# Exponential sums over Möbius convolutions with applications to partitions

Debmalya Basak  Debmalya Basak: Department of Mathematics, University of
Illinois, 1409 West Green Street, Urbana, IL 61801, USA<EMAIL_ADDRESS>,  Nicolas Robles  Nicolas Robles: RAND Corporation, Engineering and Applied
Sciences, 1200 S Hayes, Arlington, VA 22202, USA<EMAIL_ADDRESS>and
Alexandru Zaharescu  Alexandru Zaharescu: Department of Mathematics,
University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA; and
Institute of Mathematics of the Romanian Academy, P.O. BOX 1-764, Bucharest,
Ro-70700, Romania<EMAIL_ADDRESS>
###### Abstract.

We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the
generating functions
$\sum_{n=1}^{\infty}p_{w}(n)z^{n}=\prod_{m\in\mathbb{N}}(1-z^{m})^{-w(m)}$,
where the weights $w(m)$ are Möbius convolutions. We establish an upper bound
for $p_{w}(n)$ and, as a consequence, we obtain an asymptotic formula
involving the number of odd and even partitions emerging from the weights. In
order to achieve the desired bounds on the minor arcs resulting from the
Hardy-Littlewood circle method, we establish bounds on exponential sums
twisted by Möbius convolutions. In the process, we improve upon a classical
result involving the bound of $\sum_{n\leqslant
X}\mu(n)\operatorname{e}(n\alpha)$. Lastly, we provide an explicit formula
relating the contributions from the major arcs with a sum over the zeros of
the Riemann zeta-function.

###### 2020 Mathematics Subject Classification:

Primary: 11L07, 11P55. Secondary: 11L03, 11P82.
Keywords and phrases: exponential sums, Möbius convolutions, Hardy-Littlewood
circle method, weighted partitions, Riemann zeta-function.

##  1\. Introduction

###  1.1. Motivation

A partition of a positive integer $n$ is a non-decreasing sequence of positive
integers whose sum is equal to $n$. The partition function $p(n)$, which
represents the number of partitions of $n$, was first studied by Hardy and
Ramanujan [14] in 1918 by applying the Hardy-Littlewood circle method. Their
result states that

$\displaystyle
p(n)\sim\frac{1}{4n\sqrt{3}}\exp\bigg{(}\pi\sqrt{\frac{2n}{3}}\bigg{)}\quad\textnormal{as}\quad
n\to\infty.$

We consider a question about partitions of a positive integer, with certain
additional arithmetic flavors. Let $B$ (blue) and $R$ (red) denote the set of
squarefree numbers that have an even, and respectively, an odd number of prime
factors. For the convenience of the reader, we use colors to show parity of
these partitions. Thus, we write

$\displaystyle B$
$\displaystyle=\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1,6,10,14,15,21,22,26,\dots}\\},$
$\displaystyle R$
$\displaystyle=\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2,3,5,7,11,13,17,19,23,29,30,\dots}\\}.$

We note in passing that the cardinalities of the sets $B$ and $R$ are
asymptotically just as numerous. Actually, it is expected that as $x$ tends to
infinity, the number of elements of $B$ less than $x$ differs from the number
of elements of $R$ less than $x$ by at most $O_{\epsilon}(x^{1/2+\epsilon})$
for any $\epsilon>0$. As is well-known, this is equivalent to the Riemann
hypothesis.

######  Definition 1.1.

We call a partition of a positive integer $n$ admissible if each part is
either blue or red and each red part appears at most once. We say that an
admissible partition is even (respectively odd) if the number of red parts
used in the partitions is even (and respectively odd).

For example, there are exactly 15 admissible partitions in the case when
$n=10$. The partition

$\displaystyle
10={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}$

is not admissible because the red 2 appears twice. We see in Table 1.1 that
there are $8$ even admissible partitions and $7$ odd admissible partitions in
this situation. This naturally leads to the following question.

| Case for $n=10\textrm{ }$ | Admissible partitions
---|---|---
|  Odd admissible partitions |  $\begin{aligned} 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}7}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}5}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}5}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\end{aligned}$
|  |
|  Even admissible partitions |  $\begin{aligned} \quad\;\;10&={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}5}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}5}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}7}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\ 10&={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}7}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}\\\ 10&={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}10}\\\ \end{aligned}$
|  |

Question: Is the number of even admissible partitions of $n$ asymptotic to the
number of odd admissible partitions of $n$ as $n$ tends to infinity?

In this paper, our primary objective is to answer this question. In order to
do so, we will rewrite the question in terms of the Möbius function. In fact,
we will also establish results for a more general version of this problem,
where the Möbius function is replaced by convolutions of the Möbius function.
In the process, we are naturally led to prove results on exponential sums
involving Möbius convolutions, which is the other main goal of this article
and it is of independent interest.

###  1.2. Survey of previous results

In [31], Vaughan established an asymptotic formula for the number of
partitions of a number $n$ into primes when $n$ is large, by applying the
Hardy-Littlewood circle method. In [32], Vaughan then extended this approach
to study partition asymptotics regarding squares and other additive questions.

Since the publication of [31, 32], there has been considerable interest in the
study of restricted or weighted partitions over arithmetically interesting
sets. In [12], Gafni showed an asymptotic formula for the number of partitions
of $n$ into $k$-th powers, and in [13], she proved results for partitions into
prime powers. Berndt, Malik and the third author [2] studied partitions into
$k$-th powers in arithmetic progressions. Dunn and the second author [11] then
considered the more general case where the parts are values of an arbitrary
polynomial $f$ of degree $d\geq 2$. More recently, Das, Berndt, Zeindler, and
the second and third authors [3, 9, 27] studied partitions weighted by
generalized divisor functions, partitions into semiprimes and almost primes.
General strategies for dealing with asymptotic weighted partitions have been
established by Meinardus [21] and more recently by Debryune and Tenenbaum
[10]. Notice that in the case of the above results, the corresponding
partitions have non-negative weights, unlike the question discussed in Section
1.1. For such questions, where the weights take positive and negative values
(see Figure 1.2), one should be prepared in principle to have a main result
which is an upper bound rather than an asymptotic formula. This indeed happens
in our case, see Theorem 1.2 below.

To proceed, we start by considering the partition function $p_{w}(n)$
associated to a generating function of the form,

(1.1)
$\displaystyle\Psi_{w}(z)=\sum_{n=1}^{\infty}p_{w}(n)z^{n}=\prod_{n\in\mathbb{N}}(1-z^{n})^{-w(n)},\quad\lvert
z\rvert<1,$

where $w(n)\in\mathbb{Z}$ is the weight placed on each element
$n\in\mathbb{N}$. We refer to such partitions as weighted partitions. Consider
the Möbius function denoted by $\mu$, which is a multiplicative function
defined in the following way: for each prime $p$,

$\mu(p^{k}):=\begin{cases}-1,&k=1\\\ 0,&k\geqslant 2.\end{cases}$

By definition, $\mu^{2}(n)=1$ if $n$ is squarefree and $\mu^{2}(n)=0$
otherwise. Our cases of interest will have weights given by the Möbius
function, i.e., $w(n)=\mu(n)$, or more generally by convolutions of the Möbius
function, i.e., $w(n)=(\mu*\mu*\cdots*\mu)(n)$, where the Dirichlet
convolution is performed $k$ times.

It is easy to see that in the particular case when $w(n)=\mu(n)$, the
partition function $p_{\mu}(n)$ is exactly the difference between the number
of even admissible partitions and odd admissible partitions of $n$ as defined
in Section 1.1. More precisely, suppose we denote by $\mathcal{A}(n)$ the
number of admissible partitions of $n$. Additionally, let $\mathcal{E}(n)$ and
$\mathcal{O}(n)$ be the number of even and odd admissible partitions of $n$
respectively. Then

(1.2) $\displaystyle p_{\mu}(n)$
$\displaystyle=\mathcal{E}(n)-\mathcal{O}(n),$ (1.3)
$\displaystyle\textrm{and}\quad\mathcal{A}(n)$
$\displaystyle=\mathcal{E}(n)+\mathcal{O}(n).$

Therefore, in order to show that the number of even admissible partitions is
asymptotic to the number of odd admissible partitions as $n$ tends to
infinity, our goal in what follows is to provide an upper bound for their
difference, i.e., for $p_{\mu}(n)$.

Very recently, in [5, 6], Daniels investigated partitions whose generating
functions are given by

(1.4)
$\displaystyle\Psi(z,f)=1+\sum_{n=1}p(n,f)z^{n}=\prod_{n=1}^{\infty}(1-f(n)z^{n})^{-1},\quad\lvert
z\rvert<1.$

Daniels refers to such partitions as signed partitions. In particular, he
studied the case when $f(n)=\mu(n)$. Note the difference between placing
$\mu(n)$ in the exponent of the factor $(1-z^{n})^{-\mu(n)}$ in (1.1) as
opposed to the presence of $\mu(n)$ inside the factor $(1-\mu(n)z^{n})^{-1}$
in (1.4). This is a subtle difference, which surprisingly leads to the
existence of main terms in Daniel’s works and the non-existence of main terms
in our context. For comparison, the reader is referred to Theorem 1.1 in [5]
and Theorem 1.2 below. We remark that Theorem 1.2 below does lead to an
asymptotic formula connecting the number of odd and even admissible partitions
from Section 1.1.

###  1.3. Main Results

We start by answering the question raised in Section 1.1.

######  Theorem 1.1.

The number of odd admissible partitions of $n$ is asymptotic to the number of
even admissible partitions of $n$, as $n$ tends to infinity.

More generally, for any positive integer $k$, we define the Popovici-Möbius
function (see [28, $\mathsection$2.2] as well as [4] and [19, 26] for related
applications) with parameter $k$ by

(1.5) $\displaystyle\mu_{k}(n):=(\mu*\mu*\cdots*\mu)(n),$

where the Dirichlet convolution is performed $k$ times. The Dirichlet series
for $\mu_{k}(n)$ is given by

$\sum_{n=1}^{\infty}\frac{\mu_{k}(n)}{n^{s}}=\frac{1}{\zeta(s)^{k}},$

for $\operatorname{Re}(s)>1$. Let $p_{\mu_{k}}(n)$ be the number of partitions
of $n$ weighted by $\mu_{k}(n)$, that is,

$\displaystyle\sum_{n=1}^{\infty}p_{\mu_{k}}(n)z^{n}=\prod_{n=1}^{\infty}(1-z^{n})^{-\mu_{k}(n)}.$

In the present paper, we will establish results on exponential sums twisted by
Möbius convolutions for any positive integer $k$. Concerning applications to
partitions, for simplicity, we will focus only on the case when $k=2$. This is
more involved than the case when $k=1$, and already demonstrates all the
ingredients necessary for a general $k$. For $k=2$, we prove the following
result.

######  Theorem 1.2.

For any fixed $B>0$ and for all $n\geqslant 2$, we have

(1.6) $\displaystyle\log p_{\mu*\mu}(n)=O_{B}\left(\frac{\sqrt{n}}{(\log
n)^{B}}\right).$

######  Remark 1.1.

As we shall see in the proof of Theorem 1.2, the method can be generalized to
any $k\geqslant 1$, with the same upper bound as in (1.6). In particular, when
$k=1$, we have

(1.7) $\displaystyle\log p_{\mu}(n)=O_{B}\left(\frac{\sqrt{n}}{(\log
n)^{B}}\right),$

for any fixed $B>0$ and all $n\geqslant 2$. Theorem 1.1 is an immediate
consequence of (1.2) and (1.7). See Figure 1.1 for a plot of the logarithms of
$\mathcal{A}(n)$ and $\log p_{\mu}(n)$.

######  Remark 1.2.

Results similar to Theorems 1.1 and 1.2 hold true if the Möbius function
$\mu(n)$ is replaced by the Liouville function $\lambda(n)$, where

$\lambda(n)=\prod_{i=1}^{k}(-1)^{a_{i}},\quad\textrm{where}\quad
n=\prod_{i=1}^{k}p_{i}^{a_{i}}.$

Figure 1.1. Plot of $\log\mathcal{A}(n)$ in blue and $\log{p}_{\mu}(n)$ in
orange for $1\leqslant n\leqslant 10000$.

Figure 1.2. Left: Plot of partition function $p_{w}(n)$ with $w(n)=\mu(n)$,
Right: Plot of partition function $p_{w}(n)$ with $w(n)=(\mu*\mu)(n)$.

In order to prove Theorem 1.2, we will employ the Hardy-Littlewood method. As
we shall see in Section 8, the contribution from the major arcs is closely
related to the zeros of the Riemann zeta-function $\zeta(s)$. To be precise,
following (1.1), let us write

(1.8)
$\displaystyle\Psi_{\mu*\mu}(z)=\sum_{n=1}^{\infty}p_{\mu*\mu}(n)z^{n}=\prod_{n=1}^{\infty}(1-z^{n})^{-(\mu*\mu)(n)},\quad\lvert
z\rvert<1.$

Set

(1.9)
$\displaystyle\Phi_{\mu*\mu}(z):=\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\frac{(\mu*\mu)(n)}{j}z^{jn},\quad\lvert
z\rvert<1.$

Then $\Psi_{\mu*\mu}(z)=\exp(\Phi_{\mu*\mu}(z))$. Let $X\geqslant 1$ and

$\rho=\exp(-1/X).$

Choose $z=\rho\operatorname{e}(\theta)$ in (1.9) and let

$\Delta:=\frac{1}{(1+4\pi^{2}X^{2}\theta^{2})^{1/2}}.$

Here, and in what follows, we write $\operatorname{e}(x)=\exp(2\pi ix)$. Then
we have the following theorem.

######  Theorem 1.3.

Suppose that $\theta\in\mathbb{R}$ and $X\geq 1$. If $X\Delta^{3}\geq 1$, then
there exists as sequence $T_{\nu}$ with $\nu\leqslant T_{\nu}\leqslant\nu+1$
such that

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))$
$\displaystyle=4\log\frac{X}{1-2\pi iX\theta}-8\log(2\pi)+72\frac{1-2\pi
iX\theta}{X}$ (1.10)
$\displaystyle\quad+\lim_{\nu\to\infty}\sum_{|\operatorname{Im}(\rho)|<T_{\nu}}f(X,\theta,\rho)+\sum_{n=1}^{\infty}g(X,\theta,n),$

where the functions $f$ and $g$ in (1.3) are given by

$\displaystyle f(X,\theta,\rho)$ $\displaystyle:=\bigg{(}\frac{X}{1-2\pi
iX\theta}\bigg{)}^{\rho}\frac{\zeta(1+\rho)\Gamma(\rho)}{{\zeta^{\prime}{{(\rho)}^{2}}}}\bigg{(}\log\frac{X}{1-2\pi
iX\theta}+\psi(\rho)-\frac{{\zeta^{\prime\prime}}}{{\zeta^{\prime}}}(\rho)+\frac{{\zeta^{\prime}}}{\zeta}(1+\rho)\bigg{)},$
$\displaystyle g(X,\theta,n)$ $\displaystyle:=\bigg{(}\frac{X}{1-2\pi
iX\theta}\bigg{)}^{-2n}\bigg{[}\mathfrak{c}_{1}(n)\bigg{(}\log\frac{X}{1-2\pi
iX\theta}\bigg{)}^{2}+\mathfrak{c}_{2}(n)\log\frac{X}{1-2\pi
iX\theta}+\mathfrak{c}_{3}(n)\bigg{]}.$

The coefficients $\mathfrak{c}_{1}(n),\mathfrak{c}_{2}(n)$ and
$\mathfrak{c}_{3}(n)$ are defined in (8.3), (8.4) and (8) respectively. The
sum is taken over the non-trivial zeros $\rho$ of $\zeta(s)$ under the
assumption, for notational ease, of simplicity.

######  Remark 1.3.

As Titchmarsh puts it in [29, $\mathsection$14.27]: ‘obvious modifications are
required if’ the zeros are not simple. Indeed, this assumption does not
represent a technical difficulty and it can be relaxed at the expense of
cluttering the expression $f(X,\theta,\rho)$ for the non-trivial zeros.

######  Remark 1.4.

Interestingly, (1.3) is an instance of an explicit formula in the theory of
partitions. It is reminiscent of the classical explicit formula

$\displaystyle\psi_{0}(x)=\frac{1}{2}\lim_{\delta\to
0}(\psi(x+\delta)-\psi(x-\delta))=x-\lim_{T\to\infty}\sum_{\lvert\operatorname{Im}(\rho)\rvert<T}\frac{x^{\rho}}{\rho}-\log(2\pi)+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}.$

Here $\rho$ runs over the non-trivial zeros of $\zeta(s)$ and

(1.11) $\displaystyle\psi(x)=\sum_{p^{k}\leqslant x}\log p,$

where the sum in (1.11) is over prime powers.

######  Remark 1.5.

Consider the truncated arithmetic sum

$\displaystyle\Phi_{1}:=\Phi_{1}(X,\theta,J,N)=\sum_{j=1}^{J}\sum_{n=1}^{N}\frac{(\mu*\mu)(n)}{j}\exp\bigg{(}-jn\bigg{(}\frac{1}{X}-2\pi
i\theta\bigg{)}\bigg{)},$

and its analytic truncated counterpart

$\displaystyle\Phi_{2}:=\Phi_{2}(X,\theta,T,N)$
$\displaystyle=4\log\frac{X}{1-2\pi iX\theta}-8\log(2\pi)+72\frac{1-2\pi
iX\theta}{X}$
$\displaystyle\quad+\sum_{\lvert\operatorname{Im}(\rho)\rvert<T}f(X,\theta,\rho)+\sum_{n=1}^{N}g(X,\theta,n).$

We set $\theta=\theta(X)$ as

$\displaystyle\theta(X):=\frac{1}{2\pi}\sqrt{\frac{1}{X^{4/3}}-\frac{1}{X^{2}}},$

such that $X\Delta^{3}=1$. In Figures 1.4 and 1.4, we numerically illustrate
our result from Theorem 1.3 in terms of the functions $\Phi_{1}$ and
$\Phi_{2}$.

Figure 1.3. Left: Plot of $\operatorname{Re}(\Phi_{1}(X,\theta,120,800))$ in
blue and $\operatorname{Re}(\Phi_{2}(X,\theta,20,10))$ in pink. Right: Plot of
$\operatorname{Re}(\Phi_{1}(X,\theta,120,800))-\operatorname{Re}(\Phi_{2}(X,\theta,20,10))$.

Figure 1.4. Left: Plot of $\operatorname{Im}(\Phi_{1}(X,\theta,120,800))$ in
blue and $\operatorname{Im}(\Phi_{2}(X,\theta,20,10))$ in pink. Right: Plot of
$\operatorname{Im}(\Phi_{1}(X,\theta,120,800))-\operatorname{Im}(\Phi_{2}(X,\theta,20,10))$.

We will also need to estimate the minor arc contributions arising from the
Hardy-Littlewood method. This leads us to the second major goal of the present
paper, which is to study exponential sums twisted by general Möbius
convolutions and related questions. With this in mind, we start by considering
the Möbius function itself. Let us write

$\displaystyle S_{\mu}(X,\alpha)=\sum_{n\leqslant
X}\mu(n)\operatorname{e}(n\alpha).$

Then we have the following result.

######  Theorem 1.4.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Then for $X\geqslant 2$, we have

$\displaystyle
S_{\mu}(X,\alpha)\ll\bigg{(}X^{5/6}+\dfrac{X}{q^{1/2}}+X^{1/2}q^{1/2}\bigg{)}(\log
X)^{4}.$

######  Remark 1.6.

Note that the estimate in Theorem 1.4 improves the exponent presented by
Iwaniec and Kowalski in [16, Theorem 13.9] from $9/10$ to $5/6$.

More generally, let

$\displaystyle S_{\mu_{k}}(X,\alpha)=\sum_{n\leqslant
X}\mu_{k}(n)\operatorname{e}(n\alpha),$

where $\mu_{k}(n)$ is as defined in 1.5. We prove the following theorem.

######  Theorem 1.5.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Consider the sequences
$\\{a_{m}\\},\\{b_{m}\\}$ and $\\{c_{m}\\}$ given by

$\displaystyle a_{1}$ $\displaystyle=\frac{5}{6}\quad\textrm{and}\quad\hskip
9.95863pta_{m+1}=\frac{4-a_{m}}{5-2a_{m}}\quad\textrm{for}\quad m=1,2,\ldots,$
$\displaystyle b_{1}$ $\displaystyle=\frac{1}{2}\quad\textrm{and}\quad\quad
b_{m+1}=\frac{b_{m}}{3}\quad\quad\quad\hskip 1.42271pt\textrm{for}\quad
m=1,2,\ldots,$ $\displaystyle\textrm{as well as}\quad c_{1}$
$\displaystyle=\frac{1}{2}\quad\textrm{and}\quad\quad
c_{m+1}=\frac{4-c_{m}}{5-2c_{m}}\quad\textrm{for}\quad m=1,2,\ldots.$

Then for $X\geqslant 2$ and $k\geqslant 2$, we have

$\displaystyle
S_{\mu_{k}}(X,\alpha)\ll\bigg{(}X^{a_{k}}+\dfrac{X}{q^{b_{k}}}+X^{c_{k}}q^{1-c_{k}}\bigg{)}(\log
X)^{k^{2}}.$

We also establish results for exponential sums involving Dirichlet
convolutions of the Möbius function with the characteristic function of
squarefull and squarefree integers respectively. To this end, we define the
arithmetic function $\hat{\mu}$ by

$\displaystyle\hat{\mu}(n):=(\mu*f)(n),$

where $f$ denotes the indicator function for the squarefull numbers. Writing

$\displaystyle S_{\hat{\mu}}(X,\alpha)=\sum_{n\leqslant
X}\hat{\mu}(n)\operatorname{e}(n\alpha),$

we are able to prove the following estimate.

######  Theorem 1.6.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Then for $X\geqslant 2$, we have

$\displaystyle
S_{\hat{\mu}}(X,\alpha)\ll\bigg{(}X^{7/8}+q^{1/2}X^{1/2}+\dfrac{X}{q^{1/2}}\bigg{)}(\log
X)^{5}.$

Considering the convolution of the Möbius function with the characteristic
function of squarefree integers, we define $\tilde{\mu}_{k}(n)$ in terms of
coefficients of the following Dirichlet series. Let $k\geqslant 1$ and

$\displaystyle\sum_{n=1}^{\infty}\dfrac{\tilde{\mu}_{k}(n)}{n^{s}}=\dfrac{1}{\zeta(ks)},$

for $\operatorname{Re}(s)>1/k$. As an example, for $k=2$, one has

$\displaystyle\tilde{\mu}_{2}(n)=(\mu*\mu^{2})(n),$

where $\mu^{2}$ is the indicator function for the squarefree integers. Our
next result provides an upper bound for the exponential sum

$\displaystyle S_{\tilde{\mu}_{2}}(X,\alpha)=\sum_{n\leqslant
X}\tilde{\mu}_{2}(n)\operatorname{e}(n\alpha).$

######  Theorem 1.7.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Then for $X\geqslant 2$, we have

$\displaystyle
S_{\tilde{\mu}_{2}}(X,\alpha)\ll\bigg{(}X^{5/6}+\dfrac{X}{q^{1/4}}+X^{3/4}q^{1/4}\bigg{)}(\log
X)^{10}.$

######  Remark 1.7.

One can apply the method from the proof of Theorem 1.5 to generalize the
result in Theorem 1.7 to any $k\geqslant 1$.

###  1.4. Structure of the paper

The paper is organized as follows. In Section 2, we introduce some preliminary
lemmas, such as Type I and Type II estimates on exponential sums involving
Dirichlet convolutions, a result on the distribution of the Möbius function in
arithmetic progressions, a divisor sum estimate and an unconditional upper
bound on $1/\zeta(s)$ in the critical strip along specific vertical line
segments. Section 3 is devoted to the proofs of Theorems 1.4, 1.6 and 1.7. The
proof of Theorem 1.5 is singled out in Section 4. In Section 5, we move on to
the theory of partitions and set up the proof of Theorem 1.2. Section 6 and
Section 7 will contain the details of the major and minor arc estimates
respectively. Finally, in 8, we present the proof of Theorem 1.3.

##  2\. Preliminary Lemmas

In this section, we start by establishing some standard notations. Then we
collect some necessary results from the literature and proceed to prove some
preliminary lemmas.

###  2.1. Notations

We employ some standard notation that will be used throughout the paper.

  * •

Throughout the paper, the expressions $f(X)=O(g(X))$, $f(X)\ll g(X)$, and
$g(X)\gg f(X)$ are equivalent to the statement that $|f(X)|\leq C|g(X)|$ for
all sufficiently large $X$, where $C>0$ is an absolute constant. A subscript
of the form $\ll_{\alpha}$ means the implied constant may depend on the
parameter $\alpha$. Dependence on several parameters is indicated in an
analogous manner, as in $\ll_{\alpha,\lambda}$.

  * •

The notation $f=o(g)$ as $x\to a$ means that $\lim_{x\to a}f(x)/g(x)=0$ and
$f\sim g$ as $x\to a$ denotes $\lim_{x\to a}f(x)/g(x)=1$.

  * •

We denote the set of real numbers by $\mathbb{R}$, the set of rational numbers
by $\mathbb{Q}$, the set of integers by $\mathbb{Z}$, the set of natural
numbers by $\mathbb{N}$ and the set of complex numbers by $\mathbb{C}$.

  * •

For any set $\mathcal{A}$, $\operatorname{card}(\mathcal{A})$ denotes the
cardinality of the set $\mathcal{A}$.

  * •

The notation $\operatorname{e}(x)$ stands for $\exp(2\pi ix)$.

  * •

Dyadic sums are represented by $\sum_{n\sim N}f(n)=\sum_{N<n\leqslant
2N}f(n)$.

  * •

The divisor function is denoted by $d(n)$. The three-fold divisor function
$\tau_{3}(n)$ is defined by the coefficients of the Dirichlet series

$\zeta^{3}(s)=\sum_{n=1}^{\infty}\frac{\tau_{3}(n)}{n^{s}},\quad\textrm{for
}\operatorname{Re}(s)>1.$

###  2.2. Exponential sums and Dirichlet convolutions

We first state the following two lemmas due to Koukoulopoulos, that provide
bounds on exponential sums involving Dirichlet convolutions.

######  Lemma 2.1 (Type I estimate).

Let $f\colon\mathbb{N}\to\mathbb{C}$ be supported on $[1,y]$, $v\geqslant 2$,
$X\geqslant 2$, $\alpha\in\mathbb{R}$ and $a/q$ be a reduced fraction such
that $|\alpha-a/q|\leqslant 1/q^{2}$. Then

$\displaystyle\sum_{n\leqslant
X}(f*\log^{v})(n)\operatorname{e}(n\alpha)\ll\bigg{(}y+\dfrac{X}{q}+q\bigg{)}(\log
X)^{v+1}\|f\|_{\infty}.$

###### Proof.

See [18, Theorem 23.5]. ∎

######  Lemma 2.2 (Type II estimate).

Let $f,g\colon\mathbb{N}\to\mathbb{C}$ be two arithmetic functions supported
on $[1,y]$ and $[1,z]$, respectively. Let $X\geqslant 2$,
$\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Then

$\displaystyle\sum_{n\leqslant
X}(f*g)(n)\operatorname{e}(n\alpha)\ll\bigg{(}q+y+z+\dfrac{yz}{q}\bigg{)}^{1/2}\sqrt{\log
2q}\cdot\|f\|_{2}\|g\|_{2}.$

###### Proof.

See [18, Theorem 23.6]. ∎

We would like to establish more general versions of Lemmas 2.1 and 2.2, in the
case when $\alpha\in\mathbb{R}$ and there exists some reduced fraction $a/q$
such that $|\alpha-a/q|\leqslant\gamma/q^{2}$, for some $\gamma\geqslant 1$.
In this regard, we first prove the following proposition.

######  Proposition 2.1.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant\gamma/q^{2}$ for some $\gamma\geqslant 1$. Then for
each $m_{1}\in\\{1,2,\dots,q\\}$, there exists at most $14\gamma$ choices of
$m_{2}\in\\{1,2,\dots,q\\}$ such that

(2.1) $\displaystyle\bigg{\lvert}\left\lVert
m_{1}\alpha\right\rVert-\left\lVert
m_{2}\alpha\right\rVert\bigg{\rvert}<\frac{1}{q},$

where $\left\lVert x\right\rVert$ denotes the smallest distance of $x$ to an
integer.

###### Proof.

Fix $m_{1}\in\\{1,2,\dots,q\\}$ and suppose $m_{2}\in\\{1,2\dots,q\\}$ such
that (2.1) is satisfied. Writing $\alpha=a/q+\beta$ with
$\lvert\beta\rvert\leqslant\gamma/q^{2}$, we have

$m_{1}\alpha=\frac{m_{1}a}{q}+m_{1}\beta\quad\textrm{ and}\quad
m_{2}\alpha=\frac{m_{2}a}{q}+m_{2}\beta.$

Therefore, we obtain

(2.2) $\displaystyle\bigg{\lvert}\left\lVert
m_{1}\alpha\right\rVert-\left\lVert\frac{m_{1}a}{q}\right\rVert\bigg{\rvert}\leqslant\lvert
m_{1}\beta\rvert\leqslant\frac{\gamma}{q}\quad\textnormal{and}\quad\bigg{\lvert}\left\lVert
m_{2}\alpha\right\rVert-\left\lVert\frac{m_{2}a}{q}\right\rVert\bigg{\rvert}\leqslant\lvert
m_{2}\beta\rvert\leqslant\frac{\gamma}{q}.$

From (2.1) and (2.2), it follows that

(2.3)
$\displaystyle\bigg{\lvert}\left\lVert\frac{m_{1}a}{q}\right\rVert-\left\lVert\frac{m_{2}a}{q}\right\rVert\bigg{\rvert}\leqslant\frac{2\gamma+1}{q}\leqslant\frac{3\gamma}{q}.$

Next, we write

$\left\lVert\frac{m_{1}a}{q}\right\rVert=\frac{\ell_{1}}{q}\quad\textnormal{and}\quad\left\lVert\frac{m_{2}a}{q}\right\rVert=\frac{\ell_{2}}{q},$

where $\ell_{1},\ell_{2}\in\\{0,1,\dots,\lfloor q/2\rfloor\\}$. Note that when
$m_{1}$ is fixed, $\ell_{1}$ is fixed. Also, for each choice of $\ell_{2}$,
there’s at most two different choices of $m_{2}$. Hence, in order to satisfy
(2.3), there’s at most $7\gamma$ choices of $\ell_{2}$. This implies there are
at most $14\gamma$ choices of $m_{2}$ which satisfy (2.1). ∎

With Proposition 2.1 in hand, we have the following two lemmas.

######  Lemma 2.3.

Let $f\colon\mathbb{N}\to\mathbb{C}$ be supported on $[1,y]$, $v\geqslant 2$,
$X\geqslant 2$, $\alpha\in\mathbb{R}$ and $a/q$ be a reduced fraction such
that $|\alpha-a/q|\leqslant\gamma/q^{2}$ for some $\gamma\geqslant 1$. Then

(2.4) $\displaystyle\sum_{n\leqslant
X}(f*\log^{v})(n)\operatorname{e}(n\alpha)\ll\gamma\bigg{(}y+\dfrac{X}{q}+q\bigg{)}(\log
x)^{v+1}\|f\|_{\infty},$

where the implied constant in (2.4) is absolute.

######  Lemma 2.4.

Let $f,g\colon\mathbb{N}\to\mathbb{C}$ be two arithmetic functions supported
on $[1,y]$ and $[1,z]$, respectively. Let $X\geqslant 2$,
$\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant\gamma/q^{2}$ for some $\gamma\geqslant 1$. Then

(2.5) $\displaystyle\sum_{n\leqslant
X}(f*g)(n)\operatorname{e}(n\alpha)\ll\gamma\bigg{(}q+y+z+\dfrac{yz}{q}\bigg{)}^{1/2}\sqrt{\log
2q}\cdot\|f\|_{2}\|g\|_{2},$

where the implied constant in (2.5) is absolute.

We only prove Lemma 2.3. The proof of Lemma 2.4 is similar.

###### Proof of Lemma 2.3.

Following the proof of Theorem 23.5 in [18], it suffices to bound the sum

$\sum_{k\leqslant y}\min\bigg{\\{}\frac{X}{k},\frac{1}{\left\lVert
k\alpha\right\rVert}\bigg{\\}}.$

To do so, we divide the interval $[1,y]$ into subintervals of length $q$ and
write

$S_{m}:=\sum_{mq<k\leqslant(m+1)q}\min\bigg{\\{}\frac{X}{k},\frac{1}{\left\lVert
k\alpha\right\rVert}\bigg{\\}}.$

Note that

$\sum_{k\leqslant y}\min\bigg{\\{}\frac{X}{k},\frac{1}{\left\lVert
k\alpha\right\rVert}\bigg{\\}}\leqslant\sum_{m=0}^{\lfloor y/q\rfloor}S_{m}.$

We write $\alpha=a/q+\beta$ with $\lvert\beta\rvert\leqslant\gamma/q^{2}$ and
consider two cases.

Case 1: $m\geqslant 1$. Writing $k=mq+r$ where $1\leqslant r\leqslant q$, we
have $\left\lVert k\alpha\right\rVert=\left\lVert
r\alpha+mq\beta\right\rVert$. Define the intervals

$I_{j}=\bigg{[}\frac{j-1}{2q},\frac{j}{2q}\bigg{)},\quad
j\in\\{1,2,\dots,\lceil q\rceil\\}.$

Fix some $k_{1}=mq+r_{1}$, such that $\left\lVert k_{1}\alpha\right\rVert\in
I_{j}$ for some $j$. Since $I_{j}$ is an interval of length $1/2q$, it follows
that $\left\lVert r_{1}\alpha\right\rVert$ is contained in the union of at
most two intervals, say $T_{1}$ and $T_{2}$, each of length at most $1/2q$.
Suppose for some $k_{2}=mq+r_{2}$, $\left\lVert k_{2}\alpha\right\rVert\in
I_{j}$ as well. Then $\left\lVert r_{2}\alpha\right\rVert$ is also contained
in $T_{1}\cup T_{2}$. But using Proposition 2.1, each of $T_{1}$ or $T_{2}$
can contain at most $14\gamma$ choices of $r_{2}$. Hence, there can be at most
$28\gamma$ different $k_{2}$ such that both $\left\lVert
k_{1}\alpha\right\rVert$ and $\left\lVert k_{2}\alpha\right\rVert$ lies in
$I_{\ell}$. Therefore, the contribution from those $k$’s for which
$\left\lVert k\alpha\right\rVert\in I_{j}$ to the sum $S_{m}$ is

$\leqslant\frac{60\gamma q}{j-1},$

when $j>1$. For the specific case when $j=1$, we trivially bound the
contribution by $30\gamma/mq$. Therefore, for $m\geqslant 1$, we obtain

$S_{m}\leqslant 30\gamma\bigg{(}\frac{X}{mq}+\sum_{2\leqslant j\leqslant\lceil
q\rceil}\frac{2q}{j-1}\bigg{)}\ll\gamma\left(\frac{X}{mq}+q\log q\right).$

Case 2: $m=0$. For all $r\geqslant q/2\gamma$, an argument similar to the
previous case shows that the contribution from such terms is
$\ll\gamma(X/q+q\log q)$. Finally, we are left with the cases when
$r<q/2\gamma$. Then $\lvert r\beta\rvert\leqslant 1/2q$, which implies
$\left\lVert r\alpha\right\rVert\geqslant 1/2q$. Therefore, the total
contribution from the cases when $r<q/2\gamma$ is

$\ll\gamma(2q+q\log q)\ll\gamma q\log q.$

The rest of the proof follows along the lines of the proof of [18, Theorem
23.5], and noting that there are $\ll y/q$ choices of $m$. ∎

###  2.3. Exponential sums and the divisor function

We record the following result achieved by Mikawa that provide bounds on
exponential sums involving the three-fold divisor function.

######  Lemma 2.5.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Then, for $0<M,$ $J\leqslant X$, we have

$\sum_{m\sim M}\sum_{j\sim
J}\tau_{3}(j)\min\bigg{(}\frac{X}{m^{2}j},\frac{1}{\|\alpha
m^{2}j\|}\bigg{)}\ll MJ(\log
X)^{3}+\dfrac{X^{3/4}}{M}\bigg{(}Xq^{-1}+XM^{-1}+q\bigg{)}^{1/4}(\log X)^{8},$

where $\tau_{3}$ denotes the three-fold divisor function.

###### Proof.

See [22]. ∎

###  2.4. The Möbius function in arithmetic progressions

We collect below a result on the Möbius function in arithmetic progressions,
due to Davenport, which is an application of the Siegel-Walfisz theorem with
the standard zero-free region for Dirichlet $L$-functions.

######  Lemma 2.6.

Let $X\geqslant 2$ and $B>0$. Let $r,q\geqslant 1$ be two integers with
$q\leqslant(\log X)^{B}$. Then there exists a constant $c_{B}>0$ such that

$\displaystyle\sum_{\begin{subarray}{c}n\leqslant X\\\ n\equiv
r\operatorname{mod}q\end{subarray}}\mu(n)\ll_{B}X\exp(-c_{B}\sqrt{\log X}),$

where $c_{B}>0$ is a constant depending only on $B$.

###### Proof.

See [7, Lemma 6]. ∎

###  2.5. A divisor sum estimate

Let $k\in\mathbb{N}$ and define

$d_{k}(n):=(1*1*\cdots*1)(n),$

where the Dirichlet convolution of the identity function $1$ is performed $k$
times. The Dirichlet series for $d_{k}(n)$ is given by $\zeta(s)^{k}$. Norton
established the following result.

######  Lemma 2.7.

For $X\geqslant 1$, and $k,r\geqslant 2$, we have

$\displaystyle\sum_{n\leqslant X}(d_{k}(n))^{r}\ll X(\log X)^{k^{r}-1}.$

###### Proof.

See [24, Eq. 1.10]. ∎

###  2.6. Bounding $1/\zeta(s)$ in the critical strip.

In order to prove the explicit formula in Section 8, we will require a contour
integration involving the term $1/\zeta^{2}(s)$. To prove our results
unconditionally, we shall use the following result due to works of Ramachandra
and Sankaranarayanan, and Inoue, which provides unconditional upper bounds for
$1/\zeta(s)$ inside the critical strip along a specific vertical line segment.

######  Lemma 2.8.

Let $T>0$ be a sufficiently large positive number and $H=T^{1/3}$. Then one
has

$\displaystyle\min_{T\leqslant t\leqslant
T+H}\max_{\frac{1}{2}\leqslant\sigma\leqslant
2}|\zeta(\sigma+it)|^{-1}\leqslant\exp(C(\log\log T)^{2})$

with an absolute constant $C>0$. In particular, there exists a real
$T_{*}\in[T,T+T^{1/3}]$ such that

$\displaystyle\frac{1}{\zeta(\sigma+iT_{*})}\leqslant
T_{*}^{\varepsilon}\quad\textnormal{where}\quad\frac{1}{2}\leqslant\sigma\leqslant
2,$

for any $\varepsilon>0$.

###### Proof.

See [25, Theorem 2] and [15, Lemma 1]. ∎

##  3\. Proofs of Theorems 1.4, 1.6 and 1.7

In this section we present the proofs of Theorems 1.4, 1.6 and 1.7.

###  3.1. Proof of Theorem 1.4

We can assume $q\leqslant X$, otherwise, the conclusion follows trivially. Let
$y\geqslant 2$ be a parameter to be chosen later on. We start by writing

(3.1) $\displaystyle\mu(n)=\mu_{\leqslant y}(n)+\mu_{>y}(n).$

Applications of the Möbius inversion formula yield

(3.2) $\displaystyle\mu_{>y}(n)$
$\displaystyle=(\mu_{>y}*\mu*1)(n)=(\mu_{>y}*\mu_{>y}*1)(n)+(\mu_{>y}*\mu_{\leqslant
y}*1)(n)$

for the second term in (3.1) and

(3.3) $\displaystyle(\mu_{>y}*\mu_{\leqslant y}*1)(n)$
$\displaystyle=(\mu*\mu_{\leqslant y}*1)(n)-(\mu_{\leqslant y}*\mu_{\leqslant
y}*1)(n)=\mu_{\leqslant y}(n)-(\mu_{\leqslant y}*\mu_{\leqslant y}*1)(n).$

Combining (3.1), (3.2) and (3.3), we arrive at

$\displaystyle\mu(n)=2\mu_{\leqslant
y}(n)+(\mu_{>y}*\mu_{>y}*1)(n)-(\mu_{\leqslant y}*\mu_{\leqslant y}*1)(n).$

We write $F=\mu_{\leqslant y}*\mu_{\leqslant y}$ and note that $F$ is
supported on $[1,y^{2}]$ and $F(n)\leqslant d(n)$, where, we recall, $d(n)$
denotes the divisor function. Further decomposition allows us to write

$\displaystyle\mu(n)$ $\displaystyle=2\mu_{\leqslant
y}(n)+(\mu_{>y}*\mu_{>y}*1)(n)-(F_{\leqslant y}*1)(n)-(F_{(y,y^{2}]}*1)(n)$
$\displaystyle=\mu^{(1)}+\mu^{(2)}-\mu^{(3)}-\mu^{(4)},$

say. Consider $\mu^{(1)}$ first. Since our choice of parameter $y$ will be
small compared to $X$, bounding trivially, we arrive at

(3.4) $\displaystyle\sum_{n\leqslant
X}\mu^{(1)}(n)\operatorname{e}(n\alpha)\ll y.$

For $\mu^{(2)}(n)$, we will use Lemma 2.2. However, applying the lemma
directly doesn’t suffice our purpose. We consider breaking our range into
dyadic intervals. By definition, we have

$\mu^{(2)}(n)=\sum_{\begin{subarray}{c}k\ell=n\\\
k>y,\ell>y\end{subarray}}\mu_{>y}(k)(\mu_{>y}*1)(\ell).$

To obtain better control of the support of the variables $k$ and $\ell$, we
break the interval $(y,X/y]$ into dyadic intervals
$\left(2^{j-1},2^{j}\right]$, where $2^{j}\in(y,2X/y]$. If
$k\in\left(2^{j-1},2^{j}\right]$, then $\ell=n/k\leqslant X/2^{j-1}$. With
this more accurate decomposition, we have

$\mu^{(2)}(n)=\sum_{y<2^{j}\leqslant
2X/y}\left(f_{j}*g_{j}\right)(n),\quad\text{ for }n\leqslant X,$

where $f_{j}(k)=\mu_{>y}(k)\cdot 1_{2^{j-1}<k\leqslant 2^{j}}$ and
$g_{j}(\ell)=(\mu_{>y}*1)(\ell)\cdot 1_{y<\ell\leqslant X/2^{j-1}}$. Then,
applying Lemma 2.2, we obtain

$\displaystyle\sum_{n\leqslant X}\mu^{(2)}(n)\operatorname{e}(n\alpha)$
$\displaystyle=\sum_{y<2^{j}\leqslant 2X/y}\sum_{n\leqslant
X}\left(f_{j}*g_{j}\right)(n)\operatorname{e}(n\alpha)$
$\displaystyle\ll\sum_{y<2^{j}\leqslant
2X/y}\bigg{(}q^{1/2}+\dfrac{X^{1/2}}{y^{1/2}}+\dfrac{X^{1/2}}{q^{1/2}}\bigg{)}\sqrt{\log
2q}\cdot\|f_{j}\|_{2}\|g_{j}\|_{2}$ (3.5)
$\displaystyle\ll\bigg{(}X^{1/2}q^{1/2}+\dfrac{X}{y^{1/2}}+\dfrac{X}{q^{1/2}}\bigg{)}(\log
X)^{3}.$

Note. In the rest of this section, we will always apply the above more
accurate decomposition technique before applying Lemma 2.2.

Next, we focus on $\mu^{(3)}(n)$. Note that $\|F\|_{\infty}\leqslant
N^{\epsilon}$ since it is divisor bounded. Thus, applying Lemma 2.1 directly
will result in an extra $X^{\epsilon}$ in the estimate. In order to circumvent
this situation, we apply the Cauchy-Schwarz inequality first to remove this
function. With this in mind, we write

$\displaystyle\sum_{n\leqslant X}\mu^{(3)}(n)\operatorname{e}(n\alpha)$
$\displaystyle=\sum_{n\leqslant X}(F_{\leqslant
y}*1)(n)\operatorname{e}(n\alpha)\ll\bigg{(}\sum_{k\leqslant
y}F^{2}(k)G(k)\bigg{)}^{1/2}\bigg{(}\sum_{k\leqslant y}G(k)\bigg{)}^{1/2},$

where

$G(k)=\min\bigg{\\{}\frac{X}{k},\frac{1}{\left\lVert
k\alpha\right\rVert}\bigg{\\}}.$

Following the proof of Lemma 2.1, one can show that

$\sum_{k\leqslant y}G(k)\ll\bigg{(}y+\dfrac{X}{q}+q\bigg{)}\log X.$

Therefore, it follows that

$\displaystyle\sum_{n\leqslant X}\mu^{(3)}(n)\operatorname{e}(n\alpha)$
$\displaystyle\ll X^{1/2}(\log X)^{2}\bigg{(}y+\dfrac{X}{q}+q\bigg{)}^{1/2}$
(3.6)
$\displaystyle\ll\bigg{(}X^{1/2}y^{1/2}+\dfrac{X}{q^{1/2}}+X^{1/2}q^{1/2}\bigg{)}(\log
X)^{2}.$

Finally, applying Lemma 2.2 equipped with the dyadic decomposition, we have

$\displaystyle\sum_{n\leqslant X}\mu^{(4)}(n)\operatorname{e}(n\alpha)$
$\displaystyle=\sum_{n\leqslant
X}(F_{(y,y^{2}]}*1)(n)\operatorname{e}(n\alpha)$ (3.7)
$\displaystyle\ll\bigg{(}X^{1/2}q^{1/2}+\dfrac{X}{q^{1/2}}+X^{1/2}y+\dfrac{X}{y^{1/2}}\bigg{)}(\log
X)^{4}.$

We choose $y=X^{1/3}$ and combine the estimates (3.4), (3.1), (3.1) and (3.1)
to obtain

$\displaystyle\sum_{n\leqslant
X}\mu(n)\operatorname{e}(n\alpha)\ll\bigg{(}X^{5/6}+\dfrac{X}{q^{1/2}}+X^{1/2}q^{1/2}\bigg{)}(\log
X)^{4}.$

This ends the proof. ∎

######  Remark 3.1.

The key step in improving the exponent from $9/10$ in [16, Theorem 13.9] to
$5/6$ comes from decomposing the third sum further into two sub-sums.

###  3.2. Proof of Theorem 1.6

The proof follows along the lines of the proof of Theorem 1.4. Again, we can
assume $q\leqslant X$. Let $y\geqslant 2$ be a parameter to be chosen later
on. Writing

(3.8) $\displaystyle\hat{\mu}(n)=\hat{\mu}_{\leqslant
y}(n)+\hat{\mu}_{>y}(n),$

and applying the Möbius inversion, we have

$\displaystyle\hat{\mu}_{>y}(n)$
$\displaystyle=(\hat{\mu}_{>y}*\mu_{>y}*1)(n)+(\hat{\mu}_{>y}*\mu_{\leqslant
y}*1)(n)$ (3.9)
$\displaystyle=(\hat{\mu}_{>y}*\mu_{>y}*1)(n)+(f*\mu_{\leqslant
y})(n)-(\hat{\mu}_{\leqslant y}*\mu_{\leqslant y}*1)(n),$

where $f$ denotes the indicator function for the squarefull numbers.
Therefore, it follows from (3.8) and (3.2) that

$\displaystyle\hat{\mu}(n)=\hat{\mu}_{\leqslant
y}(n)+(\hat{\mu}_{>y}*\mu_{>y}*1)(n)+(f*\mu_{\leqslant
y})(n)-(\hat{\mu}_{\leqslant y}*\mu_{\leqslant y}*1)(n).$

Trivially, we have

(3.10) $\displaystyle\sum_{n\leqslant X}\hat{\mu}_{\leqslant
y}(n)\operatorname{e}(n\alpha)\ll y\log y.$

Next, by Lemma 2.2, similar to how we obtained (3.1), we have

$\displaystyle\sum_{n\leqslant
X}(\hat{\mu}_{>y}*\mu_{>y}*1)(n)\operatorname{e}(n\alpha)$ $\displaystyle\ll
X^{1/2}(\log X)^{4}\bigg{(}q+\dfrac{X}{q}+\dfrac{X}{y}\bigg{)}^{1/2}$ (3.11)
$\displaystyle\ll\bigg{(}\dfrac{X}{y^{1/2}}+X^{1/2}q^{1/2}+\dfrac{X}{q^{1/2}}\bigg{)}(\log
X)^{4}.$

Observing that $\|f\cdot 1_{n\leqslant X}\|_{2}\ll X^{1/4}$, again, an
application of Lemma 2.2 shows that

$\displaystyle\sum_{n\leqslant X}(f*\mu_{\leqslant
y})(n)\operatorname{e}(n\alpha)$
$\displaystyle\ll\bigg{(}q+X+y+\dfrac{X}{q}\bigg{)}^{1/2}X^{1/4}y^{1/2}(\log
X)^{3}$ (3.12)
$\displaystyle\ll\bigg{(}X^{3/4}y^{1/2}+X^{1/4}y^{1/2}q^{1/2}+\dfrac{X^{3/4}y^{1/2}}{q^{1/2}}\bigg{)}(\log
X)^{3}.$

Arguing as in the proof of (3.1) for the case of the Möbius function, we have

(3.13) $\displaystyle\sum_{n\leqslant X}(\hat{\mu}_{\leqslant
y}*\mu_{\leqslant
y}*1)(n)\operatorname{e}(n\alpha)\ll\bigg{(}\dfrac{X}{y^{1/2}}+X^{1/2}y+X^{1/2}q^{1/2}+\dfrac{X}{q^{1/2}}\bigg{)}(\log
X)^{5}.$

Finally, we choose $y=X^{1/4}$ and combine (3.10), (3.2), (3.2) and (3.13) to
obtain

$\displaystyle\sum_{n\leqslant
X}\hat{\mu}(n)\operatorname{e}(n\alpha)\ll\bigg{(}X^{7/8}+q^{1/2}X^{1/2}+\dfrac{X}{q^{1/2}}\bigg{)}(\log
X)^{5},$

which is the desired result. ∎

###  3.3. Proof of Theorem 1.7

Similar to the previous proofs, we can assume $q\leqslant X$. For some
parameter $y\geqslant 2$, we start with the following identity

$\displaystyle\tilde{\mu}_{2}(n)$ $\displaystyle=\tilde{\mu}_{2}(n)\cdot
1_{n\leqslant y}+(\tilde{\mu}_{2}\cdot 1_{n>y}*\mu_{>y}*1)(n)$
$\displaystyle\quad+(\mu_{\leqslant y}*\mu^{2})(n)-(\mu_{\leqslant
y}*\tilde{\mu}_{2}\cdot 1_{n\leqslant y}*1)(n).$

Arguing as in the previous proofs, we can show that

(3.14) $\displaystyle\sum_{n\leqslant X}\tilde{\mu}_{2}(n)\cdot 1_{n\leqslant
y}\operatorname{e}(n\alpha)$ $\displaystyle\ll y\log y,$ (3.15)
$\displaystyle\sum_{n\leqslant X}(\tilde{\mu}_{2}\cdot 1_{n>y}*\mu\cdot
1_{n>y}*1)(n)\operatorname{e}(n\alpha)$ $\displaystyle\ll X(\log
X)^{3}\bigg{(}\dfrac{1}{q^{1/2}}+\dfrac{1}{y^{1/2}}+\dfrac{q^{1/2}}{X^{1/2}}\bigg{)},$
(3.16) $\displaystyle\textrm{and}\quad\sum_{n\leqslant X}(\mu\cdot
1_{n\leqslant y}*\tilde{\mu}_{2}\cdot 1_{n\leqslant y}*1)(n)e(n\alpha)$
$\displaystyle\ll X(\log
X)^{3}\bigg{(}\dfrac{1}{q^{1/2}}+\dfrac{y}{X^{1/2}}+\dfrac{q^{1/2}}{X^{1/2}}\bigg{)}.$

It remains to estimate the sum

$\displaystyle\sum_{n\leqslant X}(\mu\cdot 1_{\leqslant
y}*\mu^{2})(n)\operatorname{e}(n\alpha).$

Using the relation $\mu^{2}(m)=\sum_{ab^{2}=n}\mu(b)$, we can write

$\displaystyle\sum_{n\leqslant X}(\mu\cdot 1_{\leqslant
y}*\mu^{2})(n)\operatorname{e}(n\alpha)$
$\displaystyle=\sum_{\begin{subarray}{c}nb^{2}\leqslant X\\\ n\leqslant
y\end{subarray}}\mu(b)\mu(n)\sum_{a\leqslant
X/(nb^{2})}\operatorname{e}(anb^{2}\alpha)$ (3.17)
$\displaystyle\ll\sum_{b\leqslant X^{1/2}}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}.$

We need to break the range of $b$ in (3.17) into two parts: for large $b$’s,
we will use the bound from Lemma 2.5, and for small $b$’s, we will use the
Cauchy-Schwarz inequality along with Lemma 2.1. Indeed, we write

$\displaystyle\sum_{b\leqslant X^{1/2}}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}=\sum_{b\leqslant
B}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}+\sum_{B<b\leqslant
X^{1/2}}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}.$

Then, by applying the Cauchy-Schwarz inequality in a similar way how we
established (3.1),

$\displaystyle\sum_{b\leqslant B}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}$
$\displaystyle\leqslant\sum_{m\leqslant
B^{2}y}\tau_{3}(m)\min\bigg{\\{}\dfrac{X}{m},\dfrac{1}{\|m\alpha\|}\bigg{\\}}$
$\displaystyle\ll X^{1/2}(\log
X)^{6}\bigg{(}B^{2}y+q+\dfrac{X}{q}\bigg{)}^{1/2}$ (3.18)
$\displaystyle\ll\bigg{(}X^{1/2}By^{1/2}+X^{1/2}q^{1/2}+\dfrac{X}{q^{1/2}}\bigg{)}(\log
X)^{5}.$

Next, by Lemma 2.5, we have

(3.19) $\displaystyle\sum_{B<b\leqslant X^{1/2}}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}\ll\bigg{(}X^{1/2}y+\dfrac{X}{q^{1/4}B}+\dfrac{X}{B^{5/4}}+\dfrac{X^{3/4}q^{1/4}}{B}\bigg{)}(\log
X)^{10}.$

Choose $B=(X/y)^{2/9}$ and combining (3.18) and (3.19), we obtain

(3.20) $\displaystyle\sum_{b\leqslant X^{1/2}}\sum_{n\leqslant
y}\min\bigg{\\{}\dfrac{X}{b^{2}n},\dfrac{1}{\|b^{2}n\alpha\|}\bigg{\\}}\ll\bigg{(}X^{13/18}y^{5/18}+X^{1/2}y+\dfrac{X}{q^{1/4}}+X^{3/4}q^{1/4}\bigg{)}(\log
X)^{10}.$

Finally, we choose $y=X^{1/3}$ and combine (3.14), (3.15), (3.16) and (3.20)
to obtain

$\displaystyle\sum_{n\leqslant
X}\tilde{\mu}_{2}(n)\operatorname{e}(n\alpha)\ll\bigg{(}X^{5/6}+\dfrac{X}{q^{1/4}}+X^{3/4}q^{1/4}\bigg{)}(\log
X)^{10},$

as it was to be shown. ∎

##  4\. Proof of Theorem 1.5

We explicitly show the proof for the case when $k=2$. The argument then
follows similarly for higher values of $k$ using induction. We can assume
$q\leqslant X$, otherwise, the conclusion is trivial. We break our proof into
several steps.

###  4.1. Initial step

Let

$\displaystyle S_{\mu_{2}}(X,\alpha):=\sum_{n\leqslant
X}(\mu*\mu)(n)\operatorname{e}(n\alpha).$

Then, for some parameters $2\leqslant M,N\leqslant X$ to be chosen later, we
can write

$\displaystyle S_{\mu_{2}}(X,\alpha)=$
$\displaystyle\>\sum_{\begin{subarray}{c}mn\leqslant X\\\
m>M,n>N\end{subarray}}\mu(m)\mu(n)\operatorname{e}(mn\alpha)+\sum_{m\leqslant
M}\mu(m)\sum_{n\leqslant X/m}\mu(n)\operatorname{e}(mn\alpha)$
$\displaystyle+\sum_{n\leqslant N}\mu(n)\sum_{m\leqslant
X/n}\mu(m)\operatorname{e}(mn\alpha)-\sum_{\begin{subarray}{c}m\leqslant M\\\
n\leqslant N\end{subarray}}\mu(m)\mu(n)\operatorname{e}(mn\alpha)$ (4.1)
$\displaystyle=$
$\displaystyle\>S_{\mu_{2}}^{(1)}(X,\alpha)+S_{\mu_{2}}^{(2)}(X,\alpha)+S_{\mu_{2}}^{(3)}(X,\alpha)-S_{\mu_{2}}^{(4)}(X,\alpha),$

say. We now turn our attention to each of these terms.

###  4.2. Estimation of $S_{\mu_{2}}^{(1)}(X,\alpha)$ and
$S_{\mu_{2}}^{(4)}(X,\alpha)$.

Applying Lemma 2.2 after the dyadic decomposition following the proof of
Theorem 1.4, we deduce that

$\displaystyle S_{\mu_{2}}^{(1)}(X,\alpha)$ $\displaystyle\ll
X^{1/2}\bigg{(}\dfrac{X}{M}+\dfrac{X}{N}+\dfrac{X}{q}+q\bigg{)}^{1/2}(\log
X)^{2}$ (4.2)
$\displaystyle\ll\bigg{(}\dfrac{X}{M^{1/2}}+\dfrac{X}{N^{1/2}}+\dfrac{X}{q^{1/2}}+X^{1/2}q^{1/2}\bigg{)}(\log
X)^{2}.$

Trivially, we have

(4.3) $\displaystyle S_{\mu_{2}}^{(4)}(X,\alpha)\ll MN.$

###  4.3. Estimation of $S_{\mu_{2}}^{(2)}(X,\alpha)$ and
$S_{\mu_{2}}^{(3)}(X,\alpha)$.

Writing $\alpha=a/q+\beta$, one has

$m\alpha=\dfrac{ma}{q}+m\beta=\dfrac{ma/(m,q)}{q/(m,q)}+m\beta\quad\text{and}\quad|m\beta|\leqslant\dfrac{m/(m,q)^{2}}{q^{2}/(m,q)^{2}}\leqslant\dfrac{m/(m,q)}{q^{2}/(m,q)^{2}}.$

Therefore, using Lemma (2.4), we arrive at

$\displaystyle S_{\mu_{2}}^{(2)}(X,\alpha)$ $\displaystyle\ll\sum_{m\leqslant
M}\dfrac{m}{(m,q)}\bigg{\\{}\bigg{(}\dfrac{X}{m}\bigg{)}^{5/6}+\dfrac{X(m,q)^{1/2}}{mq^{1/2}}+\bigg{(}\dfrac{X}{m}\bigg{)}^{1/2}\dfrac{q^{1/2}}{(q,m)^{1/2}}\bigg{\\}}\bigg{(}\log\frac{X}{m}\bigg{)}^{4}$
(4.4)
$\displaystyle\ll\bigg{(}X^{5/6}M^{7/6}+\dfrac{XM}{q^{1/2}}+X^{1/2}q^{1/2}M^{3/2}\bigg{)}(\log
X)^{4}.$

Similarly, we can show that

(4.5) $\displaystyle
S_{\mu_{2}}^{(3)}(X,\alpha)\ll\bigg{(}X^{5/6}N^{7/6}+\dfrac{XN}{q^{1/2}}+X^{1/2}q^{1/2}N^{3/2}\bigg{)}(\log
X)^{4}.$

###  4.4. Parameter optimization

Substituting the estimates (4.2), (4.4), (4.5) and (4.3) in (4.1) and choosing
$M=N$, we deduce that

$S_{\mu_{2}}(X,\alpha)\ll\bigg{(}\dfrac{X}{M^{1/2}}+\dfrac{X}{q^{1/2}}+X^{1/2}q^{1/2}+M^{2}+X^{5/6}M^{7/6}+\dfrac{XM}{q^{1/2}}+X^{1/2}q^{1/2}M^{3/2}\bigg{)}(\log
X)^{4}.$

We are now left with the crucial step, to choose our parameter $M$ optimally.
Since $M\geqslant 2$, we can omit the terms ${X}/{q^{1/2}}$ and
$X^{1/2}q^{1/2}$. Also, we ignore the term $M^{2}$ since $M\leqslant X$. Let
$M=y^{6}$, $A=\sqrt{q}$, $B=x^{1/6}$ and $C=\sqrt{{x}/{q}}$. Then it suffices
to find an upper bound for the expression

(4.6) $\displaystyle
H(y):=\max\bigg{\\{}\dfrac{1}{y^{3}},\dfrac{y^{6}}{A},\dfrac{y^{7}}{B},\dfrac{y^{9}}{C}\bigg{\\}}.$

Note that as a function of $y$, only $1/y^{3}$ is decreasing. Hence, the
maximum in (4.6) is attained when $1/y^{3}$ is equal to one of the other three
functions. We now consider the following cases.

Case 1: Consider the case when

(4.7)
$\displaystyle\dfrac{y^{6}}{A}\geqslant\max\bigg{\\{}\dfrac{y^{7}}{B},\dfrac{y^{9}}{C}\bigg{\\}}.$

Then the maximum in (4.6) is attained when

$\dfrac{1}{y^{3}}=\dfrac{y^{6}}{A}\implies y=A^{1/9},$

and therefore we obtain

$H(y)=\dfrac{1}{A^{1/3}}=\dfrac{1}{q^{1/6}}.$

Now we are only left to find the appropriate range of $q$ where (4.7) holds.
For that, we need

$\dfrac{1}{A^{1/3}}\geqslant\max\bigg{\\{}\dfrac{A^{7/9}}{B},\frac{A}{C}\bigg{\\}}.$

Solving for $A^{-1/3}\geqslant A^{7/9}/B$, we get that $q\leqslant X^{3/10}$.
On the other hand, the condition $A^{-1/3}\geqslant A/C$ implies that
$q\leqslant X^{3/7}$. Therefore, for both conditions to hold, we need
$1\leqslant q\leqslant X^{3/10}$ and in this range, we have

(4.8) $\displaystyle S_{\mu_{2}}(X,\alpha)\ll\dfrac{X(\log X)^{4}}{q^{1/6}}.$

Case 2: Now, assume the scenario when

(4.9)
$\displaystyle\dfrac{y^{7}}{B}\geqslant\max\bigg{\\{}\dfrac{y^{6}}{A},\dfrac{y^{9}}{C}\bigg{\\}}.$

The maximum in (4.6) is achieved when

$\dfrac{1}{y^{3}}=\dfrac{y^{7}}{B}\implies y=B^{1/10}.$

Hence, in this case,

$H(y)=\dfrac{1}{B^{3/10}}=\dfrac{1}{x^{1/20}}.$

Again, similar to Case 1, for the appropriate range of $q$ such that (4.9)
holds, we require

$\dfrac{B^{7/10}}{B}\geqslant\max\bigg{\\{}\dfrac{B^{3/5}}{A},\frac{B^{9/10}}{C}\bigg{\\}}.$

Solving for $B^{7/10}/B\geqslant B^{3/5}/A$, it follows that $q\geqslant
x^{3/10}$. On the other hand, $B^{7/10}/B\geqslant B^{9/10}/C$ implies that
$q\leqslant X^{3/5}$. Hence, we need $x^{3/10}\leqslant q\leqslant X^{3/5}$
and in this range, we obtain

(4.10) $\displaystyle S_{\mu_{2}}(X,\alpha)\ll X^{19/20}(\log X)^{4}.$

Case 3: Finally, we arrive at the case when

(4.11)
$\displaystyle\dfrac{y^{9}}{C}\geqslant\max\bigg{\\{}\dfrac{y^{6}}{A},\dfrac{y^{7}}{B}\bigg{\\}}.$

Now, the maximum in (4.6) is attained when

$\dfrac{1}{y^{3}}=\dfrac{y^{9}}{C}\implies y=C^{1/12}.$

Therefore, we arrive at

$H(y)=\dfrac{1}{C^{1/4}}=\dfrac{q^{1/8}}{x^{1/8}}.$

Once again, for the appropriate range of $q$ where (4.11) holds, we require

$\dfrac{C^{3/4}}{C}\geqslant\max\bigg{\\{}\dfrac{C^{1/2}}{A},\frac{C^{7/12}}{B}\bigg{\\}}.$

The condition $C^{3/4}/C\geqslant C^{1/2}/A$ implies that $q\geqslant
x^{3/7}$. On the other hand, solving for $C^{3/4}/C\geqslant C^{7/12}/B$, it
follows that $q\geqslant x^{3/5}$. Therefore, for both conditions to hold, we
need $x^{3/5}\leqslant q\leqslant X$ and in this range, we have

(4.12) $\displaystyle S_{\mu_{2}}(X,\alpha)\ll X^{7/8}q^{1/8}(\log X)^{4}.$

Combining the estimates (4.8), (4.10) and (4.12), we conclude that

$\sum_{n\leqslant
X}(\mu*\mu)(n)\operatorname{e}(n\alpha)\ll\bigg{(}X^{19/20}+\dfrac{X}{q^{1/6}}+X^{7/8}q^{1/8}\bigg{)}(\log
X)^{4}.$

This completes the proof for the case when $k=2$.

###  4.5. Induction step

Assume that for $X\geqslant 2$,

$\displaystyle\sum_{n\leqslant
X}\mu_{k}(n)\operatorname{e}(n\alpha)\ll\bigg{(}X^{a_{k}}+\dfrac{X}{q^{b_{k}}}+X^{c_{k}}q^{1-c_{k}}\bigg{)}(\log
X)^{k^{2}},$

for some $k=\ell\geqslant 2$ and we want to conclude for the case when
$k=\ell+1$. Similar to the proof for the case when $k=2$, one can show that if
$\alpha\in\mathbb{R}$ such that $|\alpha-a/q|\leqslant\gamma/q^{2}$ for some
$(a,q)=1$ and $\gamma\geqslant 1$, then for $X\geqslant 2$,

(4.13) $\displaystyle\sum_{n\leqslant
X}\mu_{\ell}(n)\operatorname{e}(n\alpha)\ll\gamma\bigg{(}X^{a_{\ell}}+\dfrac{X}{q^{b_{\ell}}}+X^{c_{\ell}}q^{1-c_{\ell}}\bigg{)}(\log
X)^{\ell^{2}},$

where the implied constant in (4.13) is absolute. To establish (4.13), one can
exactly follow the case for $k=2$, except that while bounding the analogue of
$S_{\mu_{2}}^{(1)}(X,\alpha)$, we use Lemma 2.4 instead of Lemma 2.2. Equipped
with (4.13), let

$\displaystyle S_{\mu_{\ell+1}}(X,\alpha):=\sum_{n\leqslant
X}\mu_{\ell+1}(n)\operatorname{e}(n\alpha).$

Then, for some parameters $2\leqslant M,N\leqslant X$ to be chosen later, we
can write

$\displaystyle S_{\mu_{\ell+1}}(X,\alpha)=$
$\displaystyle\>\sum_{\begin{subarray}{c}mn\leqslant X\\\
m>M,n>N\end{subarray}}\mu(m)\mu_{\ell}(n)\operatorname{e}(mn\alpha)+\sum_{m\leqslant
M}\mu(m)\sum_{n\leqslant X/m}\mu_{\ell}(n)\operatorname{e}(mn\alpha)$
$\displaystyle+\sum_{n\leqslant N}\mu_{\ell}(n)\sum_{m\leqslant
X/n}\mu(m)\operatorname{e}(mn\alpha)-\sum_{\begin{subarray}{c}m\leqslant M\\\
n\leqslant N\end{subarray}}\mu(m)\mu_{\ell}(n)\operatorname{e}(mn\alpha)$
(4.14) $\displaystyle=$
$\displaystyle\>S_{\mu_{\ell+1}}^{(1)}(X,\alpha)+S_{\mu_{\ell+1}}^{(2)}(X,\alpha)+S_{\mu_{\ell+1}}^{(3)}(X,\alpha)-S_{\mu_{\ell+1}}^{(4)}(X,\alpha).$

Again, applying Lemma 2.2, we obtain

(4.15) $\displaystyle
S_{\mu_{\ell+1}}^{(1)}(X,\alpha)\ll\bigg{(}\dfrac{X}{M^{1/2}}+\dfrac{X}{N^{1/2}}+\dfrac{X}{q^{1/2}}+X^{1/2}q^{1/2}\bigg{)}(\log
X)^{(\ell+1)^{2}}.$

Trivially, we have

(4.16) $\displaystyle S_{\mu_{\ell+1}}^{(4)}(X,\alpha)\ll MN(\log
N)^{\ell-1}.$

Choosing $\gamma=m/(m,q)$ and applying (4.13), we arrive at

(4.17) $\displaystyle
S_{\mu_{\ell+1}}^{(2)}(X,\alpha)\ll\bigg{(}X^{a_{\ell}}M^{2-a_{\ell}}+\dfrac{XM}{q^{b_{\ell}}}+X^{c_{\ell}}q^{1-c_{\ell}}M^{2-c_{\ell}}\bigg{)}(\log
X)^{(\ell+1)^{2}}.$

Note that Lemma 2.7 shows that

$\sum_{n\leqslant N}d_{\ell}^{2}(n)\ll N(\log N)^{\ell^{2}-1}.$

Hence applying Lemma 2.4 and Cauchy-Schwartz inequality in conjunction, we can
write

$\displaystyle S_{\mu_{\ell+1}}^{(3)}(X,\alpha)$
$\displaystyle\ll\sum_{n\leqslant
N}d_{\ell}(n)\cdot\dfrac{n}{(n,q)}\bigg{\\{}\bigg{(}\dfrac{X}{n}\bigg{)}^{5/6}+\dfrac{X(n,q)^{1/2}}{nq^{1/2}}+\bigg{(}\dfrac{X}{n}\bigg{)}^{1/2}\dfrac{q^{1/2}}{(n,q)^{1/2}}\bigg{\\}}\bigg{(}\log\frac{X}{n}\bigg{)}^{4}$
(4.18)
$\displaystyle\ll\bigg{(}X^{5/6}N^{7/6}+\dfrac{XN}{q^{1/2}}+X^{1/2}q^{1/2}N^{3/2}\bigg{)}(\log
X)^{(\ell+1)^{2}}.$

Substituting the estimates (4.15), (4.16), (4.17) and (4.5) in (4.14) and
choosing $M=N$, we have

$S_{\mu_{\ell+1}}(X,\alpha)\ll\bigg{(}\dfrac{X}{M^{1/2}}+X^{a_{\ell}}M^{2-a_{\ell}}+\dfrac{XM}{q^{b_{\ell}}}+X^{c_{\ell}}q^{1-c_{\ell}}M^{2-c_{\ell}}+X^{1/2}q^{1/2}M^{3/2}\bigg{)}(\log
X)^{(\ell+1)^{2}}.$

An optimization procedure similar to that for the case $k=2$ then shows that

$\displaystyle
S_{\mu_{\ell+1}}(X,\alpha)\ll\bigg{(}X^{a_{\ell+1}}+\dfrac{X}{q^{b_{\ell+1}}}+X^{c_{\ell+1}}q^{1-c_{\ell+1}}\bigg{)}(\log
X)^{(\ell+1)^{2}},$

thereby completing the proof. ∎

##  5\. Proof of Theorem 1.2

We now set up the structure of the proof of Theorem 1.2. Recall from (1.8) and
(1.9) that

$\displaystyle\Psi_{\mu*\mu}(z)$
$\displaystyle=\sum_{n=1}^{\infty}p_{\mu*\mu}(n)z^{n}=\prod_{n=1}^{\infty}(1-z^{n})^{-(\mu*\mu)(n)},\quad\lvert
z\rvert<1,$ $\displaystyle\Phi_{\mu*\mu}(z)$
$\displaystyle=\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\frac{(\mu*\mu)(n)}{j}z^{jn},\quad\lvert
z\rvert<1,$

and $\Psi_{\mu*\mu}(z)=\exp(\Phi_{\mu*\mu}(z))$. The strategy to prove Theorem
1.2 is to employ the Hardy-Littlewood circle method. An application of
Cauchy’s theorem yields

(5.1) $\displaystyle
p_{\mu*\mu}(n)=\rho^{-n}\int_{0}^{1}\exp(\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))-2\pi
in\theta)d\theta,$

where $0<\rho<1$. We write $\rho=\exp(-1/X)$, for some parameter $X>0$. As we
shall see, for each value of $n$, we will choose $X=X(n)$ and therefore,
$\rho=\rho(n)$ in (5.1) to be such that as $n\to\infty$, $X\to\infty$ and
$\rho\to 1$. Therefore, in what follows, we can assume $X$ to be sufficiently
large. Invoking the periodicity of the integrand in (5.1), we can replace the
unit interval by the interval $\mathcal{U}=[-X^{-1}(\log X)^{A},1-X^{-1}(\log
X)^{A})$, where $A>0$ is a sufficiently large constant. This choice
$\mathcal{U}$ keeps the region around the origin from being split into two
regions.

Now we are ready to set up the major and minor arcs. Let
$a\in\mathbb{N}\cup\\{0\\}$ and $q\in\mathbb{N}$ such that $0\leqslant a<q$
and $(a,q)=1$. Choose

(5.2) $\displaystyle\delta_{q}=q^{-1}X^{-1}(\log
X)^{A}\quad\textnormal{and}\quad Q=(\log X)^{A},$

and define

(5.3)
$\displaystyle\mathfrak{M}(q,a)=\bigg{\\{}\theta\in\mathcal{U}:\bigg{|}\theta-\frac{a}{q}\bigg{|}\leqslant\delta_{q}\bigg{\\}}.$

The major arcs $\mathfrak{M}$ and the minor arcs $\mathfrak{m}$ are

(5.4) $\displaystyle\mathfrak{M}=\bigcup_{\begin{subarray}{c}1\leqslant
a\leqslant q\leqslant Q\\\
(a,q)=1\end{subarray}}\mathfrak{M}(q,a)\quad\textnormal{and}\quad\mathfrak{m}=\mathcal{U}\backslash\mathfrak{M}.$

We will show that if $\theta\in\mathfrak{M}$, then

(5.5)
$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))\ll_{A}\frac{X}{(\log
X)^{A}}.$

On the other hand, if $\theta\in\mathfrak{m}$, then one has

(5.6)
$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))\ll_{A}\frac{X}{(\log
X)^{A/9}}.$

With (5.5) and (5.6) in mind, let us first prove Theorem 1.2.

###### Proof of Theorem 1.2.

We rewrite (5.1) as

(5.7) $\displaystyle
p_{\mu*\mu}(n)=\rho^{-n}\bigg{(}\int_{\mathfrak{M}}+\int_{\mathfrak{m}}\bigg{)}\exp(\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))-2\pi
in\theta)d\theta.$

First, we consider the integral over the minor arcs . Let $X=\sqrt{n}(\log
n)^{A/18}$. Since $\rho=\exp(-1/X)$, using (5.6), we can write

(5.8)
$\displaystyle\rho^{-n}\int_{\mathfrak{m}}\exp(\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))-2\pi
in\theta)d\theta\ll\exp\bigg{(}\frac{n}{X}+\frac{X}{(\log
X)^{A/9}}\bigg{)}\ll\exp\bigg{(}\frac{\sqrt{n}}{(\log n)^{A/20}}\bigg{)}.$

A concomitant argument holds for the major arcs. Therefore, choosing $A$
sufficiently large, we obtain

$\log p_{\mu*\mu}(n)\ll_{B}\frac{\sqrt{n}}{(\log n)^{B}},$

for any fixed $B>0$. ∎

Therefore, it suffices to establish (5.5) and (5.6), which we do in Sections 6
and 7 respectively.

##  6\. Major arcs

We begin our discussion on the major arcs by estimating Möbius convolutions in
arithmetic progressions. The key ingredient required for the following result
is Lemma 2.6.

######  Lemma 6.1.

Let $X\geqslant 2$. Suppose $B>0$ is fixed. Let $r,q\geqslant 1$ be positive
integers with $q\leqslant(\log X)^{B}$. Then there exists a constant $c_{B}>0$
depending only on $B$ such that

$\displaystyle S(X):=\sum_{\begin{subarray}{c}n\leqslant X\\\ n\equiv
r\operatorname{mod}q\end{subarray}}(\mu*\mu)(n)\ll_{B}X\exp(-c_{B}\sqrt{\log
X}).$

###### Proof.

By Dirichlet’s hyperbola method, we break $S(X)$ into three sub-sums, i.e.,
$S(X)=S_{1}(X)+S_{2}(X)-S_{3}(X)$, where

$\displaystyle S_{1}(X)$
$\displaystyle=\sum_{n_{1}\leqslant\sqrt{X}}\mu(n_{1})\sum_{\begin{subarray}{c}n_{2}\leqslant
X/n_{1}\\\ n_{1}n_{2}\equiv r\operatorname{mod}q\end{subarray}}\mu(n_{2}),$
$\displaystyle S_{2}(X)$
$\displaystyle=\sum_{n_{2}\leqslant\sqrt{X}}\mu(n_{2})\sum_{\begin{subarray}{c}n_{1}\leqslant
X/n_{2}\\\ n_{1}n_{2}\equiv r\operatorname{mod}q\end{subarray}}\mu(n_{2}),$
$\displaystyle S_{3}(X)$
$\displaystyle=\sum_{n_{1}\leqslant\sqrt{X}}\mu(n_{1})\sum_{\begin{subarray}{c}n_{2}\leqslant\sqrt{X}\\\
n_{1}n_{2}\equiv r\operatorname{mod}q\end{subarray}}\mu(n_{2}).$

We treat $S_{3}(X)$ first. Consider the inner sum over $n_{2}$. Note that if
$n_{1}n_{2}\equiv r\operatorname{mod}q$, then $n_{1}(n_{2}+q)\equiv
r\operatorname{mod}q$. With this in mind, we write

(6.1) $\displaystyle\mathcal{B}:=\big{\\{}x\operatorname{mod}q\colon
n_{1}x\equiv r\operatorname{mod}q\big{\\}}.$

Then using Lemma 2.6, it follows that

(6.2) $\displaystyle\sum_{\begin{subarray}{c}n_{2}\leqslant\sqrt{X}\\\
n_{1}n_{2}\equiv
r\operatorname{mod}q\end{subarray}}\mu(n_{2})=\sum_{b\in\mathcal{B}}\sum_{\begin{subarray}{c}n_{2}\equiv
b\operatorname{mod}q\\\
n_{2}\leqslant\sqrt{X}\end{subarray}}\mu(n_{2})\ll_{B}\operatorname{card}(\mathcal{B})\sqrt{X}\exp(-c_{1}\sqrt{\log
X}),$

for some constant $c_{1}=c_{1}(B)>0$. The bound in (6.2) holds for all
$q\leqslant(\log X)^{B}$. Therefore, by trivially bounding the sum over
$n_{1}$, we obtain

(6.3) $\displaystyle
S_{3}(X)\ll_{B}\bigg{(}\sum_{n_{1}\leqslant\sqrt{X}}|\mu(n_{1})|\bigg{)}\operatorname{card}(\mathcal{B})\sqrt{X}\exp(-c_{1}\sqrt{\log
X})\ll_{B}X\exp(-c_{2}\sqrt{\log X}),$

for some constant $c_{2}=c_{2}(B)>0$.

Note that $S_{1}(X)=S_{2}(X)$ by symmetry. Therefore, it is enough to estimate
the sum $S_{1}(X)$. We analyze the inner sum over $n_{2}$ first. Again
applying Lemma 2.6, we obtain

$\displaystyle\sum_{\begin{subarray}{c}n_{2}\leqslant X/n_{1}\\\
n_{1}n_{2}\equiv
r\operatorname{mod}q\end{subarray}}\mu(n_{2})=\sum_{b\in\mathcal{B}}\sum_{\begin{subarray}{c}n_{2}\equiv
b\operatorname{mod}q\\\ n_{2}\leqslant
X/n_{1}\end{subarray}}\mu(n_{2})\ll_{B}\operatorname{card}(\mathcal{B})\frac{X}{n_{1}}\exp\bigg{(}-c_{3}\sqrt{\log\frac{X}{n_{1}}}\bigg{)},$

for some constant $c_{3}=c_{3}(B)>0$, where $\mathcal{B}$ is as in (6.1). This
leaves us with

(6.4) $\displaystyle S_{1}(X)$
$\displaystyle\ll_{B}\sum_{n_{1}\leqslant\sqrt{X}}\operatorname{card}(\mathcal{B})\frac{X}{n_{1}}\exp\bigg{(}-c_{3}\sqrt{\log\frac{X}{n_{1}}}\bigg{)}\ll_{B}X\exp\big{(}-c_{4}\sqrt{\log
X}\big{)},$

for some constant $c_{4}=c_{4}(B)>0$. Combining (6.3) and (6.4), we arrive at
the conclusion. ∎

Applying Lemma 6.1, one can quickly conclude the following.

######  Lemma 6.2.

Let $X\geqslant 2$ and suppose $B>0$ is fixed. Let $\theta\in\mathbb{R}$ such
that $\theta=a/q+\beta$ where $a,q\in\mathbb{Z}$ with $0\leqslant
a<q\leqslant(\log X)^{B}$, $(a,q)=1$ and $\lvert\beta\rvert\leqslant
X^{-1}(\log X)^{B}$. Then there exists a constant $c_{B}>0$ depending only on
$B$ such that

$\displaystyle\sum_{n\leqslant
X}(\mu*\mu)(n)\operatorname{e}(n\theta)\ll_{B}X\exp(-c_{B}\sqrt{\log X}).$

###### Proof.

By partial summation, we can write

(6.5) $\displaystyle\sum_{n\leqslant
X}(\mu*\mu)(n)\operatorname{e}(n\theta)\ll_{B}(1+|\beta|X)\max_{1\leqslant
y\leqslant X}\bigg{|}\sum_{n\leqslant
y}(\mu*\mu)(n)\operatorname{e}(na/q)\bigg{|}.$

Consequently, it is enough to estimate the sum

(6.6) $\displaystyle\sum_{n\leqslant
y}(\mu*\mu)(n)\operatorname{e}(na/q)=\sum_{b=1}^{q}\operatorname{e}\bigg{(}\dfrac{ab}{q}\bigg{)}\sum_{\begin{subarray}{c}n\leqslant
y\\\ n\equiv b\operatorname{mod}q\end{subarray}}(\mu*\mu)(n).$

An immediate application of Lemma 6.1 shows that for some constant
$c_{1}=c_{1}(B)>0$ depending only on $B$, one has

(6.7) $\displaystyle\sum_{\begin{subarray}{c}n\leqslant y\\\ n\equiv
b\operatorname{mod}q\end{subarray}}(\mu*\mu)(n)\ll_{B}y\exp(-c_{1}\sqrt{\log
y}).$

Combining (6.5), (6.6) and (6.7), we conclude that

$\displaystyle\sum_{n\leqslant
X}(\mu*\mu)(n)\operatorname{e}(n\theta)\ll_{B}(1+|\beta|X)qX\exp(-c_{1}\sqrt{\log
X})\ll_{B}X\exp(-c_{B}\sqrt{\log X}),$

for some constant $c_{B}>0$ depending only on $B$. ∎

Recall from (5.4) that if $\theta\in\mathfrak{M}$, we can write
$\theta=a/q+\beta$ with $q\leqslant(\log X)^{A}$ and
$|\beta|\leqslant\delta_{q}\leqslant X^{-1}(\log X)^{A}$, where $A>0$ is
sufficiently large. Using Lemmas 6.1 and 6.2, we can now prove (5.5) for
$\theta\in\mathfrak{M}$.

######  Lemma 6.3.

Let $\theta\in\mathfrak{M}$. Then

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))\ll_{A}\frac{X}{(\log
X)^{A}},$

for some fixed $A>0$.

###### Proof.

Since $\rho=\exp(-1/X)$, it follows from (1.9) that

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))=\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{j}(\mu*\mu)(n)e^{-nj/X}\operatorname{e}(jn\theta).$

We replace the exponentially damped term above by the integral

$\displaystyle e^{-nj/X}=\int_{n}^{\infty}jX^{-1}e^{-tj/X}\,dt,$

leaving us with

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))=\sum_{j=1}^{\infty}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt.$

Suppose $J=(\log X)^{2A}$. Since

$\sum_{n\leqslant t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\ll\sum_{n\leqslant
t}d(n)\ll t\log t,$

we obtain

$\displaystyle\sum_{j>J}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}$
$\displaystyle\sum_{n\leqslant t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt$
$\displaystyle\ll\sum_{j>J}^{\infty}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}t\log
t\,dt$ (6.8) $\displaystyle\ll\sum_{j>J}^{\infty}\frac{X\log
X}{j^{2}}\int_{j/X}^{\infty}ue^{-u}\log u\,du\ll_{A}\frac{X}{(\log X)^{A}}.$

Hence, in what follows, it suffices to work with the sum

(6.9)
$\displaystyle\sum_{j=1}^{J}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt.$

Writing $\theta=a/q+\beta$ with $q\leqslant(\log X)^{A}$ and
$|\beta|\leqslant\delta_{q}\leqslant X^{-1}(\log X)^{A}$, one has

$\theta_{j}:=j\theta=\frac{aj/(j,q)}{q/(j,q)}+j\beta=\frac{a_{j}}{q_{j}}+\beta_{j},$

where $(a_{j},q_{j})=1$, $1\leqslant q_{j}\leqslant q\leqslant(\log X)^{A}$
and $\beta_{j}\leqslant X^{-1}(\log X)^{3A}$. We can now break the inner
integral in (6.9) as

$\displaystyle I$
$\displaystyle=\int_{1}^{\infty}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt$
$\displaystyle=\bigg{(}\int_{1}^{X\log X}+\int_{X\log
X}^{\infty}\bigg{)}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt=I_{1}+I_{2},$

say. For $I_{2}$, one has

(6.10) $\displaystyle I_{2}$ $\displaystyle\ll\int_{X\log
X}^{\infty}jX^{-1}e^{-tj/X}t\log t\,dt\ll\frac{X\log X}{j}\int_{\log
X}^{\infty}ue^{-u}\log u\,du\ll_{A}\frac{X}{(\log X)^{3A}}.$

For $I_{1}$, an application of Lemma 6.2 shows that

$\displaystyle I_{1}$ $\displaystyle\ll_{A}X\log X\exp(-C_{A}\sqrt{\log
X})\int_{1}^{X\log X}jX^{-1}e^{-tj/X}\,dt$ (6.11) $\displaystyle\ll_{A}X\log
X\exp(-C_{A}\sqrt{\log X})\int_{1}^{X\log X}e^{-u}\,du\ll_{A}\frac{X}{(\log
X)^{3A}},$

where $C_{A}>0$ is a constant depending only on $A$. Putting together (6.10)
and (6), we arrive at

$I\ll_{A}\frac{X}{(\log X)^{3A}},$

which implies that

$\sum_{j=1}^{J}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt\ll_{A}\frac{X}{(\log X)^{A}}.$

Combining this with (6), we arrive at our desired conclusion. ∎

######  Remark 6.1.

The above arguments and the statements of Lemmas 6.1, 6.2 and 6.3 hold for a
general $\mu_{k}$ with appropriate changes.

##  7\. Minor Arcs

In general, being able to show that the contribution of the minor arcs is of
lower order than the principal arcs is the most challenging aspect of
successfully deploying the circle method. In situations such as [2, 12] which
involved partitioning with respect to powers or powers in arithmetic
progressions, one requires Weyl’s inequality [30, Lemma 2.4] in conjunction
with Hua’s inequality [30, Theorem 4.1]. Adapting Weyl’s inequality was also a
key aspect for partitioning with respect to polynomials in [11]. Restricted
partitions with respect to primes [31] necessitated Vinogradov’s inequality
[30, Chapter 3]. However, Vinogradov’s inequality was not sufficient for
functions of primes. In [13], Gafni employed a bound by Kawada and Wooley
[17], and in [9], a new bound altogether for exponential sums dealing with
semiprimes was introduced and proved. This was also the case in [3,
$\mathsection$6] where a bound for exponential sums twisted by generalized
divisor functions had to be supplied. In our current situation, the
exponential sums discussed in Section 1 and proved in Sections 3 and 4 will
serve as the main technology to bound the contributions from the minor arcs.
It is worth remarking that interesting exponential sums for minor arcs can
also be found in [20].

We begin by recording the immediate corollary from Theorem 1.5 for the case
when $k=2$.

######  Corollary 7.1.

Let $\alpha\in\mathbb{R}$ and consider a reduced fraction $a/q$ such that
$|\alpha-a/q|\leqslant 1/q^{2}$. Then for $X\geqslant 2$, we have

$\displaystyle\sum_{n\leqslant
X}(\mu*\mu)(n)\operatorname{e}(n\alpha)\ll\bigg{(}X^{19/20}+\dfrac{X}{q^{1/6}}+X^{7/8}q^{1/8}\bigg{)}(\log
X)^{4}.$

Now, we are ready to prove (5.6) for $\theta\in\mathfrak{m}$.

######  Lemma 7.2.

Let $\theta\in\mathfrak{m}$. Then one has

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))\ll_{A}\frac{X}{(\log
X)^{A/9}},$

for some fixed $A>0$.

###### Proof.

As in the proof of Lemma 6.3, it suffices to only consider the sum

$\sum_{j=1}^{J}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt,$

where $J=(\log X)^{2A}$. For each $j\leqslant J$, we employ Dirichlet’s
theorem (see [30, Lemma 2.1]) to choose $a\in\mathbb{Z}$ and $q\in\mathbb{N}$
with $(a,q)=1$ such that

$\displaystyle\bigg{|}j\theta-\frac{a}{q}\bigg{|}\leqslant q^{-1}X^{-1}(\log
X)^{A}\quad\textnormal{and}\quad q<X(\log X)^{-A}.$

We now set $a_{j}:=a/(a,j)$ and $q_{j}:=jq/(a,j)$. The definition of
$\delta_{q}$ in (5.2) imply that

$\displaystyle\bigg{|}\alpha-\frac{a_{j}}{q_{j}}\bigg{|}\leqslant\delta_{q_{j}}.$

Since $\alpha\in\mathfrak{m}$, the definition of $\mathfrak{m}$ implies that
$q_{j}>Q$, where $Q=(\log X)^{A}$. Now, applying Corollary 7.1, we obtain

(7.1) $\displaystyle\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\ll\bigg{(}t^{19/20}+\dfrac{t}{q^{1/6}}+t^{7/8}q^{1/8}\bigg{)}(\log
t)^{4}.$

Integrating each of the three terms on the right hand side of (7.1) yields

$\displaystyle E_{1}(X,j)$
$\displaystyle=\int_{1}^{\infty}t^{\frac{19}{20}}(\log
t)^{4}jX^{-1}e^{-tj/X}\,dt\ll\bigg{(}\frac{X}{j}\bigg{)}^{\frac{19}{20}}\bigg{(}\log\frac{X}{j}\bigg{)}^{4},$
$\displaystyle E_{2}(X,j)$
$\displaystyle=\int_{1}^{\infty}\frac{t}{q^{1/6}}(\log
t)^{4}jX^{-1}e^{-tj/X}\,dt\ll\frac{X}{{j{q^{1/6}}}}\left({\log\frac{X}{j}}\right)^{4},$
$\displaystyle E_{3}(X,j)$
$\displaystyle=\int_{1}^{\infty}t^{\frac{7}{8}}q^{\frac{1}{8}}(\log
t)^{4}jX^{-1}e^{-tj/X}\,dt\ll{\left({\frac{X}{j}}\right)^{\frac{7}{8}}}{q^{1/8}}\left({\log\frac{X}{j}}\right)^{4}.$

Note that $jq\geqslant q_{j}>(\log X)^{A}$, and by our choice, $q<X(\log
X)^{-A}$. Therefore, we can conclude that

$\displaystyle\sum_{j=1}^{J}\frac{1}{j}\int_{1}^{\infty}jX^{-1}e^{-tj/X}\sum_{n\leqslant
t}(\mu*\mu)(n)\operatorname{e}(jn\theta)\,dt\ll\sum_{j=1}^{J}\frac{1}{j}(E_{1}+E_{2}+E_{3})\ll\frac{X}{(\log
X)^{A/9}},$

which is the bound that we wanted to show. ∎

######  Remark 7.1.

The statement of Lemma 7.2 holds more generally for $\mu_{k}$ by applying
Theorem 1.5 appropriately in place of Corollary 7.1.

##  8\. Proof of Theorem 1.3

In order to prove Theorem 1.3, we need to investigate how $\Phi_{\mu*\mu}(z)$
behaves in the major arc and examine the effect that the non-trivial zeros of
the Riemann zeta-function have in its behavior. We break our proof into
several steps.

### Initial setup

Recall that

(8.1)
$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))=\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\frac{(\mu*\mu)(n)}{j}\exp\bigg{(}-jn\bigg{(}\frac{1}{X}-2\pi
i\theta\bigg{)}\bigg{)}.$

where $\rho=\exp(-1/X)$. We first apply the Cahen-Mellin formula and rewrite
(8.1) to obtain

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))=\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\frac{(\mu*\mu)(n)}{j}\frac{1}{2\pi
i}\int_{(c)}\Gamma(s)j^{-s}n^{-s}\bigg{(}\frac{X}{1-2\pi
iX\theta}\bigg{)}^{s}ds,$

for $c>0$. Interchanging the summations and the integration, for any real
$c>1$, we have

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))$
$\displaystyle=\frac{1}{2\pi
i}\int_{(c)}\Gamma(s)\bigg{(}\sum_{j=1}^{\infty}\frac{1}{j^{s+1}}\bigg{)}\bigg{(}\sum_{n=1}^{\infty}\frac{(\mu*\mu)(n)}{n^{s}}\bigg{)}\bigg{(}\frac{X}{1-2\pi
iX\theta}\bigg{)}^{s}ds$ (8.2) $\displaystyle=\frac{1}{2\pi
i}\int_{(c)}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}\bigg{(}\frac{X}{1-2\pi
iX\theta}\bigg{)}^{s}ds.$

### Contour Shifting.

The path of integration in 8.2 is a vertical line with
$\operatorname{Re}(s)=c>1$. We move the portion $|t|\leqslant T$ of this path
to the left, replacing it by a rectangular path joining the points $c\pm iT$
and $-u\pm iT$, where $u=2N+1$, for some large $N\in\mathbb{N}$. Thus, the new
path of integration is of the form $L=\bigcup_{i=1}^{5}L_{i}$, with

$\displaystyle L_{1}=(c-i\infty,c-iT],$ $\displaystyle L_{2}=[c-iT,-u-iT],$
$\displaystyle L_{3}=[-u-iT,-u+iT],$ $\displaystyle L_{4}=[-u+iT,c+iT],$
$\displaystyle L_{5}=[c+iT,c+i\infty).$

With this change of the path of integration, an application of the Cauchy
residue theorem shows that

$\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))=M+\frac{1}{2\pi
i}\sum_{j=1}^{5}I_{j},$

where $M$ denotes the contribution of the residues at singularities of the
integrand in the region enclosed by the two paths and $I_{j}$ denotes the
integral over the path $L_{j}$. Our method will require an analysis of the
$I_{j}$’s. Equipped with Lemma 2.8, we will establish our results
unconditionally. For similar techniques, the reader is referred to the works
of Bartz [1], Ramachandra and Sankaranarayanan [25] and Inoue [15].

### Residue computations

Set

$y:=\frac{X}{1-2\pi iX\theta}.$

A singularity analysis regarding the integrand in (8.2) shows the following.
We have a double pole at $s=0$ coming from $\Gamma(s)\zeta(s+1)$ for which

$\displaystyle\mathop{\operatorname{res}}\limits_{s=0}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}=4\log
y-8\log(2\pi).$

The simple pole at $s=-1$ yields

$\displaystyle\mathop{\operatorname{res}}\limits_{s=-1}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}=-\frac{\zeta(0)}{\zeta^{2}(-1)y}=\frac{72}{y}.$

Unlike previous cases in the literature [2, 3, 9, 11, 12, 13], we now capture
residues from the trivial and non-trivial zeros of the Riemann zeta-function.
We start with the non-trivial zeros. Those take place at $s=\rho$ and yield
double poles for which

$\displaystyle\mathop{\operatorname{res}}\limits_{s=\rho}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}$
$\displaystyle=y^{\rho}\Gamma(\rho)\frac{\zeta(1+\rho)}{\zeta^{\prime}(\rho)^{2}}\bigg{(}\log
y+\psi(\rho)-\frac{\zeta^{\prime\prime}}{\zeta^{\prime}}(\rho)+\frac{\zeta^{\prime}}{\zeta}(1+\rho)\bigg{)},$

where we have used the simplicity of the zeros to alleviate the notation. For
the trivial zeros, we have triple poles at $s=-2n$, for $n=1,2,3,\dots$,
coming from $\Gamma(s)/\zeta^{2}(s)$. These residues can be written as

$\displaystyle\mathop{\operatorname{res}}\limits_{s=-2n}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}=y^{-2n}(\mathfrak{c}_{1}(n)(\log
y)^{2}+\mathfrak{c}_{2}(n)\log y+\mathfrak{c}_{3}(n)),$

where the coefficients are given by

(8.3) $\displaystyle\mathfrak{c}_{1}(n)$
$\displaystyle:=-\frac{B_{2n}}{8n^{2}\Gamma(2n)\zeta^{\prime}(-2n)^{2}},$
(8.4) $\displaystyle\mathfrak{c}_{2}(n)$
$\displaystyle:=\frac{B_{2n}\zeta^{\prime\prime}(-2n)+\zeta^{\prime}(-2n)\left(2n\zeta^{\prime}(1-2n)-B_{2n}\psi(2n+1)\right)}{2n(2n)!\zeta^{\prime}(-2n)^{3}},$

and

$\displaystyle\mathfrak{c}_{3}(n)$
$\displaystyle:=\frac{1}{48n^{2}\Gamma(2n)\zeta^{\prime}(-2n)^{4}}$
$\displaystyle\quad\times\big{[}B_{2n}\big{\\{}-9\zeta^{\prime\prime}(-2n)^{2}-2\left(3\psi(2n+1)^{2}-3\psi^{\prime}(2n+1)+\pi^{2}\right)\zeta^{\prime}(-2n)^{2}$
$\displaystyle\quad\quad\quad+4\zeta^{\prime}(-2n)(\zeta^{\prime\prime\prime}(-2n)+3\psi(2n+1)\zeta^{\prime\prime}(-2n))\big{\\}}$
(8.5)
$\displaystyle\quad\quad+12n\zeta^{\prime}(-2n)\left(\zeta^{\prime}(-2n)\zeta^{\prime\prime}(1-2n)+2\zeta^{\prime}(1-2n)\left(\psi(2n+1)\zeta^{\prime}(-2n)-\zeta^{\prime\prime}(-2n)\right)\right)\big{]}.$

Here $\psi(x)=\Gamma^{\prime}/\Gamma(x)$ denotes the digamma function and
$B_{n}$ denote the Bernoulli numbers.

### Estimating the vertical integral $I_{3}$

Using Stirling’s formula, for $\operatorname{Re}(s)\leqslant-3/2$,

(8.6)
$\displaystyle\Gamma(s)\zeta(s+1)\zeta(s)\ll(2\pi)^{2\operatorname{Re}(s)}\lvert
s\rvert^{-\operatorname{Re}(s)-1/2}e^{-\pi\lvert t\rvert/2}.$

Therefore, we have

$\displaystyle
I_{3}=\int_{-u-iT}^{-u+iT}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds$
$\displaystyle\ll\int_{-u-iT}^{-u+iT}\frac{(2\pi)^{-2u}\lvert
s\rvert^{u-1/2}e^{-\pi\lvert t\rvert/2}}{\zeta^{3}(s)}y^{s}\,ds$ (8.7)
$\displaystyle\ll\frac{1}{(2\pi)^{2u}}\int_{-u-iT}^{-u+iT}\frac{(u^{2}+\lvert
t\rvert^{2})^{u/2-1/4}e^{-\pi\lvert t\rvert/2}}{\zeta^{3}(s)}y^{s}\,ds.$

We apply the functional equation for $\zeta(s)$ in (8) alongside the
observation that $\zeta^{-1}(1-s)\ll 1$. Also, following [32] (See Lemma 2.8),

$\lvert y\rvert^{s}\leqslant(X\Delta)^{-u}e^{-\lvert t\rvert(\Delta-\pi/2)}.$

Hence, we obtain

(8.8) $\displaystyle I_{3}$
$\displaystyle\ll\frac{2^{3u}\pi^{3(u-1)}}{(2\pi)^{2u}(X\Delta)^{u}}\int_{-u-iT}^{-u+iT}\frac{(u^{2}+\lvert
t\rvert^{2})^{u/2-1/4}e^{-\lvert t\rvert\Delta}}{\sin^{3}(\pi
s/2)\Gamma^{3}(1-s)}\,ds.$

Next, we note that in our range of integration,

$\displaystyle\frac{1}{\Gamma(1-s)}\ll
e^{(u+1)-(u+\frac{1}{2})\log(u+1)+\frac{1}{2}\pi|t|}\quad\textnormal{and}\quad\frac{1}{\sin(\pi
s/2)}\ll e^{-\frac{1}{2}\pi|t|}.$

Inserting these two bounds in (8.8), we arrive at

$\displaystyle I_{3}$
$\displaystyle\ll\frac{(2\pi)^{u}e^{3(u+1)-3(u+\frac{1}{2})\log(u+1)}}{(X\Delta)^{u}}\int_{-T}^{T}(u^{2}+\lvert
t\rvert^{2})^{u/2-1/4}e^{-\lvert t\rvert\Delta}\,dt.$ (8.9)
$\displaystyle\ll\frac{(2\pi)^{u}e^{3(u+1)-3(u+\frac{1}{2})\log(u+1)}}{(X\Delta)^{u}}\int_{0}^{T}(u^{2}+t^{2})^{u/2-1/4}e^{-t\Delta}\,dt.$

We consider the inside integral in (8). We divide this into two cases. When
$t\leqslant u$, we have

(8.10) $\displaystyle\int_{0}^{T}(u^{2}+t^{2})^{u/2-1/4}e^{-t\Delta}\,dt\ll
2^{u/2}u^{u-1/2}\int_{0}^{\infty}e^{-t\Delta}\,dt\ll
2^{u/2}u^{u-1/2}\Delta^{-1}.$

On the other hand, when $t>u$,

(8.11) $\displaystyle\int_{0}^{T}(u^{2}+t^{2})^{u/2-1/4}e^{-t\Delta}\,dt\ll
2^{u/2}\int_{0}^{\infty}t^{u-1/2}e^{-t\Delta}\,dt\ll
2^{u/2}\Gamma(u+1/2)\Delta^{-u-1/2}.$

Therefore, applying (8.10) and (8.11) in (8) and using Stirling’s
approximation for $\Gamma(u+1/2)$, we obtain

$\displaystyle I_{3}$
$\displaystyle\ll\frac{(2\sqrt{2}\pi)^{u}e^{3(u+1)-3(u+\frac{1}{2})\log(u+1)}}{(X\Delta)^{u}}(u^{u-1/2}\Delta^{-1}+\Gamma(u+1/2)\Delta^{-u-1/2})$
$\displaystyle\ll\frac{(2\sqrt{2}\pi)^{u}e^{3(u+1)-3(u+\frac{1}{2})\log(u+1)}}{(eX\Delta^{2})^{u}}(u^{u}\Delta^{-1/2})$
(8.12) $\displaystyle\ll\bigg{(}\frac{2\sqrt{2}\pi
e^{2}}{X\Delta^{2}}\bigg{)}^{u}(u^{-2u-3/2}\Delta^{-1/2}).$

### Estimating the horizontal integrals $I_{2}$ and $I_{4}$.

We consider $I_{4}$ only. The treatment for $I_{2}$ is similar. Dividing the
contour into two parts, we write

$I_{4}=\int_{-u+iT}^{-3/2+iT}+\int_{-3/2+iT}^{c+iT}=I_{4,1}+I_{4,2},$

say. We work with $I_{4,1}$ first. Applying (8.6), we have

$\displaystyle
I_{4,1}=\int_{-u+iT}^{-3/2+iT}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds$
$\displaystyle\ll\int_{-u+iT}^{-3/2+iT}\frac{(2\pi)^{2\sigma}\lvert
s\rvert^{-\sigma-1/2}e^{-\pi T/2}}{\zeta^{3}(s)}y^{s}\,ds$ (8.13)
$\displaystyle\ll e^{-\pi
T/2}\int_{-u+iT}^{-3/2+iT}\frac{(2\pi)^{2\sigma}(\sigma^{2}+T^{2})^{-\sigma/2-1/4}}{\zeta^{3}(s)}y^{s}\,ds.$

Proceeding as in the case of $I_{3}$, we apply the functional equation for
$\zeta(s)$ in (8) and use the bounds

$\zeta^{-1}(1-s)\ll 1,\quad\textrm{and}\quad\lvert
y\rvert^{s}\leqslant(X\Delta)^{\sigma}e^{-T(\Delta-\pi/2)}.$

Then

$\displaystyle I_{4,1}$ $\displaystyle\ll
e^{-T\Delta}\int_{-u+iT}^{-3/2+iT}\frac{(2\pi)^{2\sigma}(X\Delta)^{\sigma}(\sigma^{2}+T^{2})^{-\sigma/2-1/4}}{2^{3\sigma}\pi^{3(\sigma-1)}\sin^{3}(\pi
s/2)\Gamma^{3}(1-s)}\,ds.$

In our range of integration,

$\displaystyle\frac{1}{\Gamma(1-s)}\ll
e^{1-\sigma-(\frac{1}{2}-\sigma)\log(1-\sigma)+\frac{1}{2}\pi
T}\quad\textnormal{as well as}\quad\frac{1}{\sin(\pi s/2)}\ll
e^{-\frac{1}{2}\pi T}.$

Therefore, we arrive at

$\displaystyle I_{4,1}$ $\displaystyle\ll
e^{-T\Delta}\int_{-u}^{-3/2}(2\pi)^{-\sigma}(X\Delta)^{\sigma}(\sigma^{2}+T^{2})^{-\sigma/2-1/4}e^{3(1-\sigma)-3(\frac{1}{2}-\sigma)\log(1-\sigma)}\,d\sigma$
(8.14) $\displaystyle\ll e^{-T\Delta}\int_{3/2}^{u}\bigg{(}\frac{2\pi
e^{3}}{X\Delta}\bigg{)}^{\sigma}\frac{(\sigma^{2}+T^{2})^{\sigma/2-1/4}}{(\sigma+1)^{3(\sigma+\frac{1}{2})}}\,d\sigma.$

We write (8.14) as

$e^{-T\Delta}\int_{3/2}^{u}\bigg{(}\frac{2\pi
e^{3}}{X\Delta}\bigg{)}^{\sigma}\frac{(\sigma^{2}+T^{2})^{\sigma/2-1/4}}{(\sigma+1)^{3(\sigma+\frac{1}{2})}}\,d\sigma=e^{-T\Delta}\int_{3/2}^{T}+e^{-T\Delta}\int_{T}^{u}=J_{1}+J_{2},$

say. Consider $J_{1}$ first. We have

$J_{1}\ll e^{-T\Delta}\int_{3/2}^{T}\bigg{(}\frac{2\sqrt{2}\pi
e^{3}T}{\sigma^{3}}\bigg{)}^{\sigma}\,d\sigma.$

Let $f$ be defined as

$f(\sigma)=\bigg{(}\frac{2\sqrt{2}\pi e^{3}T}{\sigma^{3}}\bigg{)}^{\sigma},$

and suppose $g(\sigma)=\log f(\sigma)$. The maximum of $g(\sigma)$ is attained
at $\sigma_{0}=(2\sqrt{2}\pi T)^{1/3}$, and thus, the maximum value attained
by $f$ in our range of integration is $\exp(3(2\sqrt{2}\pi T)^{1/3})$.
Therefore, we obtain

(8.15) $\displaystyle J_{1}\ll e^{-T\Delta}e^{3(2\sqrt{2}\pi T)^{1/3}}T\ll
e^{-T\Delta/2}.$

For $J_{2}$, we obtain

(8.16) $\displaystyle e^{-T\Delta}\int_{T}^{u}\bigg{(}\frac{2\pi
e^{3}}{X\Delta}\bigg{)}^{\sigma}\frac{(\sigma^{2}+T^{2})^{\sigma/2-1/4}}{(\sigma+1)^{3(\sigma+\frac{1}{2})}}\,d\sigma\ll
e^{-T\Delta}\int_{T}^{u}\bigg{(}\frac{2\sqrt{2}\pi
e^{3}}{\sigma^{2}}\bigg{)}^{\sigma}\,d\sigma\ll e^{-T\Delta/2}.$

Now, we consider $I_{4,2}$, which is given by

$\displaystyle
I_{4,2}=\int_{-3/2+iT}^{c+iT}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds.$

In our range of integration,

(8.17) $\displaystyle\Gamma(s)\ll T^{\sigma-1/2}e^{-\pi
T/2},\quad\textrm{and}\quad\zeta(s+1)\ll T^{\eta},$

for some fixed constant $\eta>0$. We write

$I_{4,2}=\bigg{(}\int_{-3/2+iT}^{-1+iT}+\int_{-1+iT}^{c+iT}\bigg{)}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds.$

For the first part, we apply the functional equation for $\zeta(s)$ to obtain

$\displaystyle\int_{-3/2+iT}^{-1+iT}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds$
$\displaystyle\ll\frac{T^{\eta-1/2}}{e^{T\Delta}}\int_{-3/2}^{-1}\bigg{(}\frac{X\Delta
T}{2\pi e}\bigg{)}^{\sigma}\frac{1}{(1-\sigma)^{1/2-\sigma}}\,d\sigma$
$\displaystyle\ll\frac{T^{\eta-1/2}}{e^{T\Delta}}\int_{1}^{3/2}\bigg{(}\frac{2\pi
e}{X\Delta T}\bigg{)}^{\sigma}\frac{1}{(1+\sigma)^{1/2+\sigma}}\,d\sigma$
(8.18) $\displaystyle\ll e^{-T\Delta/2}.$

For the second part, using Lemma 2.8, for
$-1\leqslant\operatorname{Re}(s)=\sigma\leqslant 2$ and any $\epsilon>0$, for
sufficiently large $V>0$, there exists some $V\leqslant T_{*}\leqslant 2V$
such that

(8.19) $\displaystyle\frac{1}{\zeta^{2}(\sigma+iT_{*})}\ll T_{*}^{\epsilon}.$

Therefore, using the bounds (8.17) and (8.19) and choosing $T=T_{*}$, we
arrive at

(8.20)
$\displaystyle\int_{-1+iT}^{c+iT}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds$
$\displaystyle\ll\frac{T^{-1/2+\eta+\epsilon}}{e^{T\Delta}}\int_{-1}^{c}(X\Delta
T)^{\sigma}\,d\sigma\ll e^{-T\Delta/2}.$

Finally combining the estimates (8.15), (8.16), (8.18) and (8.20), we conclude
that

(8.21) $\displaystyle I_{4}\ll e^{-T\Delta/2}.$

### Estimating the horizontal integrals $I_{1}$ and $I_{5}$.

Writing

$I_{1}=\int_{c+iT}^{c+i\infty}\Gamma(s)\zeta(s+1)\frac{1}{\zeta^{2}(s)}y^{s}\,ds,$

we note that in our range of integration,

$\displaystyle\Gamma(s)$ $\displaystyle\ll e^{-\pi T/2},\quad\zeta(s+1)\ll
1,\quad\zeta^{-2}(s)\ll 1,\quad\textrm{and}\quad\lvert
y\rvert^{s}\leqslant(X\Delta)^{\sigma}e^{-T(\Delta-\pi/2)}.$

Combining these estimates, we obtain

(8.22) $\displaystyle I_{1}\ll e^{-T\Delta/2}.$

A similar argument holds true for $I_{5}$.

### Final steps

Combining (8.12), (8.21) and (8.22), and letting $N\to\infty$, we obtain

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))$
$\displaystyle=4\log\frac{X}{1-2\pi iX\theta}-8\log(2\pi)+72\frac{1-2\pi
iX\theta}{X}$ (8.23)
$\displaystyle\quad+\sum_{|\operatorname{Im}(\rho)|<T_{\nu}}f(X,\theta,\rho)+\sum_{n=1}^{\infty}g(X,\theta,n)+O(e^{-T_{\nu}\Delta/2}),$

where $\nu\leqslant T_{\nu}\leqslant\nu+1$, $\nu\in\mathbb{N}$, the sum over
$\rho$ runs along all the non-trivial zeros of $\zeta$, and $f,g$ are as
defined in the statement of the theorem. Finally, we let $\nu\to\infty$ and
choose our sequence of $T_{\nu}$ appropriately such that (8.19) is satisfied.
We conclude that

$\displaystyle\Phi_{\mu*\mu}(\rho\operatorname{e}(\theta))$
$\displaystyle=4\log\frac{X}{1-2\pi iX\theta}-8\log(2\pi)+72\frac{1-2\pi
iX\theta}{X}$
$\displaystyle\quad+\lim_{\nu\to\infty}\sum_{|\operatorname{Im}(\rho)|<T_{\nu}}f(X,\theta,\rho)+\sum_{n=1}^{\infty}g(X,\theta,n),$

thereby finishing the last step of the proof. ∎

##  9\. Acknowlegements

We thank Kunjakanan Nath for many comments and discussions on the subject of
the paper. We also thank Dorian Goldfeld for useful suggestions. NR wishes to
acknowledge support from Vikram Kilambi and from Christopher Pernin.

## References

  * [1] K. M. Bartz, _On some complex explicit formulae connected with the Möbius function, II_, Acta Arith. 57 (1991) 295–305.
  * [2] B. C. Berndt, A. Malik, and A. Zaharescu, _Partitions into $k$th powers of terms in an arithmetic progression_, Math. Z. (2018), 290:1277-1307.
  * [3] B. C. Berndt, N. Robles, A. Zaharescu, and D. Zeindler, _Partitions with multiplicities associated with divisor functions_ , J. Math. Anal. Appl., Vol. 533, Issue 1, 1 May 2024.
  * [4] C. Calderón, M. J. Zárate, _A generalization of Selberg’s asymptotic formula._ Arch. Math. (Basel) 56 (1991), no. 5, 465–470.
  * [5] T. Daniels, _Bounds on the Möbius-signed partition numbers._ Preprint, arXiv:2310.10609, 2023.
  * [6] T. Daniels, _Biasymptotics of the Möbius- and Liouville-signed partition numbers._ Preprint, arXiv:2310.10617, 2023.
  * [7] H. Davenport, _On some infinite series involving arithmetical functions. II_. Quart. J. Math. Oxf. 8 (1937), 313–320.
  * [8] H. Davenport, _Multiplicative Number Theory._ Springer, Graduate Texts in Mathematics v.74, 3rd edition, 2000.
  * [9] M. Das, N. Robles, A. Zaharescu, and D. Zeindler, _Partitions into semiprimes._ Preprint, arXiv:2212.12489, 2022.
  * [10] G. Debryune and G. Tenenbaum.  The saddle-point method for general partition functions.  Indag. Math., (31) no. 4: 728–738, 2020.
  * [11] A. Dunn and N. Robles, _Polynomial partition asymptotics_ , J. Math. Anal. Appl., Vol. 459, 2018, pp. 359-384.
  * [12] A. Gafni, _Power partitions_ , Journal of Number Theory, 163 (2016), 19-42.
  * [13] A. Gafni, _Partitions into prime powers_ , Mathematika, volume 67, issue 2, April 2021, 368-388.
  * [14] G. H. Hardy and S. Ramanujan, _Asymptotic Formulaæ in Combinatory Analysis_ , Proceedings of the London Mathematical Society, (2) 17, 1918, 75-115.
  * [15] S. Inoue, Relations among some conjectures on the Möbius function and the Riemann zeta-function, Acta Arith. 191 (2019), no. 1, 1-32.
  * [16] H. Iwaniec, E. Kowalski, _Analytic Number Theory._ American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004.
  * [17] K. Kawada and T. Wooley.  On the Waring-Goldbach Problem for Fourth and Fifth Powers.  Proc. London Math. Soc., (83) no. 1: 1–50, 2001.
  * [18] D. Koukoulopoulos, _The distribution of prime numbers._ Graduate Studies in Mathematics, vol. 203, American Mathematical Society, Providence, RI, 2019.
  * [19] P. Kühn and N. Robles, _Explicit formulas of a generalized Ramanujan sum_ , Int. J. Number Theory, Vol. 12, No. 02, pp. 383-408 (2016).
  * [20] A. Kumchev.  On Weyl Sums over Primes and Almost Primes.  Michigan Math. J., (54):243–268, 2006.
  * [21] G. Meinardus, _Asymptotische Aussagen über Partitionen_ , Math. Z. 59 (1954) 388–398.
  * [22] H. Mikawa, _On exponential sums over primes in arithmetic progressions._ Tsukuba J. Math. 24 (2000), no. 2, 351–360.
  * [23] H. L. Montgomery, _Extreme values of the Riemann zeta-function_ , Comment. Math. Helv. 52 (1977) 511–518.
  * [24] K. K. Norton, _Upper Bounds for Sums of Powers of Divisor Functions_ , J. Number Theory 40, 60-85 (1992).
  * [25] K. Ramachandra and A. Sankaranarayanan, _Notes on the Riemann zeta-function_ , J. Indian Math. Soc. 57 (1991), 67-77.
  * [26] N. Robles and A. Roy, _Unexpected average values of generalized von Mangoldt functions in residue classes._ J. Aust. Math. Soc. 111 (2021), no. 1, 127–144.
  * [27] N. Robles and D. Zeindler, _Partitions asymptotics over almost primes and functions of primes_ , in preparation.
  * [28] J. Sándor and B. Crstici, _Handbook of number theory II_, Kluwer Academic Press, 2004.
  * [29] E. C. Titchmarsh, _The Theory of the Riemann Zeta Function_ , Clarendon Press, Oxford, 1986.
  * [30] R. C. Vaughan. The Hardy-Littlewood Method, Cambridge tracts in Mathematics; 125, 2nd Ed. 1997.
  * [31] R. C. Vaughan. On the number of partitions into primes, Ramanujan J., (15), 109-121, 2008.
  * [32] R. C. Vaughan. Squares: Additive questions and partitions, Int. J. Number Theory, Vol 11, No 5: 1-43, 2015.
  * [33] G. Valiron, _Sur les fonctions entieres d’ordre nul et ordre fini_ , Ann. Toulouse (3) 5 (1914) 117–257.