\varepsilon}x^{u}\,\,\mathrm{d}u\ll x/T\ll
h_{2},$

which is also negligible.

When $\chi=\chi_{0}$ and $|T^{*}|\leqslant X+X/h_{1}^{1-\varepsilon}$, it can
be seen from the notion of $t_{0}$ that

$t_{0}=\begin{cases}X\qquad&\text{if }X<T^{*}\leqslant
X+X/h_{1}^{1-\varepsilon},\\\ T^{*}\qquad&\text{if }|T^{*}|\leqslant X,\\\
-X&\text{if }-X-X/h_{1}^{1-\varepsilon}\leqslant T^{*}<-X.\end{cases}$

Besides, as noting that

$h^{-1}\sup_{|t_{0}|\geqslant
x}\Bigl{|}\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u\Bigr{|}\ll
h^{-1}\sup_{|t_{0}|\geqslant
x}\bigl{|}\frac{(x+h)^{1+it_{0}}-x^{1+it_{0}}}{1+it_{0}}\bigr{|}\ll h^{-1},$

which means that when $|t_{0}|\geqslant X$ the long average is also
negligible. Therefore we can assume that $|T^{*}|\leqslant X$ and the
proposition follows.

∎

######  Proposition 3.3 (Long and short averages of $\tau_{k}^{*}$).

Suppose that the assumptions are as in Proposition 3.2. Then when
$\exp(\log^{1/2}X)\leqslant h\leqslant X^{1/2-\varepsilon}\leqslant
h_{1}\leqslant X^{1-\frac{1}{100k}}$ one of the following statements holds for
all but $O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$ integers
$x\in(X,2X]$:

  1. (1)

if $\chi\neq\chi_{0}$ or $|T^{*}|>X$ then

$\bigl{|}\sum_{x<n\leqslant
x+h}\tau_{k}^{*}(n)\chi(n)n^{iT^{*}}\bigr{|}\ll\delta h\log^{k-1}X;$

  2. (2)

otherwise we have

$\biggl{|}h^{-1}\sum_{x<n\leqslant
x+h}\tau_{k}^{*}(n)\chi_{0}(n)n^{iT^{*}}-h^{-1}\int_{x}^{x+h}u^{iT^{*}}\,\,\mathrm{d}u\cdot
h_{1}^{-1}\sum_{x<n\leqslant x+h_{1}}\tau_{k}^{*}(n)\chi_{0}(n)\biggr{|}\\\
\ll\delta\log^{k-1}X.$

###### Proof.

Let $t_{0}$ be as in Proposition 3.2, it suffices to prove that for arbitrary
Dirichlet character $\chi\,(\mathrm{mod}\,q)$ and any number $|T^{*}|\leqslant
X^{d+1}$

$\biggl{|}h^{-1}\sum_{x<n\leqslant
x+h}\tau_{k}^{*}(n)\chi(n)n^{iT^{*}}-h^{-1}\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u\cdot
h_{1}^{-1}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}^{*}(n)\chi(n)n^{i(T^{*}-t_{0})}\biggr{|}\\\
\ll\delta\log^{k-1}X$

holds for all but $O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$
integers $x\in(X,2X]$, since it can be observed from the proof of Proposition
3.2 that the long average is always bounded by $O(\delta\log^{k-1}X)$ if
$\chi\neq\chi_{0}$ or $|T^{*}|\geqslant X$ and we have $t_{0}=T^{*}$ when
$\chi=\chi_{0}$ and $|T^{*}|\leqslant X$.

Let’s set $D(X;1+it)=\sum_{n\sim X}\frac{\tau_{k}^{*}(n)\chi(n)}{n^{1+it}}$
and $W=h_{1}^{\frac{1}{100k}}$. It follows from Perron’s formula that the
short average $h^{-1}\sum_{x<n\leqslant x+h}\tau_{k}^{*}(n)\chi(n)n^{iT^{*}}$
equals to

$\frac{h^{-1}}{2\pi i}\Bigl{(}\int_{|t-t_{0}|\leqslant
W}+\int_{|t-t_{0}|>W}\Bigr{)}D(X;1+i(t-T^{*}))\frac{(x+h)^{1+it}-x^{1+it}}{1+it}\,\,\mathrm{d}t.$

By making the change of variables $t\to t+t_{0}$, the first integral is equal
to

$\frac{h^{-1}}{2\pi i}\int_{|t|\leqslant
W}D(X;1+i(t+t_{0}-T^{*}))\int_{x}^{x+h}u^{i(t+t_{0})}\,\,\mathrm{d}u\,\mathrm{d}t.$

Taking note, from Taylor expansion, that $u^{it}=x^{it}+O(h|t|/x)$ whenever
$x\leqslant u\leqslant x+h$, this integral then equals to

$h^{-1}\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u\frac{1}{2\pi
i}\int_{|t|\leqslant
W}D(X;1+i(t+t_{0}-T^{*}))x^{it}\,\,\mathrm{d}t+O(W^{3/2}h\log^{2k}X/X),$

as from Cauchy-Schwarz inequality and [13, Lemma 4.3] one has

$\int_{|t|\leqslant W}|t||D(X;1+i(t+t_{0}-T^{*})|\,\,\mathrm{d}t\ll
W^{3/2}\log^{2k}X.$

On the other hand, for the long average, Perron’s formula also reveals that

$h^{-1}\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u\cdot
h_{1}^{-1}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}^{*}(n)\chi(n)n^{i(T^{*}-t_{0})}\\\
=\frac{\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u}{h}\cdot\frac{h_{1}^{-1}}{2\pi
i}\biggl{(}\int_{|t|\leqslant
W}+\int_{|t|>W}\biggr{)}D(X;1+i(t+t_{0}-T^{*}))\frac{(x+h_{1})^{1+it}-x^{1+it}}{1+it}\,\,\mathrm{d}t.$

The application of Taylor expansion yields that the first term equals to

$h^{-1}\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u\cdot\frac{1}{2\pi
i}\int_{|t|\leqslant
W}D(X;1+i(t+t_{0}-T^{*}))x^{it}\frac{(1+h_{1}/x)^{1+it}-1}{(1+it)h_{1}/x}\,\,\mathrm{d}t\\\
=h^{-1}\int_{x}^{x+h}u^{it_{0}}\,\,\mathrm{d}u\cdot\frac{1}{2\pi
i}\int_{|t|\leqslant
W}D(X;1+i(t+t_{0}-T^{*}))x^{it}\,\,\mathrm{d}t+O(h^{-\frac{1}{20k}}).$

Taking note that the main terms within the integral region $|t|\leqslant W$ in
both long and short averages are equal, and the error terms are acceptable, it
suffices to show that, by taking the second moment of $x\in[X,2X]$, the
integral over $|t|>W$ is acceptable. [4, Lemma 8.1] shows that

$\frac{1}{Xh^{2}}\int_{X}^{2X}\Bigl{|}\int_{|t-t_{0}|>W}D(X;1+i(t-T^{*}))\frac{(x+h)^{1+it}-x^{1+it}}{1+it}\,\,\mathrm{d}t\Bigr{|}^{2}\,\,\mathrm{d}x\\\
\ll\max_{T\geqslant X/h}\frac{X/h}{T}\int_{W<|t-t_{0}|\leqslant
T}|D(X;1+i(t-T^{*}))|^{2}\,\,\mathrm{d}t.$

As it follows from the large sieve, for example [4, Lemma 3.4], as well as
noting that $\tau_{k}^{*}\leqslant\gamma^{1-k}\tau_{k}$ pointwise, that when
$T\geqslant X$ one has

$\int_{W<|t-t_{0}|\leqslant
T}|D(X;1+i(t-T^{*}))|^{2}\,\,\mathrm{d}t\ll\frac{Th^{-1/2}}{X/h},$

thus, it suffices to prove that

$\max_{X/h\leqslant T\leqslant X}\frac{X/h}{T}\int_{W<|t|\leqslant
T}\Bigl{|}\sum_{n\sim
X}\frac{\tau_{k}^{*}(n)\chi(n)n^{i(T^{*}-t_{0})}}{n^{1+it}}\Bigr{|}^{2}\,\,\mathrm{d}t\ll
h^{-\delta^{3}}.$

Now we may recall the definitions of the Dirichlet polynomial $D(X;1+it)$ and
the approximant $\tau_{k}^{*}$ to see that

$\displaystyle|D(X;1+i(t+$ $\displaystyle
t_{0}-T^{*}))|\ll\Bigl{|}\sum_{m\leqslant
X^{\gamma}}\frac{\tau_{k-1}(m)\chi(m)m^{i(T^{*}-t_{0})}}{m^{1+it}}\sum_{n\sim\frac{X}{m}}\frac{\chi(n)n^{i(T^{*}-t_{0})}}{n^{1+it}}\Bigr{|}$
$\displaystyle\ll\sum_{i\leqslant\log X^{\gamma}/\log 2}\biggl{|}\sum_{m\sim
M_{i}}\frac{\tau_{k-1}(m)\chi(m)m^{i(T^{*}-t_{0})}}{m^{1+it}}\biggr{|}\biggl{|}\sum_{\frac{X}{2M_{i}}<n\leqslant\frac{2X}{M_{i}}}\frac{\chi(n)n^{i(T^{*}-t_{0})}}{n^{1+it}}\biggr{|}.$

If we write $a(m)=\sum_{m_{1}m_{2}=m\atop m_{1},m_{2}\sim
M_{i}}\tau_{k-1}(m_{1})\tau_{k-1}(m_{2})\ll\tau_{3k}(m)$, Cauchy-Schwarz
inequality, [4, Lemma 3.1] and [2, Theorem 1] together leads us to

$\int_{|t|\leqslant T}\Bigl{|}\sum_{n\sim
X}\frac{\tau_{k}^{*}(n)\chi(n)n^{i(T^{*}-t_{0})}}{n^{1+it}}\Bigr{|}^{2}\,\,\mathrm{d}t\\\
\ll\sum_{i\leqslant\frac{\log X^{\gamma}}{\log 2}}\biggl{(}\int_{|t|\leqslant
T}\biggl{|}\sum_{m\asymp
M_{i}^{2}}\frac{a(m)\chi(m)m^{i(T^{*}-t_{0})}}{m^{1+it}}\biggr{|}^{2}\,\mathrm{d}t\biggr{)}^{1/2}\biggl{(}\int_{|t|\leqslant
T}\biggl{|}\sum_{\frac{X}{2M_{i}}<n\leqslant\frac{2X}{M_{i}}}\frac{\chi(n)n^{i(T^{*}-t_{0})}}{n^{1+it}}\biggr{|}^{4}\,\mathrm{d}t\biggr{)}^{1/2}\\\
\ll\frac{T\log^{A}X}{X}\sum_{i\leqslant\log X^{\gamma}/\log
2}\Bigl{(}M_{i}^{-2}\sum_{m\asymp
M_{i}^{2}}\tau_{3k}^{2}(m)+M_{i}^{-2}\sum_{0<|n|\leqslant
M_{i}^{2}/T}\sum_{m\asymp
M_{i}^{2}}\tau_{3k}(m)\tau_{3k}(m+n)\Bigr{)}^{1/2}\\\
\ll\frac{T\log^{A}X}{X}\Bigl{(}\sum_{m\ll
X^{2\gamma}}\frac{\tau_{3k}^{2}(m)}{m}+\sum_{0<|n|\leqslant
X^{2\gamma}/T}\sum_{m\leqslant
X^{2\gamma}}\frac{\tau_{3k}(m)\tau_{3k}(m+n)}{m}\Bigr{)}^{2}\ll\frac{T\log^{A}X}{X},$

the last inequality follows from Lemma 2.1, as well as noting that the first
term dominates the expression when $T\geqslant X/h$ and $X^{2\gamma}<X^{1/5}$.
This completes the proof of this proposition.

∎

######  Proposition 3.4 (Difference in long intervals).

When $1\leqslant a\leqslant q\ll 1$ and $X^{1-\frac{1}{100k}}\leqslant
h_{1}\leqslant X$, we have

$\Bigl{|}\sum_{X<n\leqslant X+h_{1}\atop n\equiv
a\,(\mathrm{mod}\,q)}(\tau_{k}(n)-\tau_{k}^{*}(n))\Bigr{|}\ll
h_{1}\log^{k-2}X.$

###### Proof.

First of all, it can be seen from the notion of $\tau_{k}^{*}$ that

$\displaystyle h_{1}^{-1}\sum_{X<n\leqslant X+h_{1}\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}^{*}(n)$
$\displaystyle=h_{1}^{-1}\gamma^{1-k}\sum_{a_{1},a_{2}\,(\mathrm{mod}\,q)\atop
a_{1}a_{2}\equiv a\,(\mathrm{mod}\,q)}\sum_{m\leqslant X^{\gamma}\atop m\equiv
a_{1}\,(\mathrm{mod}\,q)}\tau_{k-1}(m)\sum_{X<mn\leqslant X+h_{1}\atop n\equiv
a_{2}\,(\mathrm{mod}\,q)}1$
$\displaystyle=\frac{\gamma^{1-k}}{q}\sum_{a_{1},a_{2}\,(\mathrm{mod}\,q)\atop
a_{1}a_{2}\equiv a\,(\mathrm{mod}\,q)}\sum_{m\leqslant X^{\gamma}\atop m\equiv
a_{1}\,(\mathrm{mod}\,q)}\frac{\tau_{k-1}(m)}{m}+O(X^{-1/3}),$

and similarly, we also have

$X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}^{*}(n)=\frac{\gamma^{1-k}}{q}\sum_{a_{1},a_{2}\,(\mathrm{mod}\,q)\atop
a_{1}a_{2}\equiv a\,(\mathrm{mod}\,q)}\sum_{m\leqslant X^{\gamma}\atop m\equiv
a_{1}\,(\mathrm{mod}\,q)}\frac{\tau_{k-1}(m)}{m}+O(X^{-1/3}).$

Therefore, we can conclude that

$h_{1}^{-1}\sum_{X<n\leqslant X+h_{1}\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}^{*}(n)=X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}^{*}(n)+O(X^{-1/3}).$

On the other hand, by introducing the minorant as in [1, (1.11)], one can also
find that

$\displaystyle X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\gamma^{1-k}\sum_{m|n\atop m\leqslant
n^{\gamma}}\tau_{k-1}(m)$ $\displaystyle=X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\gamma^{1-k}\sum_{m|n\atop m\leqslant
X^{\gamma}}\tau_{k-1}(m)+O\Bigl{(}X^{-1}\sum_{m\sim
X^{\gamma}}\tau_{k-1}(m)\sum_{n\sim X/m}1\Bigr{)}$
$\displaystyle=X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}^{*}(n)+O(\log^{k-2}X).$

Therefore, the application of [1, Theorem 1.2] shows that

$\displaystyle X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}^{*}(n)$ $\displaystyle=X^{-1}\sum_{n\sim X\atop
n\equiv a\,(\mathrm{mod}\,q)}\gamma^{1-k}\sum_{m|n\atop m\leqslant
n^{\gamma}}\tau_{k-1}(m)+O(\log^{k-2}X)$ $\displaystyle=X^{-1}\sum_{n\sim
X\atop n\equiv a\,(\mathrm{mod}\,q)}\tau_{k}(n)+O(\log^{k-2}X).$

As noted in [9, Proposition 3.4 (ii)], when $h_{1}\geqslant
X^{1-\frac{1}{100k}}$, one has

$h_{1}^{-1}\sum_{X<n\leqslant X+h_{1}\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}(n)=X^{-1}\sum_{n\sim X\atop n\equiv
a\,(\mathrm{mod}\,q)}\tau_{k}(n)+O(\log^{-A}X)$

we can thus conclude this lemma by making use of the triangle inequality.

∎

_Proof of Proposition 3.1._

Assume that $h_{1}=X^{1-\frac{1}{100k}}$ in the proof. On combining
Propositions 3.2-3.3, the triangle inequality shows that when
$\chi\neq\chi_{0}$ or $|T^{*}|\geqslant X$ for all but
$O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$ integers
$x\in(X,2X]$ one has

$h^{-1}\bigl{|}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi(n)n^{iT^{*}}-\sum_{x<n\leqslant
x+h}\tau_{k}^{*}(n)\chi(n)n^{iT^{*}}\bigr{|}\ll\delta\log^{k-1}X;$

otherwise when $\chi=\chi_{0}$ and $|T^{*}|\leqslant X$ one has

$\displaystyle h^{-1}\bigl{|}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi_{0}(n)n^{iT^{*}}-\sum_{x<n\leqslant
x+h}\tau_{k}^{*}(n)\chi_{0}(n)n^{iT^{*}}\bigr{|}$
$\displaystyle\ll\delta\log^{k-1}X+h_{1}^{-1}\Bigl{|}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\chi_{0}(n)-\sum_{x<n\leqslant
x+h_{1}}\tau_{k}^{*}(n)\chi_{0}(n)\Bigr{|}.$

Observing that $\chi(n)=\chi(a)$ if $n\equiv a\,(\mathrm{mod}\,q)$, one may
decompose $n$ into arithmetic progressions with common difference $q$ and then
it follows from Proposition 3.4, as well as the assumption $q\ll 1$, that

$\displaystyle h^{-1}\bigl{|}\sum_{x<n\leqslant x+h\atop n\in
S}\tau_{k}(n)\chi_{0}(n)-\sum_{x<n\leqslant
x+h}\tau_{k}^{*}(n)\chi_{0}(n)\bigr{|}$ $\displaystyle\ll
h_{1}^{-1}\sum_{a\,(\mathrm{mod}\,q)}\Bigl{|}\sum_{\begin{subarray}{c}x<n\leqslant
x+h_{1}\\\ n\equiv
a\,(\mathrm{mod}\,q)\end{subarray}}(\tau_{k}(n)-\tau_{k}^{*}(n))\Bigr{|}+\delta\log^{k-1}X\ll\delta\log^{k-1}X.$

Proposition 3.1 then follows from the orthogonality of the Dirichlet
characters. To be precise,

$\displaystyle\Bigl{|}\sum_{\begin{subarray}{c}x<n\leqslant x+h\\\ n\equiv
a\,(\mathrm{mod}\,q)\end{subarray}}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))n^{iT^{*}}\Bigr{|}$
$\displaystyle\leqslant\frac{1}{\phi(q)}\sum_{\chi\,(\mathrm{mod}\,q)}\Bigl{|}\sum_{x<n\leqslant
x+h}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))\chi(n)n^{iT^{*}}\Bigr{|}\ll\delta
h\log^{k-1}X$

for all but $O(X(h^{-\delta^{3}/50}+X^{-\frac{\delta^{6}}{10^{16}}}))$
integers $x\in(X,2X]$.

∎

##  4\. Reduction and discretization

Suppose that $0<\omega,\eta<1$ and $\omega$ is sufficiently small with respect
to $\eta$. Suppose that $Q=H^{\omega}$ and $Q_{0}=Q^{\omega}$, and let
$\omega_{(Q_{0},Q]}(n)$ denote the number of prime factors of $n$ in the
interval $(Q_{0},Q]$. Additionally, as $\omega$ is sufficiently small
concerning to $\eta$ and recalling the setting of the set $S$ in (3.1), we can
assume that for all $1\leqslant i\leqslant 4$ we have
$[Q_{0},Q]\cap[P_{i},Q_{i}]=\emptyset$. To simplify the description of well-
spaced intervals, let’s introduce the notion of configurations, as defined in
[15, Definition 3.1].

######  Definition 4.1 (Configuration).

We say $\mathcal{J}$ is a $(c,H)$-configuration in
$[X,2X]\times\mathbb{T}^{d}$ if $\\#\mathcal{J}\geqslant cX/H$ and the first
coordinates are $H$-separated.

The objective of this section is to demonstrate that when the integral is
large, it is possible to identify numerous well-spaced short intervals wherein
the correlation holds among these intervals. Additionally, we can observe from
this section that almost all integers in these short intervals can extract a
small prime factor. This idea also serves as the foundation for the papers [7,
8, 15, 16, 17].

######  Lemma 4.2 (Discretization).

Suppose that $\log^{2k\log k}X\leqslant H\leqslant X^{1/2-\varepsilon}$ and
$S\subseteq[X,2X]$ is as in (3.1) with $0<\delta\leqslant\eta^{2}$. Suppose
that

$\int_{X}^{2X}\sup_{\vec{\alpha}\in\mathbb{T}^{d}}\sup_{|x_{0}-x|\leqslant
H}\Bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau^{*}_{k}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}\,\,\mathrm{d}x\geqslant\eta
XH\log^{k-1}X,$

then there is a $(c\eta,H)$-configuration
$\mathcal{J}_{-1}\subseteq[X,2X]\times\mathbb{T}^{d}$ such that the following
three events happen simultaneously.

  1. (1)

When $(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}$ we have

$\sum_{x<n\leqslant x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\ll H\log^{k-1}X;$

  2. (2)

when $(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}$ there is some $x_{0}$
satisfying $|x_{0}-x|\leqslant H$ and such that

$\bigl{|}\sum_{x<n\leqslant
x+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\bigr{|}\gg\eta H\log^{k-1}X;$

  3. (3)

suppose that $0\leqslant h_{1},h_{2}\leqslant H^{1-\sigma}$ are integers,
where $0\leqslant\sigma<1/10$ is a constant and, for each $x$ we fix four
integers $z_{1,x}^{(1)},z_{2,x}^{(1)},z_{1,x}^{(2)},z_{2,x}^{(2)}$ with
$0\leqslant H^{\varepsilon}z_{j,x}^{(i)}-x\leqslant H$, then when
$(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}$ we have

$\sum_{|n-z_{j,x}^{(i)}|\leqslant h_{i}}\tau_{k}(n)\ll
h_{i}\log^{k-1}X\quad\text{ for all }i,j\in\left\\{1,2\right\\}.$

###### Proof.

We may apply Corollary 2.3 with $H=2h_{i}(i=1,2)$ and $\delta=\eta^{2}$
respectively to conclude that there is a set $\mathcal{E}_{1}\subseteq[X,2X]$
with $\\#\mathcal{E}_{1}\ll\eta^{2}X$ and when
$z_{j,x}^{(i)}-h_{i}\not\in\mathcal{E}_{1}$ for $i,j\in\left\\{1,2\right\\}$
we have

$\sum_{|n-z_{j,x}^{(i)}|\leqslant h_{i}}\tau_{k}(n)\ll h_{i}\log^{k-1}X.$

Thus, if we restrict integers $x\in[X,2X]$ such that the corresponding
$z_{j,x}^{(i)}-h_{i}$ being outside of $\mathcal{E}_{1}$ the third conclusion
always holds. Besides, the application of Corollary 2.3 with $\delta=\eta^{2}$
also shows that there is an exceptional set $\mathcal{E}_{2}\subseteq[X,2X]$
with $\\#\mathcal{E}_{2}\ll\eta^{2}X$ such that the contribution from
$\mathcal{E}_{2}$ is negligible in the sense of

$\sum_{x\in\mathcal{E}_{2}}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\ll\eta^{2}XH\log^{k-1}X,$

and when $x\in[X,2X]\backslash\mathcal{E}_{2}$ the first conclusion holds,
that is

$\sum_{x<n\leqslant x+H}(\tau_{k}(n)+\tau_{k}^{*}(n))\ll H\log^{k-1}X.$

We can further restrict the consideration of the function $\tau_{k}$ to the
set $S$ by noting that when $1\leqslant H\leqslant X^{1/2-\varepsilon}$, for
any $1\leqslant i\leqslant 4$, Lemma 2.1 reveals that

$\sum_{x\sim X}\sum_{x<n\leqslant
x+H}1_{(n,\mathcal{P}(P_{i},Q_{i}))=1}\tau_{k}(n)\ll
H^{2}X^{\varepsilon}+H\frac{X}{\log X}\exp\bigl{(}\sum_{p\leqslant
P_{i}}\frac{k}{p}+\sum_{Q_{i}<p\leqslant 2X}\frac{k}{p}\bigr{)}\\\
\ll\bigl{(}\frac{\log P_{i}}{\log
Q_{i}}\bigr{)}^{k}XH\log^{k-1}X\ll\eta^{k}XH\log^{k-1}X.$

On the other hand, it is directly from the assumption that for every
$x\in[X,2X]$ there exists a frequency
$\vec{\alpha}_{x}=(\alpha_{x}^{(d)},\dots,\alpha_{x}^{(1)})\in\mathbb{T}^{d}$
and a number $|x_{0}-x|\leqslant H$ such that

$\sum_{x\sim X}\Bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}\gg\eta XH\log^{k-1}X.$

Therefore, by combining the above two hands together we can conclude that

$\sum_{x\in[X,2X]\backslash\mathcal{E}_{2}}\Bigl{|}\sum_{x<n\leqslant
x+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}\\\
\geqslant\sum_{x\in(X,2X]}\Bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}-\sum_{n\in\mathcal{E}_{2}}(\tau_{k}(n)+\tau_{k}^{*}(n))\\\
-\sum_{1\leqslant i\leqslant 4}\sum_{x\sim X}\sum_{x<n\leqslant
x+H}1_{(n,\mathcal{P}(P_{i},Q_{i}))=1}\tau_{k}(n)\gg\eta XH\log^{k-1}X.$

Noting that the pointwise upper bound when $x\not\in\mathcal{E}_{2}$, one can
deduce from the pigeonhole principle that there is a set
$\mathcal{A}\subseteq[X,2X]\backslash\mathcal{E}_{2}$ with cardinality
$\\#\mathcal{A}\gg\eta X$ such that when $x\in\mathcal{A}$ we have

$\Bigl{|}\sum_{x<n\leqslant
x+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}\gg\eta H\log^{k-1}X.$

One may remove elements in $\mathcal{A}$ such that the corresponding
$z_{j,x}^{(i)}-h_{i}$ ($i,j\in\left\\{1,2\right\\})$ being in the set
$\mathcal{E}_{1}$ and rename the set of the remaining elements to
$\mathcal{A}$, then we still have $\\#\mathcal{A}\gg\eta X$. Now we may
decompose the elements of $\mathcal{A}$ into $H$-separated subsets, for
example, assume that $0\leqslant i<H$ and denote
$\mathcal{A}_{i}=\left\\{x\in\mathcal{A}:x\equiv
i\,(\mathrm{mod}\,H)\right\\}$. The pigeonhole principle implies that there is
some $\mathcal{A}_{i}$ satisfies $\\#\mathcal{A}_{i}\gg\eta X/H$. Taking
$\mathcal{J}_{-1}=\left\\{(x,\vec{\alpha}_{x}):x\in\mathcal{A}_{i}\right\\}$,
then $\mathcal{J}_{-1}\subseteq[X,2X]\times\mathbb{T}^{d}$, and according to
Definition 4.1, $\mathcal{J}_{-1}$ is a $(c\eta,H)$-configuration. ∎

######  Lemma 4.3 (Extracting a small prime factor).

Suppose that $\exp(\log^{\varepsilon}X)\leqslant H\leqslant
X^{1/2-\varepsilon}$ and

$\int_{X}^{2X}\sup_{\vec{\alpha}\in\mathbb{T}^{d}}\sup_{|x_{0}-x|\leqslant
H}\Bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau^{*}_{k}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}\,\,\mathrm{d}x\geqslant\eta
XH\log^{k-1}X.$

Then there is a $(c\eta^{2},H)$-configuration
$\mathcal{J}\subseteq[X,2X]\times\mathbb{T}^{d}$ and a number
$P\in[Q_{0},Q/2]$ such that the following statement holds.

For each $(x,\vec{\alpha}_{x})\in\mathcal{J}$ there is some number $x_{0}$
with $|x_{0}-x|\leqslant H$ such that

$\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}p^{j}\alpha_{x}^{(j)}(n-x_{0}/p)^{j}\bigr{)}\Bigr{|}\gg\eta^{4}\frac{H\log^{k-1}X}{p}$

holds for $\gg\eta^{6}P/\log P$ primes $p\in[P,2P]$, here, $0<t_{n}<1$ for
each integer $n\in[X,2X]$ and $S\subseteq[X,2X]$ is the set defined in (3.1).

###### Proof.

Building on the second conclusion of Lemma 4.2 there is a
$(c\eta,H)$-configuration
$\mathcal{J}_{-1}\subseteq[X,2X]\times\mathbb{T}^{d}$ for which when
$(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}$ there is some $x_{0}$ with
$|x_{0}-x|\leqslant H$ such that

$\bigl{|}\sum_{x<n\leqslant
x+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\bigr{|}\gg\eta H\log^{k-1}X.$

Recalling Definition 4.1, we may sum over
$(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}$ to conclude that

$\eta^{2}X\log^{k-1}X\ll\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}}\Bigl{|}\sum_{x<n\leqslant
x+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}.$

Based on whether or not $n$ has a prime factor in the interval $(Q_{0},Q]$,
the right-hand side summation is directly bounded by the sum of the following
two expressions:

$\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}}\Bigl{|}\sum_{x<n\leqslant
x+H}1_{(n,\mathcal{P}(Q_{0},Q))>1}(1_{S}(n)\tau_{k}(n)-\tau^{*}_{k}(n))e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}$

and

$\sum_{\begin{subarray}{c}x\sim X\\\
x:H\text{-separated}\end{subarray}}\sum_{x<n\leqslant
x+H}\bigl{(}1_{(n,\mathcal{P}(Q_{0},Q))=1}\tau_{k}(n)+1_{(n,\mathcal{P}(Q_{0},Q))=1}\tau^{*}_{k}(n)\bigr{)}\ll\sum_{n\sim
X}1_{(n,\mathcal{P}(Q_{0},Q))=1}\tau_{k}(n),$

by noting that $\tau_{k}^{*}\ll\tau_{k}$ pointwise. We are going to show that
the second expression is negligible based on our chosen parameters $Q_{0}$ and
$Q$. In practice, Lemma 2.1 implies that

$\sum_{n\sim X}1_{(n,\mathcal{P}(Q_{0},Q))=1}\tau_{k}(n)\ll\frac{X}{\log
X}\exp\Bigl{(}\sum_{p\leqslant Q_{0}}\frac{k}{p}+\sum_{Q\leqslant p\leqslant
2X}\frac{k}{p}\Bigr{)}\ll X\log^{k-1}X\bigl{(}\frac{\log Q_{0}}{\log
Q}\bigr{)}^{k}.$

Recalling that $Q_{0}=Q^{\omega}$, this term is thereby bounded by
$O(\eta^{3}X\log^{k-1}X)$ as $\omega$ is sufficiently small concerning $\eta$,
which is negligible.

Therefore, we may assume that

$\eta^{2}X\log^{k-1}X\ll\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}}\Bigl{|}\sum_{x<n\leqslant
x+H}1_{(n,\mathcal{P}(Q_{0},Q))>1}(1_{S}(n)\tau_{k}(n)-\tau^{*}_{k}(n))e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\Bigr{|}.$

The application of Ramaré’s identity
$1_{(n,\mathcal{P}(Q_{0},Q))>1}=\sum_{Q_{0}<p\leqslant
Q}\sum_{pm=n}\frac{1}{\omega_{(Q_{0},Q]}(pm)}$ yields that

$\eta^{2}X\log^{k-1}X\ll\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}}\sum_{Q_{0}<p\leqslant
Q}\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(pn)\tau_{k}(pn)-\tau^{*}_{k}(pn)}{\omega_{(Q_{0},Q]}(pn)}e\bigl{(}\sum_{1\leqslant
j\leqslant d}p^{j}\alpha_{x}^{(j)}(n-\frac{x_{0}}{p})^{j}\bigr{)}\Bigr{|}.$

Noting that the contribution of integers $n$ in the range
$(\frac{x}{p},\frac{x+H}{p}]$ which are not coprime with $p$ can be estimated
by

$\sum_{\begin{subarray}{c}x\sim X\\\
x:H\text{-separated}\end{subarray}}\sum_{Q_{0}<p\leqslant
Q}\sum_{x<p^{2}m\leqslant x+H}\tau_{k}(p^{2}n)\ll
X\log^{3k}X\sum_{Q_{0}<p\leqslant Q}p^{-2}\ll Q_{0}^{-1}X\log^{3k}X,$

which is negligible. Taking note that $1_{S}(pn)=1_{S}(n)$ when
$Q_{0}<p\leqslant Q$, along with
$\omega_{(Q_{0},Q]}(pn)=\omega_{(Q_{0},Q]}(n)+1$, $\tau_{k}(pn)=k\tau_{k}(n)$
and

$\tau_{k}^{*}(pn)=\gamma^{1-k}\tau_{k-1}(p)\sum_{m|n\atop
m\leqslant(X/p)^{\gamma}}\tau_{k-1}(m)+\gamma^{1-k}\sum_{m|n\atop m\leqslant
X^{\gamma}}\tau_{k-1}(m)$

when $(p,n)=1$ and $p\leqslant Q<X^{\gamma}$, therefore, by assuming that
$0<t_{n}<1$ is the number such that $\tau_{k}^{*}(pn)=kt_{n}\tau_{k}^{*}(n)$,
we can conclude that

$\eta^{2}X\log^{k-1}X\ll\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}}\sum_{Q_{0}<p\leqslant
Q}\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}p^{j}\alpha_{x}^{(j)}(n-\frac{x_{0}}{p})^{j}\bigr{)}\Bigr{|}.$

On the other hand, taking note that when
$(x,\vec{\alpha}_{x})\in\mathcal{J}_{-1}$, it can be deduced from Ramaré’s
identity and the first conclusion of Lemma 4.2 that

$\sum_{Q_{0}<p\leqslant
Q}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{\tau_{k}(n)+\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}\ll\sum_{x<m\leqslant
x+H}\sum_{Q_{0}<p\leqslant
Q}\sum_{m=pn}\frac{\tau_{k}(m)+\tau^{*}_{k}(m)}{\omega_{(Q_{0},Q]}(n)+1}\\\
\ll\sum_{x<m\leqslant x+H}(\tau_{k}(m)+\tau^{*}_{k}(m))\ll H\log^{k-1}X.$

Combining the above two estimates together yields a set
$\mathcal{J}\subset\mathcal{J}_{-1}$ with $\\#\mathcal{J}\gg\eta^{2}X/H$ such
that when $(x,\vec{\alpha}_{x})\in\mathcal{J}$ we have

$\eta^{2}H\log^{k-1}X\ll\sum_{Q_{0}<p\leqslant
Q}\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}p^{j}\alpha_{x}^{(j)}(n-\frac{x_{0}}{p})^{j}\bigr{)}\Bigr{|}.$

Besides, it is obvious from Definition 4.1 that this
$\mathcal{J}\subseteq[X,2X]\times\mathbb{T}^{d}$ is a
$(c\eta^{2},H)$-configuration.

We now claim that there is a subset of primes $\mathcal{P}\subseteq(Q_{0},Q]$
satisfying $\sum_{p\in\mathcal{P}}\frac{1}{p}\gg\eta^{6}\sum_{Q_{0}<p\leqslant
Q}\frac{1}{p}$ and such that for each $p\in\mathcal{P}$ the following estimate
holds uniformly for $(x,\vec{\alpha}_{x})\in\mathcal{J}$

$\displaystyle\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}p^{j}\alpha_{x}^{(j)}(n-\frac{x_{0}}{p})^{j}\bigr{)}\Bigr{|}\gg\eta^{4}\frac{H\log^{k-1}X}{p}.$
(4.1)

It then follows from the pigeonhole principle that, if we decompose the
interval $(Q_{0},Q]$ into dyadic intervals $(P_{j},2P_{j}]$ with
$P_{j}=2^{j}Q_{0}$, there must be some $P_{j}$ such that
$\sum_{p\in\mathcal{P}\cap(P_{i},2P_{i}]}\frac{1}{p}\gg\eta^{6}\sum_{p\sim
P_{i}}\frac{1}{p}$. We can thus deduce from prime number theorem that there
are $\gg\eta^{6}\frac{P_{i}}{\log P_{i}}$ primes $p\in(P_{i},2P_{i}]$ such
that (4.1) holds. The lemma then follows by taking $P$ as this $P_{j}$.

It remains to prove the claim (4.1). It is easy to see that when
$(x,\vec{\alpha}_{x})\in\mathcal{J}$ one has

$\sum_{p\in\mathcal{P}}\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}p^{j}\alpha_{x}^{(j)}(n-\frac{x_{0}}{p})^{j}\bigr{)}\Bigr{|}+\sum_{Q_{0}<p\leqslant
Q}\eta^{4}\frac{H\log^{k-1}X}{p}\\\ \gg\eta^{2}H\log^{k-1}X.$

Given our assumptions on the parameters $Q_{0}$ and $Q$, Mertens’ theorem
implies that

$\sum_{Q_{0}<p\leqslant Q}\frac{1}{p}\ll\log\log Q-\log\log Q_{0}\ll\log\log
H^{\omega}-\log\log H^{\omega^{2}}\ll\eta^{-2},$

if $\omega$ is sufficiently small with respect to $\eta$. This implies the
second summation term has a negligible contribution. By summing over
$(x,\vec{\alpha}_{x})\in\mathcal{J}$ one thus has

$\eta^{4}X\log^{k-1}X\ll\sum_{p\in\mathcal{P}}\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}}\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}p^{j}\alpha_{x}^{(j)}(n-\frac{x_{0}}{p})^{j}\bigr{)}\Bigr{|}.$

By combining the above estimate with the upper bound below

$\sum_{\begin{subarray}{c}x\sim X\\\
x:H\text{-separated}\end{subarray}}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}(\tau_{k}(n)+t_{n}\tau^{*}_{k}(n))\ll\frac{X\log^{k-1}X}{p},$

one has
$\sum_{p\in\mathcal{P}}\frac{1}{p}\gg\eta^{4}\gg\eta^{4}\frac{\sum_{Q_{0}<p\leqslant
Q}\frac{1}{p}}{\eta^{-2}}$. ∎

##  5\. The configuration is somewhat fully connected

In this section, we aim to show that, to some extent, the configuration we
constructed in Lemma 4.2 is fully connected. Specifically, we will define a
type of relationship between any two elements in a configuration and
ultimately identify numerous pairs of elements in $\mathcal{J}$ that are
related. Before delving into this, we require some auxiliary results
concerning exponential sum estimates.

######  Lemma 5.1 (Only a few frequencies are inherently dominant).

Suppose that $\log^{10000k\log k}X\leqslant H\leqslant X^{1/2-\varepsilon}$
and $x,z\in[X,2X]$ with $|z-x|\leqslant H$. Suppose that there is a set
$\mathcal{E}\subseteq[-H,H]$ with $\\#\mathcal{E}\ll\eta^{6}H$ satisfies

$\sum_{h\in\mathcal{E}}\sum_{|n-x|\leqslant
H}\tau_{k}(n)\tau_{k}(n+h)\ll\eta^{6}H^{2}\log^{2k-2}X,$

and $\sum_{|n-x|\leqslant H}\tau_{k}(n)\tau_{k}(n+h)\ll\eta^{-3}H\log^{2k-2}X$
whenever $h\not\in\mathcal{E}$. Then there is a set
$\left\\{\vec{\beta}_{i}\right\\}_{1\leqslant i\leqslant
K^{\prime}}\subset\mathbb{T}^{d}$ with cardinality $K^{\prime}\ll\eta^{-O(1)}$
such that the following holds. If $\vec{\alpha}\in\mathbb{T}^{d}$ is a
frequency such that

$\eta H\log^{k-1}X\leqslant\Bigl{|}\sum_{x<n\leqslant
x+H}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha^{(j)}(n-z)^{j}\bigr{)}\Bigr{|},$

where $S\subseteq[X,2X]$ is defined in (3.1) and $0<t_{n}<1$ for each
$n\in[X,2X]$, then there is some
$\vec{\beta}\in\left\\{\vec{\beta}_{i}\right\\}_{1\leqslant i\leqslant
K^{\prime}}$ and some integer $1\leqslant q\ll\eta^{-O(1)}$ such that

$\left\|q(\beta^{(j)}-\alpha^{(j)})\right\|\ll\eta^{-O(1)}H^{-j}\qquad(1\leqslant
j\leqslant d).$

###### Proof.

Without loss of generality, we may assume that
$\vec{\alpha}_{1},\dots,\vec{\alpha}_{K}$ are all of the frequencies
satisfying this inequality and also satisfying
$\left\|\alpha_{i}^{(d)}-\alpha_{i^{\prime}}^{(d)}\right\|\gg H^{-d}$ for
distinct $i\neq i^{\prime}$. Summing over
$\vec{\alpha}_{1},\dots,\vec{\alpha}_{K}$ obtains that

$\eta HK\log^{k-1}X\leqslant\sum_{1\leqslant i\leqslant
K}\Bigl{|}\sum_{x<n\leqslant
x+H}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha_{i}^{(j)}(n-z)^{j}\bigr{)}\Bigr{|}.$

Square both sides, on noting that $\tau_{k}^{*}\ll\tau_{k}$ pointwise, one can
get from Cauchy-Schwarz inequality and changing the order of summation that

$\eta^{2}H^{2}K\log^{2k-2}X\ll\sum_{x<n\leqslant x+H}\sum_{x<n+h\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\\\ \Bigl{|}\sum_{1\leqslant i\leqslant
K}e\bigl{(}\alpha_{i}^{(d)}((n+h-z)^{d}-(n-z)^{d})+\dots+\alpha_{i}^{(1)}((n+h-z)-(n-z))\bigr{)}\Bigr{|}\\\
\ll\sum_{|h|\leqslant H}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\Bigl{|}\sum_{1\leqslant i\leqslant
K}e\bigl{(}\alpha^{(d)}_{i}h^{d}+P(h;\vec{\alpha}_{i},n-z)\bigr{)}\Bigr{|},$

where $P(h;\vec{\alpha}_{i},n-z)$ is a polynomial of degree at most $d-1$ with
respect to the variable $h$. Since the assumption asserts that the
contribution from $h\in\mathcal{E}$ can be bounded by
$O(\eta^{6}H^{2}K\log^{2k-2}X)$ which is negligible, we can conclude that

$\eta^{2}H^{2}K\log^{2k-2}X\ll\sum_{h\in[-H,H]\backslash\mathcal{E}}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\Bigl{|}\sum_{1\leqslant i\leqslant
K}e\bigl{(}\alpha^{(d)}_{i}h^{d}+P(h;\vec{\alpha}_{i},n-z)\bigr{)}\Bigr{|}\\\
\ll\eta^{-3}H\log^{2k-2}X\sum_{h\in[-H,H]\backslash\mathcal{E}}\Bigl{|}\sum_{1\leqslant
i\leqslant
K}e\bigl{(}\alpha^{(d)}_{i}h^{d}+P(h;\vec{\alpha}_{i},n-z)\bigr{)}\Bigr{|},$

by the pointwise bound for $\sum_{x<n\leqslant x+H}\tau_{k}(n)\tau_{k}(n+h)$
whenever $h\notin\mathcal{E}$. Taking note that $\\#\mathcal{E}\ll\eta^{6}H$,
we thus have

$\eta^{5}HK\ll\sum_{|h|\leqslant H}\Bigl{|}\sum_{1\leqslant i\leqslant
K}e\bigl{(}\alpha^{(d)}_{i}h^{d}+P(h;\vec{\alpha}_{i},n-z)\bigr{)}\Bigr{|}.$

Taking square both sides, one may find from Cauchy-Schwarz inequality that

$\eta^{10}K^{2}H\ll\sum_{1\leqslant i,i^{\prime}\leqslant
K}\Bigl{|}\sum_{|h|\leqslant
H}e\bigl{(}h^{d}(\alpha_{i}^{(d)}-\alpha_{i^{\prime}}^{(d)})+P(h;\vec{\alpha}_{i},\vec{\alpha}_{i^{\prime}},n-z)\bigr{)}\Bigr{|},$

where $P(h;\vec{\alpha}_{i},\vec{\alpha}_{i^{\prime}},n-z)$ is a polynomial of
degree at most $d-1$ with respect to $h$. Pigeonhole principle then shows that
there are $\gg\eta^{10}K^{2}$ of pairs $(i,i^{\prime})\in[1,K]^{2}$ such that

$\eta^{10}H\ll\Bigl{|}\sum_{|h|\leqslant
H}e\bigl{(}h^{d}(\alpha_{i}^{(d)}-\alpha_{i^{\prime}}^{(d)})+P(h;\vec{\alpha}_{i},\vec{\alpha}_{i^{\prime}},n-z)\bigr{)}\Bigr{|}.$

Unless $K\ll\eta^{-10}$ — in which case we can choose
$\left\\{\vec{\gamma}_{d,j}\right\\}_{1\leqslant j\leqslant K^{\prime}}$ as
the initial sequence $\left\\{\vec{\alpha}_{i}\right\\}_{1\leqslant i\leqslant
K}$, there must be a pair $(\vec{\alpha}_{i},\vec{\alpha}_{i^{\prime}})$ with
$i\neq i^{\prime}$ such that the above inequality holds. By Weyl’s bound [14,
Lemma 1.1.16] there is some integer $1\leqslant q^{\prime}\ll\eta^{-O(1)}$
such that
$\left\|q^{\prime}(\alpha_{i}^{(d)}-\alpha_{i^{\prime}}^{(d)})\right\|\ll\eta^{-O(1)}H^{-d}$.
Therefore, in this case, we may rule either $\vec{\alpha}_{i}$ or
$\vec{\alpha}_{i^{\prime}}$ out of the set
$\left\\{\vec{\alpha}_{i}\right\\}_{1\leqslant i\leqslant K}$. It is possible
to continue the excluding process until there are only $O(\eta^{-10})$
elements remaining. And by taking
$\left\\{\vec{\gamma}_{d,j}\right\\}_{1\leqslant j\leqslant K^{\prime}}$ as
the remaining frequencies, the desired conclusion holds for degree $j=d$.

We claim that the desired set for which the entire conclusion holds can be
found by induction of the argument on degrees $1\leqslant j\leqslant d$. Let
$1\leqslant j^{\prime}\leqslant d-1$ and
$\left\\{\vec{\gamma}_{j^{\prime}+1,i}\right\\}_{1\leqslant
i\ll\eta^{-O(1)}}\subset\mathbb{T}^{d}$ be a set of frequencies. Suppose that
$\vec{\alpha}\in\mathbb{T}^{d}$ is a frequency such that the following two
conditions hold:

  1. (1)

$\displaystyle\Bigl{|}\sum_{x<n\leqslant
x+H}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau_{k}^{*}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha^{(j)}(n-z)^{j}\bigr{)}\Bigr{|}\geqslant\eta H\log^{k-1}X$
(5.1)

  2. (2)

there is an integer $q^{\prime}\ll\eta^{-O(1)}$, a large constant $A>1$ and a
frequency
$\vec{\gamma}\in\left\\{\vec{\gamma}_{j^{\prime}+1,i}\right\\}_{1\leqslant
i\ll\eta^{-O(1)}}$ such that

$\displaystyle\left\|q^{\prime}(\gamma^{(j)}-\alpha^{(j)})\right\|\ll\eta^{-A}H^{-j}\text{
for all }j^{\prime}<j\leqslant d.$ (5.2)

Our goal is to find another set of frequencies
$\left\\{\vec{\gamma}_{j^{\prime},i}\right\\}_{1\leqslant i\ll\eta^{-O(1)}}$
such that (5.2) holds for $j^{\prime}\leqslant j\leqslant d$. To do this, let
us fix an integer $q^{\prime}\ll\delta^{-O(1)}$ and a frequency
$\vec{\gamma}\in\left\\{\vec{\gamma}_{j^{\prime}+1,i}\right\\}_{1\leqslant
i\ll\eta^{-O(1)}}$. Suppose that $\vec{\alpha}_{1},\dots,\vec{\alpha}_{K}$ are
all of the frequencies such that (5.1) and (5.2) hold with this particular
$\vec{\gamma}$ and $q^{\prime}$, besides, we also assume that
$\left\|\alpha_{i}^{(j^{\prime})}-\alpha_{i^{\prime}}^{(j^{\prime})}\right\|\gg
H^{-j^{\prime}}$ whenever $i\neq i^{\prime}$.

Decomposing the interval $[x,x+H]$ into sub-intervals of length $\eta^{2A}H$,
on following the argument of degree $j=d$, one has

$\eta HK\log^{k-1}X\leqslant\sum_{x_{0}}\sum_{1\leqslant i\leqslant
K}\Bigl{|}\sum_{x_{0}<n\leqslant
x_{0}+\eta^{2A}H}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau_{k}^{*}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha_{i}^{(j)}(n-z)^{j}\bigr{)}\Bigr{|}\\\
\leqslant\sum_{x_{0}}\sum_{1\leqslant i\leqslant
K}\Bigl{|}\sum_{x_{0}<n\leqslant
x_{0}+\eta^{2A}H}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau_{k}^{*}(n)}{\omega_{(Q_{0},Q]}(n)+1}\prod_{1\leqslant
j\leqslant
d}e\bigl{(}\gamma^{(j)}(n-z)^{j})e((\alpha_{i}^{(j)}-\gamma^{(j)})(n-z)^{j}\bigr{)}\Bigr{|},$

where $x_{0}$ takes values of $\eta^{2A}H$-separated points in the interval
$(x,x+H]$ so that $(x,x+H]=\bigsqcup_{x_{0}}(x_{0},x_{0}+\eta^{2A}H]$. We may
decompose $(x_{0},x_{0}+\eta^{2A}H]$ into arithmetic progressions according to
the residue classes $\,(\mathrm{mod}\,q^{\prime})$ to get that

$\sum_{r\,(\mathrm{mod}\,q^{\prime})}\sum_{x_{0}}\sum_{1\leqslant i\leqslant
K}\Bigl{|}\sum_{x_{0}<n\leqslant x_{0}+\eta^{2A}H\atop n\equiv
r\,(\mathrm{mod}\,q^{\prime})}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau_{k}^{*}(n)}{\omega_{(Q_{0},Q]}(n)+1}\\\
\prod_{1\leqslant j\leqslant
d}e\bigl{(}\gamma^{(j)}(n-z)^{j})e((\alpha_{i}^{(j)}-\gamma^{(j)})(n-z)^{j}\bigr{)}\Bigr{|}\gg\eta
HK\log^{k-1}X.$

Noting that our assumption (5.2) implies that when $j^{\prime}<j\leqslant d$
there are integers $1\leqslant a_{j}\leqslant q^{\prime}$ such that

$\alpha_{i}^{(j)}-\gamma^{(j)}=\frac{a_{j}}{q^{\prime}}+O(\eta^{-A}H^{-j}),$

one may conclude from Taylor expansion that whenever $1\leqslant
n-x_{0}\leqslant\eta^{2A}H$, $|x_{0}-z|\ll H$ and $j^{\prime}<j\leqslant d$
that the phase functrion
$e\bigl{(}(n-z)^{j}(\alpha_{i}^{(j)}-\gamma^{(j)})\bigr{)}$ is equal to

$e\bigl{(}(x_{0}-z)^{j}(\alpha_{i}^{(j)}-\gamma^{(j)})\bigr{)}e\Bigl{(}\frac{a_{j}}{q^{\prime}}\bigl{(}(n-z)^{j}-(x_{0}-z)^{j}\bigr{)}\Bigr{)}e\Bigl{(}O\bigl{(}\eta^{-A}H^{-j}\sum_{1\leqslant
i\leqslant j}(n-x_{0})^{i}(x_{0}-z)^{j-i}\bigr{)}\Bigr{)}\\\
=e\bigl{(}(x_{0}-z)^{j}(\alpha_{i}^{(j)}-\gamma^{(j)})\bigr{)}e\Bigl{(}\frac{a_{j}}{q^{\prime}}\bigl{(}(n-z)^{j}-(x_{0}-z)^{j}\bigr{)}\Bigr{)}(1+O(\eta^{A})).$

Taking note that the first factor is independent of $n$ and the second factor
is also independent of $n$ if we restrict $n$ into the arithmetic progression
$r\,(\mathrm{mod}\,q^{\prime})$, therefore,

$\eta
HK\log^{k-1}X\ll\sum_{r\,(\mathrm{mod}\,q^{\prime})}\sum_{x_{0}}\sum_{1\leqslant
i\leqslant K}\Bigl{|}\sum_{x_{0}<n\leqslant x_{0}+\eta^{2A}H\atop n\equiv
r\,(\mathrm{mod}\,q^{\prime})}\\\
\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau_{k}^{*}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}\gamma^{(j)}(n-z)^{j}\bigr{)}e\bigl{(}(\alpha_{i}^{(j^{\prime})}-\gamma^{(j^{\prime})})(n-z)^{j^{\prime}}+\cdots+(\alpha_{i}^{(1)}-\gamma^{(1)})(n-z)\bigr{)}\Bigr{|}.$

Performing as before, one can find that

$\eta^{2}H^{2}K\log^{2k-2}X\ll\\\ \sum_{|h|\leqslant H}\sum_{x<n\leqslant
x+H}\tau_{k}(n)\tau_{k}(n+h)\Bigl{|}\sum_{1\leqslant i\leqslant
K}e\bigl{(}(\alpha_{i}^{(j^{\prime})}-\gamma^{(j^{\prime})})h^{j^{\prime}}+P(h)\bigr{)}\Bigr{|},$

where
$P(h)=P(h;\alpha_{i}^{(j^{\prime})}-\gamma^{(j^{\prime})},\cdots,\alpha_{i}^{(1)}-\gamma^{(1)},n-z)$
is a polynomial of degree at most $j^{\prime}-1$ with respect to $h$.
Therefore, the previous calculation leads us to

$\eta^{10}K^{2}H\ll\sum_{1\leqslant i,i^{\prime}\leqslant
K}\bigl{|}\sum_{h\leqslant
H}e\bigl{(}(\alpha_{i}^{(j^{\prime})}-\alpha_{i^{\prime}}^{(j^{\prime})})h^{j^{\prime}}+\tilde{P}(h)\bigr{)}\bigr{|},$

where $\tilde{P}(h)$ is a polynomial of degree at most $j^{\prime}-1$. Weyl’s
bound [14, Lemma 1.1.16] then yields that there are at most $O(\eta^{-10})$
members in $\left\\{\vec{\alpha}_{i}\right\\}_{1\leqslant i\leqslant K}$
forming a sequence $\left\\{\vec{\gamma}_{i^{\prime}}\right\\}_{1\leqslant
i^{\prime}\ll\eta^{-10}}$ such that for every $\vec{\alpha}_{i}$ there is some
$\vec{\gamma}_{i^{\prime}}$ and some integer $q^{*}\ll\eta^{-O(1)}$ such that
$\left\|q^{*}(\gamma_{i^{\prime}}^{(j^{\prime})}-\alpha_{i}^{(j^{\prime})})\right\|\ll\eta^{-O(1)}H^{-j^{\prime}}$.
Set $q_{j^{\prime}}=q^{\prime}q^{*}$, then

$\left\|q_{j^{\prime}}(\gamma_{i^{\prime}}^{(j^{\prime})}-\alpha_{i}^{(j^{\prime})})\right\|\ll\eta^{-O(1)}H^{-j^{\prime}};$

besides, it follows from (5.2) and the triangle inequality that whenever
$j^{\prime}<j\leqslant d$

$\left\|q_{j^{\prime}}(\gamma_{i^{\prime}}^{(j)}-\alpha_{i}^{(j)})\right\|\leqslant\left\|q_{j^{\prime}}(\gamma_{i^{\prime}}^{(j)}-\gamma^{(j)})\right\|+\left\|q_{j^{\prime}}(\alpha_{i}^{(j)}-\gamma^{(j)})\right\|\ll\eta^{-O(1)}H^{-j}.$

The claimed sequence $\left\\{\vec{\gamma}_{j^{\prime},i}\right\\}_{i}$ is the
union of sequences $\left\\{\vec{\gamma}_{i^{\prime}}\right\\}_{1\leqslant
i^{\prime}\ll\eta^{-10}}$ obtained by varying $\vec{\gamma}$ over
$\left\\{\vec{\gamma}_{j^{\prime}+1,i}\right\\}_{i\ll\eta^{-O(1)}}$ and
varying $q^{\prime}$ over $[1,O(\eta^{-O(1)})]$. It is easy to see that
$\\#\left\\{\vec{\gamma}_{j^{\prime},i}\right\\}\ll\eta^{-O(1)}$. And the
lemma follows by taking $\left\\{\vec{\beta}_{i}\right\\}$ as
$\left\\{\vec{\gamma}_{1,i}\right\\}_{1\leqslant i\ll\eta^{-O(1)}}$.

∎

With the assistance of Lemma 5.1, we can now build a “narrow” configuration
$\mathcal{J}_{1}$ that is contained in a narrow interval for which elements in
$\mathcal{J}$ can be related to elements in $\mathcal{J}_{1}$. We call this
step as scaling down.

######  Proposition 5.2 (Scaling down).

Suppose that $\log^{10000k\log k}X\leqslant H\leqslant X^{1/3-\varepsilon}$,
$Q_{0}\leqslant P<Q/2$, $\mathcal{J}\subseteq[X,2X]\times\mathbb{T}^{d}$ is a
$(c\eta^{2},H)$-configuration, and for each
$(x,\vec{\alpha}_{x})\in\mathcal{J}$ there exist $\gg\eta^{6}P/\log P$ primes
$p\in[P,2P]$ and some $|x_{0}-x|\leqslant H$ such that

$\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}p^{j}\alpha_{x}^{(j)}(n-x_{0}/p)^{j}\bigr{)}\Bigr{|}\geqslant\eta^{4}\frac{H\log^{k-1}X}{p},$

where $S\subseteq[X,2X]$ is defined in (3.1) and $0<t_{n}<1$ for every
$n\in[X,2X]$.

Then there is a $(c\eta^{O(1)},\frac{H}{P})$-configuration
$\mathcal{J}_{1}\subseteq[\frac{X}{2P},\frac{2X}{P}]\times\mathbb{T}^{d}$ for
which when $(y,\vec{\alpha}_{y})\in\mathcal{J}_{1}$ there exists an integer
$1\leqslant q_{y}\ll\eta^{-O(1)}$ satisfying the following two events:

  1. (1)

there is some $y_{0}$ with $|y_{0}-y|\leqslant H/P$ such that

$\Bigl{|}\sum_{y<n\leqslant
y+\frac{H}{P}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}q_{y}^{-1}\alpha_{y}^{(j)}(n-y_{0})^{j}\bigr{)}\Bigr{|}\gg\eta^{4}\frac{H\log^{k-1}X}{P};$

  2. (2)

there are $\gg\eta^{O(1)}\frac{P}{\log P}$ elements
$(p,(x,\vec{\alpha}_{x}))\in[P,2P]\times\mathcal{J}$ such that

$|py-x|\leqslant 2H,\qquad\alpha_{y}^{(j)}\equiv
p^{j}q_{y}\alpha_{x}^{(j)}+O(\eta^{-O(1)}(P/H)^{j})\,(\mathrm{mod}\,1)\qquad(1\leqslant
j\leqslant d).$

###### Proof.

Corollary 2.3 yields that outside of $\\#\mathcal{E}\ll\eta^{20}\frac{X}{P}$
numbers $s\in[\frac{X}{2P},\frac{2X}{P}]$ the pointwise upper bound below
holds

$\displaystyle\sum_{s<n\leqslant
s+\eta^{5}\frac{H}{P}}(\tau_{k}(n)+\tau^{*}_{k}(n))\ll\eta^{5}\frac{H\log^{k-1}X}{P}.$
(5.3)

Now for a fixed pair $(p,(x,\vec{\alpha}_{x}))\in[P,2P]\times\mathcal{J}$
satisfies

$\displaystyle\Bigl{|}\sum_{\frac{x}{p}<n\leqslant\frac{x+H}{p}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}p^{j}\alpha_{x}^{(j)}(n-x_{0}/p)^{j}\bigr{)}\Bigr{|}\geqslant\eta^{4}\frac{H\log^{k-1}X}{p},$
(5.4)

it is easy to find that if $I\subseteq[\frac{X}{2P},\frac{2X}{P}]$ is an
interval with $|I|\gg\frac{H}{P}$ and
$|I\bigtriangleup[\frac{x}{p},\frac{x+H}{p}]|\leqslant\eta^{5}\frac{H}{P}$, by
naming the starting point of this interval as $y$, we have $|py-x|\leqslant
2H$ and $y\in[\frac{X}{2P},\frac{2X}{P}]$, and there are
$\gg\eta^{5}\frac{H}{P}$ such $y$ created by this pair
$(p,(x,\vec{\alpha}_{x}))$. When $(p,(x,\vec{\alpha}_{x}))$ varies, there are
in total $\gg\eta^{13}\frac{H}{P}\frac{P}{\log P}\frac{X}{H}$ such $y$
produced (counting the occurrences). On the contrary, noting that each
$y\in[\frac{X}{2P},\frac{2X}{P}]$ can only relate to at most $O(P/\log P)$ of
$(p,(x,\vec{\alpha}_{x}))$, we can see these $y$ for which either
$y-\eta^{5}\frac{H}{P}$ or $y+\frac{H}{P}$ is in the exceptional set
$\mathcal{E}$ are sparse. Moreover, the application of Lemma 2.4 with
parameters $\delta=\eta^{15}$, $X=X/H$, and $H=H/P$ implies that the set of
$y$ failing to satisfy the conclusion of Lemma 2.4 is bounded by
$O(\eta^{15}\frac{P}{\log P}\frac{X}{P})$ which is also sparse. Thus, after
removing those exceptional $y$, we can assume that there are
$\gg\eta^{13}\frac{H}{P}\frac{P}{\log P}\frac{X}{H}$ of
$(y,p,(x,\vec{\alpha}_{x}))$ such that

  1. (1)

$|py-x|\leqslant 2H$;

  2. (2)

(5.3) holds for both $s=y-\eta^{5}\frac{H}{P}$ and $s=y+\frac{H}{P}$;

  3. (3)

there is an exceptional set $\mathcal{E}^{\prime}\subseteq[-H/P,H/P]$ with
$|\mathcal{E}^{\prime}|\ll\eta^{30}H/P$ and such that when
$h\not\in\mathcal{E}^{\prime}$ we have $\sum_{|n-y|\leqslant
H/P}\tau_{k}(n)\tau_{k}(n+h)\ll\eta^{-15}(H/P)\log^{2k-2}X$, and
$\sum_{h\in\mathcal{E}^{\prime}}\sum_{|n-y|\leqslant
H/P}\tau_{k}(n)\tau_{k}(n+h)\ll\eta^{30}(H/P)^{2}\log^{2k-2}X$.

Pigeonhole principle illustrates that there are $\gg\eta^{13}\frac{P}{\log
P}\frac{X}{H}$ of such triples $(y,p,(x,\vec{\alpha}_{x}))$ for which $y$ are
$\frac{H}{P}$-separated. Another application of pigeonhole principle implies
that there are $\gg\eta^{7}\frac{X}{H}$ of $\frac{H}{P}$-separated $y$ such
that the above (1–3) holds, besides, for each of those $y$ there are
$\gg\eta^{6}P/\log P$ of $(p,(x,\vec{\alpha}_{x}))$ such that $|py-x|\leqslant
2H$. We can collect these $y$ into a set $\mathcal{Y}$.

Let $y\in\mathcal{Y}$, we may suppose that $I_{y}$ is the interval such that
$\bigl{|}[y,y+|I_{y}|]\bigtriangleup[\frac{x}{p},\frac{x+H}{p}]\bigr{|}\leqslant\eta^{5}\frac{H}{P}$
and rescale the length of the interval $I_{y}$ as $\frac{H}{P}$, it then
follows from (5.4) that

$\eta^{4}\frac{H\log^{k-1}X}{p}\leqslant\Bigl{|}\sum_{y<n\leqslant
y+\frac{H}{P}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}p^{j}\alpha_{x}^{(j)}(n-x_{0}/p)^{j}\bigr{)}\Bigr{|}\\\
+\Bigl{|}\sum_{\frac{x}{p}<n\leqslant
y}(\tau_{k}(n)+\tau^{*}_{k}(n))\Bigr{|}+\Bigl{|}\sum_{y+\frac{H}{P}<n\leqslant\frac{x+H}{p}}(\tau_{k}(n)+\tau^{*}_{k}(n))\Bigr{|}.$

Since our assumption
$\bigl{|}[y,y+|I_{y}|]\bigtriangleup[\frac{x}{p},\frac{x+H}{p}]\bigr{|}\leqslant\eta^{5}\frac{H}{P}$
reveals that $|y-\frac{x}{p}|\leqslant\eta^{5}\frac{H}{P}$ and
$|\frac{x+H}{p}-y-\frac{H}{P}|\leqslant\eta^{5}\frac{H}{P}$, taking (5.3) into
account, we can conclude that the last two sums on the right-hand side are
negligible. Therefore, when $y\in\mathcal{Y}$ there is some
$|y_{0}-y|\leqslant H/P$ such that

$\eta^{4}\frac{H\log^{k-1}X}{P}\ll\Bigl{|}\sum_{y<n\leqslant
y+\frac{H}{P}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}p^{j}\alpha_{x}^{(j)}(n-y_{0})^{j}\bigr{)}\Bigr{|}$

holds for $\gg\eta^{6}P/\log P$ of pairs $(p,(x,\vec{\alpha}_{x}))$ and
$|py-x|\leqslant 2H$.

Fixing such a $y\in\mathcal{Y}$ for the moment, it is immediately from Lemma
5.1 that there are at most $O(\eta^{-O(1)})$ frequencies
$\vec{\beta}_{1},\dots,\vec{\beta}_{K}$ such that for each
$(p^{d}\alpha^{(d)}_{x},\cdots,p\alpha^{(1)}_{x})$ there is some
$\vec{\beta}_{i}\in\left\\{\vec{\beta}_{1},\dots,\vec{\beta}_{K}\right\\}$ and
some integer $1\leqslant q\ll\eta^{-O(1)}$ such that

$\left\|q(\beta_{i}^{(j)}-p^{j}\alpha^{(j)}_{x})\right\|\ll\eta^{-O(1)}(P/H)^{j}\qquad(1\leqslant
j\leqslant d).$

On noting that there are totally $O(\eta^{-O(1)})$ choices of
$(\vec{\beta}_{i},q)$, pigeonhole principle tells us that there is some
$\vec{\beta}_{i}$ and some integer $q$ such that the above inequality holds
for $\gg\eta^{O(1)}P/\log P$ pairs $(p,(x,\vec{\alpha}_{x}))$. We can set
$\vec{\alpha}_{y}$ as this $q\vec{\beta_{i}}$ and set
$\mathcal{J}_{1}=\cup_{y\in\mathcal{Y}}\left\\{(y,\vec{\alpha}_{y})\right\\}$,
then $\mathcal{J}_{1}\subset[\frac{X}{2P},\frac{2X}{P}]\times\mathbb{T}^{d}$
is a $(c\eta^{O(1)},H)$-configuration. Taking $z_{1,x}^{(1)}=y$, $h_{1}=H/P$
in Lemma 4.2 (3) and enlarging $q$ if necessary, it follow from Taylor
expansion and the above two inequalities that

$\Bigl{|}\sum_{y<n\leqslant
y+\frac{H}{P}}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant d}q^{-1}\alpha_{y}^{(j)}(n-y_{0})^{j}\bigr{)}\Bigr{|}\\\
=\Bigl{|}\sum_{y<n\leqslant
y+H/P}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}\prod_{1\leqslant
j\leqslant
d}e(p^{j}\alpha_{x}^{(j)}(n-y_{0})^{j})e\bigl{(}q^{-1}(\alpha_{y}^{(j)}-qp^{j}\alpha_{x}^{(j)})(n-y_{0})^{j})\bigr{)}\Bigr{|}\\\
\gg\Bigl{|}\sum_{y<n\leqslant
y+H/P}\frac{1_{S}(n)\tau_{k}(n)-t_{n}\tau^{*}_{k}(n)}{\omega_{(Q_{0},Q]}(n)+1}e\bigl{(}\sum_{1\leqslant
j\leqslant
d}p^{j}\alpha_{x}^{(j)}(n-y_{0})^{j}\bigr{)}\Bigr{|}-O(\eta^{O(1)}\sum_{y<n\leqslant
y+H/P}\tau_{k}(n))\\\ \gg\eta^{4}\frac{H\log^{k-1}X}{P}.$

Therefore, $(y,\vec{\alpha}_{y})\in\mathcal{J}_{1}$ satisfies the two
conditions of the lemma.

∎

The next conclusion is a generalization of [7, Proposition 3.2] and [8,
Proposition 3.6], and this is also the main result of this section.

######  Proposition 5.3 (Connectedness).

Suppose that $\exp(\log^{\varepsilon}X)\leqslant H\leqslant
X^{1/3-\varepsilon}$ and

$\int_{X}^{2X}\sup_{\vec{\alpha}\in\mathbb{T}^{d}}\sup_{|x_{0}-x|\leqslant
H}\bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau^{*}_{k}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha^{(j)}(n-x_{0})^{j}\bigr{)}\bigr{|}\,\,\mathrm{d}x\geqslant\eta
HX\log^{k-1}X,$

then there are two numbers $P,P^{\prime}$ with $H^{\omega^{2}}<P\leqslant
H^{\omega}$ and
$\bigl{(}\frac{H}{P}\bigr{)}^{\omega^{3}}<P^{\prime}<\bigl{(}\frac{H}{P}\bigr{)}^{\omega^{2}}$,
two disjoint subsets of primes $\mathcal{P}_{1},\mathcal{P}_{2}\subset[P,2P]$
with $|\mathcal{P}_{1}|,|\mathcal{P}_{2}|\gg\eta^{O(1)}\frac{P}{\log P}$, and
a $(c\eta^{2},H)$-configuration
$\mathcal{J}\subseteq[X,2X]\times\mathbb{T}^{d}$ such that the following
statement holds.

  1. (1)

There are $\gg\eta^{O(1)}(\frac{P}{\log P})^{2}\frac{X}{H}$ of quadruple
$(p,q,(x_{1},\vec{\alpha}_{x_{1}}),(x_{2},\vec{\alpha}_{x_{2}}))\in\mathcal{P}_{1}\times\mathcal{P}_{2}\times\mathcal{J}^{2}$
such that $|\frac{x_{1}}{p}-\frac{x_{2}}{q}|\ll\frac{H}{P}$ and

$p^{j}\alpha^{(j)}_{x_{1}}\equiv
q^{j}\alpha^{(j)}_{x_{2}}+O\bigl{(}\eta^{-O(1)}(P/H)^{j}\bigr{)}\,(\mathrm{mod}\,p^{\prime})\qquad(1\leqslant
j\leqslant d)$

hold for $\gg\eta^{O(1)}\frac{P^{\prime}}{\log P^{\prime}}$ primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$;

  2. (2)

for each $(x,\vec{\alpha}_{x})\in\mathcal{J}$ there is a number $x_{0}$ with
$|x_{0}-x|\leqslant H$ such that

$\bigl{|}\sum_{x<n\leqslant
x+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha_{x}^{(j)}(n-x_{0})^{j}\bigr{)}\bigr{|}\gg\eta H\log^{k-1}X.$

###### Proof.

We aim to demonstrate that the configuration $\mathcal{J}$ we established in
Lemma 4.3 satisfies the two conditons of this proposition. Drawing from the
proof of Lemma 4.3, where it is established that $\mathcal{J}$ is a subset of
$\mathcal{J}_{-1}$, the second conclusion immediately follows from Lemma 4.2
(2). Let’s now focus on proving the first conclusion. By observing that
$[H^{\omega^{2}},H^{\omega}]\cap[(H/P)^{\omega^{3}},(H/P)^{\omega^{2}}]=\emptyset$,
one can employ the scaling-down step (Proposition 5.2) twice in succession to
conclude that—there is some
$P^{\prime}\in[\bigl{(}\frac{H}{P}\bigr{)}^{\omega^{3}},\bigl{(}\frac{H}{P}\bigr{)}^{\omega^{2}}]$,
a $(c\eta^{O(1)},\frac{H}{PP^{\prime}})$-configuration
$\mathcal{J}_{2}\subset[\frac{X}{4PP^{\prime}},\frac{4X}{PP^{\prime}}]\times\mathbb{T}^{d}$
satisfying that for each $(z,\vec{\alpha}_{z})\in\mathcal{J}_{2}$ there is an
integer $0<q_{z}\ll\eta^{-O(1)}$ and $\gg\eta^{O(1)}\frac{P^{\prime}}{\log
P^{\prime}}$ of
$(p^{\prime},(y,\vec{\alpha}_{y}))\in[P^{\prime},2P^{\prime}]\times\mathcal{J}_{1}$
such that

$|p^{\prime}z-y|\ll\frac{H}{P},\qquad\alpha_{z}^{(j)}\equiv p^{\prime
j}q_{z}q_{y}^{-1}\alpha_{y}^{(j)}+O(\eta^{-O(1)}(PP^{\prime}/H)^{j})\,(\mathrm{mod}\,1)\qquad(1\leqslant
j\leqslant d),$

where $1\leqslant q_{y}\ll\eta^{-O(1)}$ is the integer in Proposition 5.2. The
application of the pigeonhole principle reveals that there is a popular
modulus $q^{\prime}\ll\eta^{-O(1)}$ such that $q^{\prime}=q_{z}$ for
$\gg\eta^{O(1)}|\mathcal{J}_{2}|$ of $(z,\vec{\alpha}_{z})$. We can narrow
down $\mathcal{J}_{2}$ to the subset of elements $(z,\vec{\alpha}_{z})$ where
the corresponding $q_{z}$ equals this popular $q^{\prime}$. Fixing the degree
$1\leqslant j\leqslant d$, we first upgrade the above congruent equation to
modulus $q^{\prime}p^{\prime j}$ by adding an integer with an appropriate
residue class mudulus $q^{\prime}p^{j}$ to $\alpha_{z}^{(j)}$ and renaming the
new frequency as $\alpha_{z}^{(j)}$ to obtain that

$\alpha_{z}^{(j)}\equiv q^{\prime}p^{\prime
j}q_{y}^{-1}\alpha_{y}^{(j)}+O(\eta^{-O(1)}(PP^{\prime}/H)^{j})\,(\mathrm{mod}\,q^{\prime}p^{\prime
j}).$

By the Chinese remainder theorem we can select some $\alpha_{z}^{(j)}$ such
that the above congruent equation holds for $\gg\eta^{O(1)}P^{\prime}/\log
P^{\prime}$ primes $p^{\prime}\in[P^{\prime},2P^{\prime}]$. Next, we combine
this result with the second conclusion of Proposition 5.2 to obtain an integer
$1\leqslant q^{\prime}\ll\eta^{-O(1)}$ such that the following statement
holds. For each $(z,\vec{\alpha}_{z})\in\mathcal{J}_{2}$ there are
$\gg\eta^{O(1)}\frac{P}{\log P}\frac{P^{\prime}}{\log P^{\prime}}$ of triples
$(p,p^{\prime},(x,\vec{\alpha}_{x}))\in[P,2P]\times[P^{\prime},2P^{\prime}]\times\mathcal{J}$
such that

$|z-\frac{x}{pp^{\prime}}|\ll\frac{H}{PP^{\prime}}$

and

$\alpha_{z}^{(j)}\equiv q^{\prime}p^{\prime
j}p^{j}\alpha_{x}^{(j)}+O(\eta^{-O(1)}(PP^{\prime}/H)^{j})\,(\mathrm{mod}\,q^{\prime}p^{\prime
j})\qquad(1\leqslant j\leqslant d).$

Using $(*)$ to denote the above relations hold. It follows from Cauchy-Schwarz
inequality that

$\eta^{O(1)}(\frac{P}{\log P})^{2}(\frac{P^{\prime}}{\log
P^{\prime}})^{2}(\frac{X}{H})^{2}\ll\Bigl{(}\sum_{(z,\vec{\alpha}_{z})\in\mathcal{J}_{2}}\sum_{(x,\vec{\alpha}_{x})\in\mathcal{J}}\sum_{p\sim
P}\sum_{p^{\prime}\sim
P^{\prime}}1_{(*)}(p,p^{\prime},(x,\vec{\alpha}_{x}),(z,\vec{\alpha}_{z}))\Bigr{)}^{2}\\\
\ll\frac{X}{H}\frac{P^{\prime}}{\log
P^{\prime}}\sum_{(z,\vec{\alpha}_{z})\in\mathcal{J}_{2}}\sum_{p^{\prime}\sim
P^{\prime}}\sum_{p_{1}\sim P\atop p_{2}\sim
P}\sum_{(x_{1},\vec{\alpha}_{x_{1}})\in\mathcal{J}\atop(x_{2},\vec{\alpha}_{x_{2}})\in\mathcal{J}}1_{(*)}(p_{1},p^{\prime},(x_{1},\vec{\alpha}_{x_{1}}),(z,\vec{\alpha}_{z}))1_{(*)}(p_{2},p^{\prime},(x_{2},\vec{\alpha}_{x_{2}}),(z,\vec{\alpha}_{z})),$

which means that, there are $\gg\eta^{O(1)}(P/\log P)^{2}(P^{\prime}\log
P^{\prime})(X/H)$ of sextuples
$(p^{\prime},p_{1},p_{2},(z,\vec{\alpha}_{z}),(x_{1},\vec{\alpha}_{x_{1}}),(x_{2},\vec{\alpha}_{x_{2}}))\in[P^{\prime},2P^{\prime}]\times[P,2P]^{2}\times\mathcal{J}_{2}\times\mathcal{J}^{2}$
such that

$|z-\frac{x_{1}}{p_{1}p^{\prime}}|\ll
H/(PP^{\prime}),\qquad|z-\frac{x_{2}}{p_{2}p^{\prime}}|\ll H/(PP^{\prime})$

and for every degree $1\leqslant j\leqslant d$ we also have

$\displaystyle\alpha_{z}^{(j)}\equiv q^{\prime}p^{\prime
j}p_{1}^{j}\alpha_{x_{1}}^{(j)}+O(\eta^{-O(1)}(PP^{\prime}/H)^{j})\,(\mathrm{mod}\,q^{\prime}p^{\prime
j})$ $\displaystyle\alpha_{z}^{(j)}\equiv q^{\prime}p^{\prime
j}p_{2}^{j}\alpha_{x_{2}}^{(j)}+O(\eta^{-O(1)}(PP^{\prime}/H)^{j})\,(\mathrm{mod}\,q^{\prime}p^{\prime
j}).$

Hence, by combining the above two congruence equations together and
subsequently dividing both sides by $q^{\prime}p^{\prime j}$, we obtain that

$p_{1}^{j}\alpha_{x_{1}}^{(j)}\equiv
p_{2}^{j}\alpha_{x_{2}}^{(j)}+O(\eta^{-O(1)}(P/H)^{j})\,(\mathrm{mod}\,1).$

It is possible to upgrade this congruent equation to modulus $p^{\prime}$ by
adding a suitable integer with appropriate residue class mudulus $p^{\prime}$
to $\alpha_{x_{1}}^{(j)}$ to get that

$p_{1}^{j}\alpha_{x_{1}}^{(j)}\equiv
p_{2}^{j}\alpha_{x_{2}}^{(j)}+O(\eta^{-O(1)}(P/H)^{j})\,(\mathrm{mod}\,p^{\prime}),$

without affecting the local correlation

$\bigl{|}\sum_{x_{1}<n\leqslant
x_{1}+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant
j\leqslant d}\alpha_{x_{1}}^{(j)}(n-x_{0})^{j}\bigr{)}\bigr{|}\gg\eta
H\log^{k-1}X.$

We then apply Chinese remainder theorem to conclude that this congruent
equation holds for $\gg\eta^{O(1)}P^{\prime}/\log P^{\prime}$ primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$. The conclusion follows by observing
that we can apply the pigeonhole principle to restrict $p,q\in[P,2P]$ into two
disjoint dense (up to a $O(\eta^{O(1)})$ factor) subsets $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$. ∎

##  6\. Long disjoint pair of paths

In this section, we are going to use the frequency relations established in
Section 2.2 to construct long split paths. In what follows are the recordings
of [16, Definitions 3.6 and 3.8].

######  Definition 6.1 (Split path).

Let $Q\in\mathbb{N}$, $\mathcal{J}^{\\#}\subseteq\mathcal{J}$ and
$\mathcal{P}_{1},\mathcal{P}_{2}\subset[P,2P]$ are disjoint subsets. We say
$(x_{1},\vec{\alpha}_{1})$ and
$(x_{k+1}\vec{\alpha}_{k+1})\in\mathcal{J}^{\\#}$ are connected by a split
path $\,(\mathrm{mod}\,Q)$ of length $k$ in $\mathcal{J}^{\\#}$ if there are
primes
$p_{1},\dots,p_{k}\in\mathcal{P}_{1},q_{1},\dots,q_{k}\in\mathcal{P}_{2}$ and
$(x_{2},\vec{\alpha}_{2}),\dots,(x_{k},\vec{\alpha}_{k})\in\mathcal{J}^{\\#}$
such that

$|\frac{x_{i}}{p_{i}}-\frac{x_{i+1}}{q_{i}}|\ll\frac{H}{P}\quad\text{ and
}\quad p_{i}^{j}\alpha^{(j)}_{i}\equiv
q_{i}^{j}\alpha_{i+1}^{(j)}+O(\eta^{-O(1)}(P/H)^{j})\,(\mathrm{mod}\,Q)\qquad(1\leqslant
j\leqslant d).$

We use $l$ to denote a path and the $3k+1$-tuple
$(p_{1},\dots,p_{k},q_{1},\dots,q_{k},(x_{1},\vec{\alpha}_{1}),\dots,(x_{k+1},\vec{\alpha}_{k+1}))$
indeed determine a path by the above two inequalities. We call
$(x_{1},\vec{\alpha}_{1})$ the starting point and
$(x_{k+1},\vec{\alpha}_{k+1})$ the endpoint of this path. If $l_{1}$ and
$l_{2}$ are two split paths $\,(\mathrm{mod}\,Q)$ of length $k$ connecting
$(x,\vec{\alpha})$ and $(y,\vec{\beta})$ and sharing no prime in common, we
say they are disjoint.

######  Definition 6.2 (Regularity).

Let $Q\in\mathbb{N}$. We say $\mathcal{J}^{\\#}\subseteq\mathcal{J}$ is
$(c,Q)$-regular if $|\mathcal{J}^{\\#}|\geqslant c\eta^{O(1)}|\mathcal{J}|$
and for every $(x,\vec{\alpha})\in\mathcal{J}^{\\#}$ we can find
$\gg\eta^{O(1)}(\frac{P}{\log P})^{2}$ of
$(p,q,(y,\vec{\beta}))\in\mathcal{P}_{1}\times\mathcal{P}_{2}\times\mathcal{J}^{\\#}$
such that $(p,q,(x,\vec{\alpha}),(y,\vec{\beta}))$ forms a split path
$\,(\mathrm{mod}\,Q)$ of length 1 in $\mathcal{J}^{\\#}$.

In view of Proposition 5.3, for $\gg\eta^{O(1)}$ proportion of primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$, we can find a
$(c,p^{\prime})$-regular subset of $\mathcal{J}$, and this is indeed [16,
Lemma 3.9].

######  Lemma 6.3.

Suppose that $P^{\prime}$ and $\mathcal{J}$ are as in Proposition 5.3, then
for $\gg\eta^{O(1)}P^{\prime}/\log P^{\prime}$ primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$ we can find a $(c,p^{\prime})$-regular
subset $\mathcal{J}_{p^{\prime}}$ of $\mathcal{J}$.

The next result is the main conclusion of this section, which denominates that
there is some $(x,\vec{\alpha})\in\mathcal{J}$ and a large subset
$\mathcal{J}^{\\#}$ of $\mathcal{J}$ such that for each
$(y,\vec{\beta})\in\mathcal{J}^{\\#}$ there are two disjoint reasonable long
split paths connecting $(x,\vec{\alpha})$ and $(y,\vec{\beta})$. The proof
essentially follows from [16, Lemma 5.4], since we show the dependence of
$\eta$ here (which was not obvious in [16]) and the major restriction on $H$
will be revealed, we’d like to provide a detailed proof.

######  Proposition 6.4 (Many pairs of disjoint split paths produced).

Let $0<\sigma<1$ be a fixed number and $k_{0}=\left\lfloor\frac{\log(X/(AH\log
X))}{2\log(2P)}\right\rfloor$ for some sufficiently large constant $1\leqslant
A\ll 1$. Suppose that $0<\omega<1$ is defined in Section 4 that is
sufficiently small with respect to $\sigma$, and $\exp\bigl{(}C(\log
X)^{1/2}(\log\log X)^{1/2}\bigr{)}\leqslant H\leqslant X^{1/3-\varepsilon}$
with $C>0$ sufficiently large with respect to $\sigma$ and $\omega$. Then
there is an element $(x_{0},\vec{\alpha}_{0})\in\mathcal{J}$, a subset
$\mathcal{J}^{\\#}\subseteq\mathcal{J}$ with cardinality
$\\#\mathcal{J}^{\\#}\gg X/H^{1+\sigma}$ such that the following holds.

For every $(y,\vec{\beta})\in\mathcal{J}^{\\#}$ there are two disjoint split
paths $\,(\mathrm{mod}\,Q_{y})$ of length $k_{0}+2$ connecting
$(x,\vec{\alpha})$ and $(y,\vec{\beta})$, where $Q_{y}$ is a product of
$\gg\eta^{O(k_{0})}\frac{P^{\prime}}{\log P^{\prime}}$ of primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$.

###### Proof.

Recording what we mean by the regularity in Definition 6.2, Lemma 6.3 shows
that

$\eta^{O(1)}\frac{P^{\prime}}{\log
P^{\prime}}\frac{X}{H}\ll\sum_{p^{\prime}\sim
P^{\prime}}\\#\mathcal{J}_{p^{\prime}}\ll\sum_{(x,\vec{\alpha})\in\mathcal{J}}\sum_{p^{\prime}\sim
P^{\prime}}1_{(x,\vec{\alpha})\in\mathcal{J}_{p^{\prime}}},$

one can see, from the pigeonhole principle, that there is some
$(x,\vec{\alpha})\in\mathcal{J}$ that is also in
$\gg\eta^{O(1)}\frac{P^{\prime}}{\log P^{\prime}}$ of regular
$\mathcal{J}_{p^{\prime}}$. This $(x,\vec{\alpha})$ is the
$(x_{0},\vec{\alpha}_{0})$ described in the statement.

Fix a prime $p^{\prime}\in[P^{\prime},2P^{\prime}]$ for which
$\mathcal{J}_{p^{\prime}}$ is regular and fix the above
$(x,\vec{\alpha})\in\mathcal{J}_{p^{\prime}}$. By the regularity of
$\mathcal{J}_{p^{\prime}}$ (see Definition 6.2), there would be
$\gg\eta^{O(k_{0})}(\frac{P}{\log P})^{2k_{0}+4}$ split paths
$\,(\mathrm{mod}\,p^{\prime})$ of length $k_{0}+2$ with starting point
$(x,\vec{\alpha})$ in $\mathcal{J}_{p^{\prime}}$. As for any given
$2k_{0}+4$-tuple ordered primes $(p_{1},\dots,q_{k_{0}+2})$, starting with
$(x,\vec{\alpha})$, there is an unique split path $\,(\mathrm{mod}\,1)$ of
length $k_{0}+2$, hence, there are in total $O((\frac{P}{\log P})^{2k_{0}+4})$
split path $\,(\mathrm{mod}\,1)$ of length $k_{0}+2$ with starting point
$(x,\vec{\alpha})$ in $\mathcal{J}$. Consequently, we can suppose that there
are $\gg\eta^{O(k_{0})}(\frac{P}{\log P})^{2k_{0}+4}$ split path
$\,(\mathrm{mod}\,1)$ of length $k_{0}+2$ with starting point
$(x,\vec{\alpha})$ in $\mathcal{J}$ that are also split paths
$\,(\mathrm{mod}\,p^{\prime})$ for $\gg\eta^{O(k_{0})}\frac{P^{\prime}}{\log
P^{\prime}}$ primes $p^{\prime}\in[P^{\prime},2P^{\prime}]$. Write
$\mathcal{R}$ for this set of paths.

Let $\mathcal{R}_{y}=\left\\{l\in\mathcal{R}:l\text{ has endpoint
}(y,\vec{\beta})\right\\}$. On recalling that for each $l\in\mathcal{R}_{y}$
there are $\gg\eta^{O(k_{0})}\frac{P^{\prime}}{\log P^{\prime}}$ primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$ such that
$l\,(\mathrm{mod}\,p^{\prime})$ is also a path in $\mathcal{J}_{p^{\prime}}$,
we may use Cauchy-Schwarz inequality to produce pair of paths with the same
endpoint $(y,\vec{\beta})$ as follows.

$\eta^{O(k_{0})}(\frac{P^{\prime}}{\log
P^{\prime}})^{2}|\mathcal{R}_{y}|^{2}\ll\Bigl{(}\sum_{l\in\mathcal{R}_{y}}\sum_{p^{\prime}\sim
P^{\prime}}1_{l\,(\mathrm{mod}\,p^{\prime})}\Bigr{)}^{2}\ll\frac{P^{\prime}}{\log
P^{\prime}}\sum_{p^{\prime}\sim
P^{\prime}}\sum_{l_{1},l_{2}\in\mathcal{R}_{y}}1_{l_{1}\,(\mathrm{mod}\,p^{\prime})}1_{l_{2}\,(\mathrm{mod}\,p^{\prime})},$

by mean of pigeonhole principle there are
$\gg\eta^{O(k_{0})}|\mathcal{R}_{y}|^{2}$ of pairs of split paths
$(l_{1},l_{2})\in\mathcal{R}_{y}^{2}$ that are also split paths
$\,(\mathrm{mod}\,p^{\prime})$ for $\gg\eta^{O(k_{0})}\frac{P^{\prime}}{\log
P^{\prime}}$ primes $p^{\prime}\in[P^{\prime},2P^{\prime}]$. We can further
count the number of pairs of paths with the same endpoint in $\mathcal{R}$. In
practice, from the assumption,

$\eta^{O(k_{0})}(\frac{P^{\prime}}{\log P^{\prime}})^{2}(\frac{P}{\log
P})^{4k_{0}+8}\ll\Bigl{(}\sum_{(y,\vec{\beta})\in\mathcal{J}}\sum_{l\in\mathcal{R}_{y}}\sum_{p^{\prime}\sim
P^{\prime}}1_{l\,(\mathrm{mod}\,p^{\prime})}\Bigr{)}^{2}\\\
\ll\frac{X}{H}\frac{P^{\prime}}{\log P^{\prime}}\sum_{p^{\prime}\sim
P^{\prime}}\sum_{(y,\vec{\beta})\in\mathcal{J}}\sum_{l_{1},l_{2}\in\mathcal{R}_{y}}1_{l_{1}\,(\mathrm{mod}\,p^{\prime})}1_{l_{2}\,(\mathrm{mod}\,p^{\prime})},$

which means that the number of pairs of split paths in $\mathcal{R}$ with the
same endpoint and also being paths $\,(\mathrm{mod}\,p^{\prime})$ for
$\gg\eta^{O(k_{0})}\frac{P^{\prime}}{\log P^{\prime}}$ primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$ is at least

$\gg\eta^{O(k_{0})}(\frac{P}{\log P})^{4k_{0}+8}\frac{H}{X}.$

Write this set of pairs of paths as $\mathcal{R}_{0}$, then
$\mathcal{R}_{0}\subseteq\mathcal{R}\times\mathcal{R}$ and
$\\#\mathcal{R}_{0}\gg\eta^{O(k_{0})}(\frac{P}{\log
P})^{4k_{0}+8}\frac{H}{X}$.

Next, we need to rule out those pairs $(l_{1},l_{2})\in\mathcal{R}_{0}$ that
are not disjoint (i.e. sharing a prime in common from Definition 6.1). Thanks
to [16, Corollary 5.3] which shows that there are not many pairs of paths of
reasonable length with the same starting point and endpoint that can share a
prime in common, we can conclude by applying [16, Corollary 5.3] with
$k=k_{0}+2$ that

$\\#\left\\{(l_{1},l_{2})\in\mathcal{R}_{0}:l_{1}\text{ and }l_{2}\text{ are
not disjoint}\right\\}\\\ \ll(\frac{P}{\log
P})^{4k_{0}+8}\frac{H}{X}\frac{\log^{4k_{0}+7}P}{P}k_{0}2^{4k_{0}+7}(2k_{0}+4)!^{2}\ll\eta^{O(k_{0})}(\frac{P}{\log
P})^{4k_{0}+8}\frac{H}{X},$

by our hypothesis on $\omega$ and $C$. Hence, there is
$\mathcal{R}^{*}\subseteq\mathcal{R}_{0}$ with
$|\mathcal{R}^{*}|\gg\eta^{O(k_{0})}(\frac{P}{\log P})^{4k_{0}+8}\frac{H}{X}$
and for each pair $(l_{1},l_{2})\in\mathcal{R}^{*}$ they are disjoint paths
$\,(\mathrm{mod}\,p^{\prime})$ for $\gg\eta^{k_{0}}\frac{P^{\prime}}{\log
P^{\prime}}$ primes $p^{\prime}\in[P^{\prime},P^{\prime}]$ (so that they are
paths $\,(\mathrm{mod}\,Q)$ with $Q$ as the product of
$\gg\eta^{k_{0}}\frac{P^{\prime}}{\log P^{\prime}}$ of primes
$p^{\prime}\in[P^{\prime},P^{\prime}]$), besides, $l_{1}$ and $l_{2}$ are of
length $k_{0}+2$ and having starting point $(x,\vec{\alpha})$ and having the
same endpoint. It is thus sufficient to count how many endpoints of the paths
within $\mathcal{R}^{*}$. Write $\mathcal{J}^{\\#}$ as the set of endpoints of
paths in $\mathcal{R}^{*}$.

Given an element $(y,\vec{\beta})\in\mathcal{J}$ we now consider how many
split paths of length $k_{0}+2$ with starting point $(x,\vec{\alpha})$ and
endpoint $(y,\vec{\beta})$ can be constructed in $\mathcal{J}$. Let $l$ be one
of such paths and decompose it as $l=l_{1}+l_{2}$, where $l_{1}$ is the path
of length $k_{0}$ with starting point $(x,\vec{\alpha})$ and $l_{2}$ is the
path of length $2$ with endpoint $(y,\vec{\beta})$. As there are only
$O((\frac{P}{\log P})^{4})$ such $l_{2}$ and each such $l_{2}$ would also give
$l_{1}$ a fixed endpoint. It follows from [16, Lemma 5.1] that given the
starting point and endpoint, there are only $O((2k_{0})!)$ split paths of
length $k_{0}$. Therefore, there are at most $O((2k_{0})!(\frac{P}{\log
P})^{4})$ split paths of length $k_{0}+2$ with starting point
$(x,\vec{\alpha})$ and endpoint $(y,\vec{\beta})$. As a consequence,

$|\mathcal{J}^{\\#}|\gg|\mathcal{R}^{*}|((2k_{0})!)^{-2}(\frac{\log
P}{P})^{8}\gg\frac{X}{H^{1+\sigma}}$

if $H\geqslant\exp\bigl{(}C(\log X)^{1/2}(\log\log X)^{1/2}\bigr{)}$.

∎

##  7\. Building a global frequency and proving the main theorem

Let $(x_{0},\vec{\alpha}_{0})\in\mathcal{J}$ be the frequency in Proposition
6.4, for every $(y,\vec{\beta})\in\mathcal{J}^{\\#}$ we can build a split path
of length $2k_{0}+4$ which goes through $(y,\vec{\beta})$ and has
$(x_{0},\vec{\alpha}_{0})$ as both its starting and ending point. By adapting
the argument in Subsection 2.2, we can obtain some information on the
frequencies $\vec{\alpha}_{0}$ and $\vec{\beta}$.

######  Lemma 7.1.

Suppose that $0<\sigma<1$, $(x_{0},\vec{\alpha}_{0})\in\mathcal{J}$ and
$(y,\vec{\beta})\in\mathcal{J}^{\\#}$ are as in Proposition 6.4, and $Q_{y}$
is the product of $\gg\eta^{O(k_{0})}\frac{P^{\prime}}{\log P^{\prime}}$
primes in $[P^{\prime},2P^{\prime}]$. Then for every degree $1\leqslant
j\leqslant d$ there is always some $T_{j,y}\in\mathbb{R}$ with
$|T_{j,y}|\ll\frac{X^{j}}{H^{j-\sigma}}$ and some constant $0<c_{j,y}\ll 1$
such that

$\alpha_{0}^{(j)}\equiv\frac{a^{(j)}_{y}}{q^{(j)}_{y}}\cdot
Q_{y}+\frac{T_{j,y}}{y^{j}}+O\bigl{(}H^{-j+\sigma}\bigr{)}\,(\mathrm{mod}\,Q_{y})$

and

$\beta^{(j)}\equiv\frac{b^{(j)}_{y}}{q^{(j)}_{y}}\cdot
Q_{y}+\frac{c_{j,y}T_{j,y}}{y^{j}}+O\bigl{(}H^{-j+\sigma}\bigr{)}\,(\mathrm{mod}\,Q_{y}),$

where $a^{(j)}_{y},b^{(j)}_{y},q^{(j)}_{y}\ll H^{\sigma/d}$ are integers.

###### Proof.

In the proof we let $k=k_{0}+2$, where $k_{0}$ is the integer defined in
Proposition 6.4. As mentioned earlier, given an element
$(y,\vec{\beta})\in\mathcal{J}_{0}$, one can construct, from Proposition 6.4,
a split path modulo $Q_{y}$ of length $2k$ with both starting and ending
points being $(x_{0},\vec{\alpha}_{0})$. Suppose that this path is associated
with the prime tuple
$(p_{1},\dots,p_{k},q_{k}^{\prime},\dots,q_{1}^{\prime};q_{1},\dots,q_{k},p_{k}^{\prime},\dots,p_{1}^{\prime})$,
the path is as the figure below.

$\textstyle{{\vec{\alpha}_{0}=\vec{\alpha}_{1,1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{q_{1}}$$\textstyle{\vec{\alpha}_{1,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{q_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{k}}$$\scriptstyle{q_{k}}$$\textstyle{{\vec{\alpha}_{1,k+1}=\vec{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{q^{\prime}_{k}}}$$\scriptstyle{{p^{\prime}_{k}}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{q^{\prime}_{1}}}$$\scriptstyle{{p^{\prime}_{1}}}$$\textstyle{{\vec{\alpha}_{1,2k+1}=\vec{\alpha}_{0}}}$

Since $Q_{y}$ is the product of primes in the interval
$[P^{\prime},2P^{\prime}]$, it is evident from the choices of $P$ and
$P^{\prime}$ in Proposition 5.3 that $(p,Q_{y})=1$ whenever $p\in[P,2P]$.
Moreover, [16, Lemma 4.1] demonstrates that $\prod_{i\leqslant
m_{1}}\frac{p_{i}}{q_{i}}\prod_{i\leqslant
m_{2}}\frac{p_{i}^{\prime}}{q_{i}^{\prime}}\asymp 1$ for $1\leqslant
m_{1},m_{2}\leqslant k$. Therefore, the conditions of Lemma 2.7 are satisfied.

Setting $\varepsilon=\eta^{-O(1)}\frac{P}{H}$ in Lemma 2.7 and assuming that
$\vec{\alpha}_{y}$ is the top element, the application of Lemma 2.7 with
$i^{\prime}=0$ and $i^{\prime}=2k$, along with the triangle inequality, shows
that

$\prod_{i\leqslant
k}(p_{i}q^{\prime}_{i})^{j}\alpha_{y}^{(j)}\equiv\prod_{i\leqslant
k}(p^{\prime}_{i}q_{i})^{j}\alpha_{y}^{(j)}+O(k\eta^{-O(1)}(P/H)^{j})\,(\mathrm{mod}\,Q_{y}).$

Now we set $T_{j,y}\in\mathbb{R}$ as the number that satisfies

$\Bigl{(}\prod_{i\leqslant k}(p_{i}q_{i}^{\prime})^{j}-\prod_{i\leqslant
k}(p_{i}^{\prime}q_{i})^{j}\Bigr{)}\alpha_{y}^{(j)}\equiv\frac{\Bigl{(}\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}-\prod_{i\leqslant
k}(p_{i}^{\prime}q_{i})^{j}\Bigr{)}T_{j,y}}{y^{j}\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}}\,(\mathrm{mod}\,Q_{y})$

and

$\Bigl{|}\frac{\Bigl{(}\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}-\prod_{i\leqslant
k}(p_{i}^{\prime}q_{i})^{j}\Bigr{)}T_{j,y}}{y^{j}\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}}\Bigr{|}=\Bigl{|}\frac{T_{j,y}}{y^{j}}\bigl{(}1-\prod_{i\leqslant
k}(p_{i}/p_{i}^{\prime})^{j}(q_{i}^{\prime}/q_{i})^{j}\bigr{)}\Bigr{|}\ll
k\eta^{-O(1)}(P/H)^{j}.$

It follows from the assumption of Proposition 6.4 that
$p_{i},p_{i}^{\prime},q_{i},q_{i}^{\prime}(1\leqslant i\leqslant k)$ are
distinct primes, and thus, $\prod_{i\leqslant
k}p_{i}q_{i}^{\prime}/(p_{i}^{\prime}q_{i})\neq 1$; on the other hand, one can
also deduce from [16, Lemma 4.1] and the setting of our $k$ that
$\prod_{i\leqslant k}p_{i}q_{i}^{\prime}/(p_{i}^{\prime}q_{i})\ll 1$.
Therefore, one has

$|T_{j,y}|\ll X^{j}k\eta^{-O(1)}(P/H)^{j}\ll X^{j}/H^{j-\sigma}$

since $P\ll H^{\omega}\ll H^{\frac{\sigma}{dk}}$ in light of $\omega$ is
sufficiently small concerning $\sigma$. Dividing $\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}-\prod_{i\leqslant k}(p_{i}^{\prime}q_{i})^{j}$
both sides from the above congruence equation, one has

$\displaystyle\alpha_{y}^{(j)}\equiv\frac{c_{y}^{(j)}}{q_{y}^{(j)}}\cdot
Q_{y}+\frac{T_{j,y}}{y^{j}\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}}\,(\mathrm{mod}\,Q_{y})$ (7.1)

where $1\leqslant c_{y}^{(j)}\leqslant q_{y}^{(j)}$ are integers and
$q_{y}^{(j)}|(\prod_{i\leqslant k}(p_{i}q^{\prime}_{i})^{j}-\prod_{i\leqslant
k}(p^{\prime}_{i}q_{i})^{j})\ll H^{\sigma/d}$ since
$p_{i},p_{i}^{\prime},q_{i},q^{\prime}_{i}\ll H^{\omega}\ll H^{\sigma/2kd}$
from Proposition 5.3.

Meanwhile, Lemma 2.7 also suggests that there is some integer $1\leqslant
a^{(j)}_{y}\leqslant q^{(j)}_{y}$ such that

$\alpha_{0}^{(j)}\equiv\prod_{i\leqslant
k}(p_{i}q^{\prime}_{i})^{j}\alpha_{y}^{(j)}+O(k\eta^{-O(1)}(P/H)^{j})\equiv\frac{a_{y}^{(j)}}{q_{y}^{(j)}}\cdot
Q_{y}+\frac{T_{j,y}}{y^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,Q_{y}).$

Moreover, on combining (7.1) with Lemma 2.7 one also has

$\beta^{(j)}\equiv\prod_{i\leqslant
k}(q_{i}q_{i}^{\prime})^{j}\alpha_{y}^{(j)}+O(k\eta^{-O(1)}(P/H)^{j})\equiv\frac{b_{y}^{(j)}}{q^{(j)}_{y}}\cdot
Q_{y}+\frac{T_{j,y}}{y^{j}}\frac{\prod_{i\leqslant
k}(q_{i}q_{i}^{\prime})^{j}}{\prod_{i\leqslant
k}(p_{i}q_{i}^{\prime})^{j}}+O(H^{-j+\sigma})\\\
\equiv\frac{b_{y}^{(j)}}{q_{y}^{(j)}}\cdot
Q_{y}+\frac{c_{j,y}T_{j,y}}{y^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,Q_{y}),$

for some constant $0<c_{j,y}\ll 1$, as one has $\prod_{i\leqslant
k}p_{i}\asymp\prod_{i\leqslant k}q_{i}$ by employing [16, Lemma 4.1]. This
completes the proof.

∎

_Proof of Theorem 1.1_.

Since for $X^{1/3-\varepsilon}\leqslant H\leqslant X$ the claim follows
immediately from [5, Theorem 1.1], let’s suppose that $\exp\bigl{(}C(\log
X)^{1/2}(\log\log X)^{1/2}\bigr{)}\leqslant H\leqslant X^{1/3-\varepsilon}$ in
the following. Assume

$\int_{X}^{2X}\sup_{\vec{\alpha}\in\mathbb{T}^{d}}\Bigl{|}\sum_{x<n\leqslant
x+H}(\tau_{k}(n)-\tau^{*}_{k}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\alpha^{(j)}(n-x)^{j}\bigr{)}\Bigr{|}\,\,\mathrm{d}x\geqslant\eta
XH\log^{k-1}X,$

it follows from the second conclusion of Proposition 5.3 that there is a
$(c\eta^{2},H)$-configuration $\mathcal{J}\subseteq[X,2X]\times\mathbb{T}$
such that when $(y,\vec{\beta}_{y})\in\mathcal{J}$ the correlation below
holds:

$\displaystyle\Bigl{|}\sum_{y<n\leqslant
y+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}\beta_{y}^{(j)}(n-y)^{j}\bigr{)}\Bigr{|}\gg\eta H\log^{k-1}X.$ (7.2)

Proposition 6.4 then gives us a set $\mathcal{J}^{\\#}\subseteq\mathcal{J}$
with $|\mathcal{J}^{\\#}|\gg\frac{|\mathcal{J}|}{H^{\sigma}}$ such that for
every $(y,\vec{\beta}_{y})\in\mathcal{J}^{\\#}$, there exists a number $Q_{y}$
that is the product of $\gg\eta^{O(k_{0})}P^{\prime}\log P^{\prime}$ primes
$p^{\prime}\in[P^{\prime},2P^{\prime}]$, and $(x_{0},\vec{\alpha}_{0})$ and
$(y,\vec{\beta}_{y})$ are connected by a split path modulo $Q_{y}$. Applying
the pigeonhole principle, there exists $\tilde{Q}\geqslant P^{\prime
c\eta^{O(k_{0})}P^{\prime}/\log
P^{\prime}}\geqslant\exp\bigl{(}c\eta^{O(k_{0})}P^{\prime}\bigr{)}$ such that
for $\gg\eta^{O(k_{0})}|\mathcal{J}^{\\#}|\gg H^{-\sigma}|\mathcal{J}^{\\#}|$
elements $(y,\vec{\beta}_{y})\in\mathcal{J}^{\\#}$, each of them can be
connected with $(x_{0},\vec{\alpha}_{0})$ by a split path modulo $\tilde{Q}$.
This implies that there exists $(y_{0},\vec{\beta}_{0})\in\mathcal{J}^{\\#}$
such that

$\tilde{Q}_{y}=\gcd(Q_{y_{0}},Q_{y})\geqslant\tilde{Q}\geqslant\exp\bigl{(}c\eta^{O(k_{0})}P^{\prime}\bigr{)}$

holds for $\gg H^{-\sigma}|\mathcal{J}^{\\#}|$ of
$(y,\vec{\beta}_{y})\in\mathcal{J}^{\\#}$. Let $T_{j}=T_{j,y_{0}}$, the
application of Lemma 7.1 with $(y_{0},\vec{\beta}_{0})$ and
$(y,\vec{\beta}_{y})$, respectively, shows that

$\alpha_{0}^{(j)}\equiv\frac{a^{(j)}_{y}}{q^{(j)}_{y}}\cdot
Q_{y}+\frac{T_{j,y}}{y^{j}}+O(H^{-j+\sigma})\equiv\frac{a^{(j)}_{y_{0}}}{q^{(j)}_{y_{0}}}\cdot
Q_{y_{0}}+\frac{T_{j}}{y_{0}^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,\tilde{Q_{y}}),$

which means that $\frac{a^{(j)}_{y}}{q^{(j)}_{y}}\cdot
Q_{y}\equiv\frac{a^{(j)}_{y_{0}}}{q^{(j)}_{y_{0}}}\cdot
Q_{y_{0}}\,(\mathrm{mod}\,\tilde{Q_{y}})$ and
$\bigl{|}\frac{T_{j,y}}{y^{j}}-\frac{T_{j}}{y_{0}^{j}}\bigr{|}\ll
H^{-j+\sigma}$. Since both $y$ and $y_{0}$ are in the interval $[X/(2P),2X/P]$
we may set $y/c_{y}=y_{0}$ with $1/2<c_{y}<2$, then

$\bigl{|}\frac{T_{j,y}}{y^{j}}-\frac{c_{y}^{j}T_{j}}{y^{j}}\bigr{|}\ll
H^{-j+\sigma}.$

Therefore, one can deduce from the above inequality and Lemma 7.1 that there
are $\gg\frac{X}{H^{1+2\sigma}}$ of $(y,\vec{\beta}_{y})\in\mathcal{J}^{\\#}$
such that for each $y$ there is some constant $0<c_{y}\ll 1$ such that

$\beta_{y}^{(j)}\equiv\frac{b^{(j)}_{y}}{q_{y}^{(j)}}\cdot
Q_{y}+\frac{c_{j,y}T_{j,y}}{y^{j}}+O(H^{-j+\sigma})\equiv\frac{b^{(j)}_{y}}{q_{y}^{(j)}}\cdot
Q_{y}+\frac{c_{j,y}c^{j}_{y}T_{j}}{y^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,\tilde{Q}_{y}).$

As there are only finite number of such $c_{j,y}c^{j}_{y}$, applying the
pigeonhole principle allows us to assume that there is one such $c>0$ and
$\gg\frac{X}{H^{1+2\sigma}}$ elements $(y,\vec{\beta}_{y})\in\mathcal{J}$ such
that

$\beta_{y}^{(j)}\equiv\frac{b^{(j)}_{y}}{q_{y}^{(j)}}\cdot
Q_{y}+\frac{cT_{j}}{y^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,\tilde{Q}_{y})$

for some $1\leqslant b_{y}^{(j)}\leqslant q_{y}^{(j)}$. We then rename
$cT_{j}$ as $T_{j}$. As it follows from Lemma 7.1 that for every
$(y,\vec{\beta}_{y})\in\mathcal{J}^{\\#}$ we have
$q_{y}^{(d)},\dots,q_{y}^{(1)}\ll H^{\sigma/d}$, the pigeonhole principle
allows us to find a $k$-tuple $q^{(d)},\dots,q^{(1)}\ll H^{\sigma/d}$ such
that $q_{y}^{(j)}=q^{(j)}$ for $\gg\frac{X}{H^{1+3\sigma}}$ of such
$(y,\vec{\beta}_{y})$. [8, Proposition 3.13] further lowers these popular
denominators $q^{(j)}$ to $O(1)$. Let’s take $q$ as the product of
$q^{(1)},\dots,q^{(d)}$, then we still have $q\ll 1$. Therefore, one may
conclude from the above analysis along with inequality (7.2) that there are
integers $1\leqslant b^{(j)}\leqslant q$ such that for
$\gg\eta^{O(1)}\frac{X}{H^{1+3\sigma}}$ of elements
$(y,\vec{\beta}_{y})\in[X,2X]\times\mathbb{T}^{d}$ satisfying $y$ are
$H$-separated and

$\Bigl{|}\sum_{y<n\leqslant
y+H}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\bigl{(}\sum_{1\leqslant j\leqslant
d}(n-y)^{j}\beta_{y}^{(j)}\bigr{)}\Bigr{|}\gg\eta H\log^{k-1}X,$

where $\beta_{y}^{(j)}\equiv\frac{b^{(j)}}{q}\cdot
Q_{y}+\frac{T_{j}}{y^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,\tilde{Q}_{y})$.

By decomposing the interval $[y,y+H]$ into disjoint subintervals of length
$\Omega(H^{1-3\sigma})$, the pigeonhole principle gives us an integer
$0\leqslant z_{y}-y\leqslant H$ such that for
$\gg\eta^{O(1)}\frac{X}{H^{1+3\sigma}}$ of elements
$(y,\vec{\beta}_{y})\in[X,2X]\times\mathbb{T}^{d}$ the following inequality
holds:

$\eta H^{1-3\sigma}\log^{k-1}X\ll\Bigl{|}\sum_{z_{y}<n\leqslant
z_{y}+H^{1-3\sigma}}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\Bigl{(}\sum_{1\leqslant
j\leqslant d}(n-y)^{j}\beta_{y}^{(j)}\Bigr{)}\Bigr{|};$

besides, by noting that $|T_{j}/z_{y}^{j}-T_{j}/y^{j}|\ll H^{-j+\sigma}$ we
can also rewrite $\beta_{y}^{(j)}$ as

$\beta_{y}^{(j)}\equiv\frac{b^{(j)}}{q}\cdot
Q_{y}+\frac{T_{j}}{z_{y}^{j}}+O(H^{-j+\sigma})\,(\mathrm{mod}\,\tilde{Q}_{y}).$

If we fix a degree $1\leqslant j\leqslant d$ for a moment and consider the
phase function
$e\bigl{(}\left\\{(n-y)^{j}-(z_{y}-y)^{j}\right\\}\beta_{y}^{(j)}\bigr{)}$. By
Taylor expansion and noting that $|n-z_{y}|\leqslant H^{1-3\sigma}$ and
$|z_{y}-y|\leqslant H$, this phase function equals to

$e\Bigl{(}\left\\{(n-y)^{j}-(z_{y}-y)^{j}\right\\}\bigl{(}\frac{b_{j}}{q}+\frac{T_{j}}{z_{y}^{j}}\bigr{)}\Bigr{)}e\Bigl{(}O\bigl{(}H^{-j+\sigma}\sum_{1\leqslant
i\leqslant j}(n-z_{y})^{i}(z_{y}-y)^{j-i}\bigr{)}\Bigr{)}\\\
=e\Bigl{(}\left\\{(n-y)^{j}-(z_{y}-y)^{j}\right\\}\bigl{(}\frac{b_{j}}{q}+\frac{T_{j}}{z_{y}^{j}}\bigr{)}\Bigr{)}\bigl{(}1+O(H^{-\sigma})\bigr{)},$

where $b_{j}$ is an integer satisfying $b_{j}\equiv
b^{(j)}Q_{y}\,(\mathrm{mod}\,q)$. Therefore,

$\eta H^{1-3\sigma}\log^{k-1}X\ll\Bigl{|}\sum_{z_{y}<n\leqslant
z_{y}+H^{1-3\sigma}}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))\prod_{1\leqslant
j\leqslant
d}e\Bigl{(}\frac{b_{j}(n-y)^{j}}{q}+\frac{T_{j}}{z_{y}^{j}}(n-y)^{j}\Bigr{)}\Bigr{|}\\\
+H^{-\sigma}\sum_{z_{y}<n\leqslant
z_{y}+H^{1-3\sigma}}(\tau_{k}(n)+\tau_{k}^{*}(n))\\\
\ll\sum_{a\,(\mathrm{mod}\,q)}\biggl{|}\sum_{\begin{subarray}{c}z_{y}<n\leqslant
z_{y}+H^{1-3\sigma}\\\ n\equiv
a\,(\mathrm{mod}\,q)\end{subarray}}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\Bigl{(}\sum_{1\leqslant
j\leqslant d}\frac{T_{j}}{z_{y}^{j}}(n-y)^{j}\Bigr{)}\biggr{|}\\\
+H^{-\sigma}\sum_{z_{y}<n\leqslant
z_{y}+H^{1-3\sigma}}(\tau_{k}(n)+\tau_{k}^{*}(n))$

for $\gg\eta^{O(1)}\frac{X}{H^{1+3\sigma}}$ of $H$-separated $z_{y}\in[X,2X]$.
If we claim that there are at most $O(\frac{X}{H^{1+3.1\sigma}})$ of
$H$-separated $z_{y}\in[X,2X]$ such that

$\displaystyle\sum_{z_{y}<n\leqslant
z_{y}+H^{1-3\sigma}}(\tau_{k}(n)+\tau_{k}^{*}(n))\gg
H^{1-3\sigma}\log^{k-1}X,$ (7.3)

recalling that $q\ll 1$, it follows from the pigeonhole principle that there
are at least $\gg\eta^{O(1)}\frac{X}{H^{1+3\sigma}}$ of $H$-separated
$z_{y}\in[X,2X]$ and an integer $1\leqslant a\leqslant q$ such that

$\Bigl{|}\sum_{z_{y}<n\leqslant z_{y}+H^{1-3\sigma}\atop n\equiv
a\,(\mathrm{mod}\,q)}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\Bigl{(}\sum_{1\leqslant
j\leqslant d}\frac{T_{j}}{z_{y}^{j}}(n-y)^{j}\Bigr{)}\Bigr{|}\gg\eta
H^{1-3\sigma}\log^{k-1}X.$

In light of the claim (7.3) above, along with the fact that $|z_{y}-y|\ll H$
and $|z_{y}-n|\ll H^{1-3\sigma}$ whenever $n\in[z_{y},z_{y}+H^{1-3\sigma}]$,
we may further assume that there are at least
$\gg\eta^{O(1)}\frac{X}{H^{1+3\sigma}}$ of $H$-separated $z_{y}\in[X,2X]$ and
an integer $1\leqslant a\leqslant q$ such that

$\Bigl{|}\sum_{z_{y}<n\leqslant z_{y}+H^{1-3\sigma}\atop n\equiv
a\,(\mathrm{mod}\,q)}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))e\Bigl{(}\sum_{1\leqslant
j\leqslant d}\frac{T_{j}}{z_{y}^{j}}(n-z_{y})^{j}\Bigr{)}\Bigr{|}\gg\eta
H^{1-3\sigma}\log^{k-1}X.$

By representing the $d$-th polynomial $\frac{T_{d}}{z_{y}^{d}}(n-z_{y})^{d}$
in the form

$\frac{(-1)^{d-1}\bigl{(}(-1)^{d-1}dT_{d}\bigr{)}}{dz_{y}^{d}}(n-z_{y})^{d}+(1-1)\sum_{1\leqslant
i\leqslant
d-1}\frac{(-1)^{i-1}\bigl{(}(-1)^{d-1}dT_{d}\bigr{)}}{iz_{y}^{i}}(n-z_{y})^{i}\\\
=(-1)^{d-1}dT_{d}(\log n-\log{z_{y}})-\sum_{1\leqslant i\leqslant
d-1}\frac{(-1)^{i-1}\bigl{(}(-1)^{d-1}dT_{d}\bigr{)}}{iz_{y}^{i}}(n-z_{y})^{i}+O(H^{-\sigma}),$

the second inequality follows from Taylor expansion and the assumption
$|T_{d}|\ll X^{d}/H^{d-\sigma}$ when $|n-z_{y}|\leqslant H^{1-3\sigma}$.
Inspired by this expansion, one can deduce that

$\sum_{1\leqslant j\leqslant
d}\frac{T_{j}}{z_{y}^{j}}(n-z_{y})^{j}=\sum_{1\leqslant j\leqslant
d}\sum_{1\leqslant i\leqslant
j}\frac{(-1)^{i-1}(j(-1)^{j-1}T_{j}-(j+1)(-1)^{j}T_{j+1})}{iz_{y}^{i}}(n-z_{y})^{i}\\\
=\sum_{1\leqslant j\leqslant
d}\bigl{(}j(-1)^{j-1}T_{j}-(j+1)(-1)^{j}T_{j+1}\bigr{)}(\log
n-\log{z_{y}})+O(H^{-\sigma}),$

with the convenience of setting $T_{d+1}=0$ and the second equation follows
from Taylor expansion and the assumption $|T_{j}+T_{j+1}|\ll
X^{j+1}/H^{j+1-\sigma}$. Therefore, considering the claim (7.3), we may assume
there is some $T^{*}$ with $|T^{*}|\leqslant X^{d+1}$ and
$\gg\eta^{O(1)}\frac{X}{H^{1+3\sigma}}$ of $H$-separated $y\in[X,2X]$ and an
integer $1\leqslant a\leqslant q$ such that

$\Bigl{|}\sum_{z_{y}<n\leqslant z_{y}+H^{1-3\sigma}\atop n\equiv
a\,(\mathrm{mod}\,q)}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))n^{iT^{*}}\Bigr{|}\gg\eta
H^{1-3\sigma}\log^{k-1}X.$

Taking $h=H^{1-3\sigma}$ in the following, we now fix a value $z_{y}\in(X,2X]$
such that the above inequality holds. Suppose that
$I\subseteq[z_{y}-h^{1-\sigma},z_{y}+h+h^{1-\sigma}]$ is an interval
satisfying $|I|=h$ and $|I\bigtriangleup(z_{y},z_{y}+h]|\ll\eta^{2}h$. It
follows from the triangle inequality and Lemma 4.2 (3) with $h_{2}=\eta^{2}h$,
$z_{1,x}^{(2)}=z_{y}$ and $z_{2,x}^{(2)}=z_{y}+h$ that

$\Bigl{|}\sum_{\begin{subarray}{c}n\in I\\\ n\equiv
a\,(\mathrm{mod}\,q)\end{subarray}}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))n^{iT^{*}}\Bigr{|}\geqslant\bigl{|}\sum_{z_{y}<n\leqslant
z_{y}+h\atop n\equiv
a\,(\mathrm{mod}\,q)}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))n^{iT^{*}}\bigr{|}\\\
-O\Bigl{(}\sum_{|n-z_{y}|\leqslant\eta^{2}h}\tau_{k}(n)\Bigr{)}-)O\Bigl{(}\sum_{|n-z_{y}-h|\leqslant\eta^{2}h}\tau_{k}(n)\Bigr{)}\gg\eta
h\log^{k-1}X.$

Due to for each $z_{y}$, there would be at least $\gg\eta^{2}h$ such intervals
$I$ produced, we can conclude that there are $\gg\eta^{O(1)}X/h^{6\sigma}$
integers $x\in(X,2X]$ such that

$\Bigl{|}\sum_{x<n\leqslant x+h\atop n\equiv
a\,(\mathrm{mod}\,q)}(1_{S}(n)\tau_{k}(n)-\tau_{k}^{*}(n))n^{iT^{*}}\Bigr{|}\gg\eta
h\log^{k-1}X,$

and this contradicts Proposition 3.1.

Now, if we assume that the claim (7.3) fails and translate the interval
$[z_{y},z_{y}+H^{1-3\sigma}]$ by any shift of size at most
$O(H^{1-3.1\sigma})$ as done above, then we can conclude that there are
$\gg\frac{X}{H^{\sigma}}$ integers $x\in[X,2X]$ such that

$\sum_{x<n\leqslant x+H^{1-3\sigma}}(\tau_{k}(n)+\tau_{k}^{*}(n))\gg
H^{1-3\sigma}\log^{k-1}X.$

However, this contradicts with Lemma 2.1 and Propositions 3.2-3.3 by noting
that, supposing $h_{1}=X^{1-\frac{1}{100k}}$, for all but
$O(\frac{X}{H^{\sigma}})$ of $x\in[X,2X]$ we have

$(H^{1-3\sigma})^{-1}\sum_{x<n\leqslant
x+H^{1-3\sigma}}(\tau_{k}(n)+\tau_{k}^{*}(n))\ll h_{1}^{-1}\sum_{x<n\leqslant
x+h_{1}}\tau_{k}(n)\ll\log^{k-1}X.$

This completes the proof of Theorem 1.1. ∎

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