# A Note on Large Sums of Divisor-Bounded Multiplicative Functions

Claire Frechette  Boston College, Department of Mathematics, Chestnut Hill, MA
02467, USA<EMAIL_ADDRESS>,  Mathilde Gerbelli-Gauthier  McGill University,
Department of Mathematics, Montréal, QC H3A 2K6, Canada  mathilde.gerbelli-
<EMAIL_ADDRESS>,  Alia Hamieh  University of Northern British Columbia,
Department of Mathematics and Statistics, Prince George, BC V2N 4Z9, Canada
<EMAIL_ADDRESS>and  Naomi Tanabe  Bowdoin College, Department of
Mathematics, Brunswick, ME 04011, USA<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the
associated partial sum. In this note, we show that lower bounds on partial
sums of divisor-bounded functions result in lower bounds on the partial sums
associated to their products. More precisely, we let $f_{j}$, $j=1,2$ be such
that $|f_{j}(n)|\leq\tau(n)^{\kappa}$ for some $\kappa\in\mathbb{N}$, and
assume their partial sums satisfy $\left|S(x_{j},f_{j})\right|\geq\eta
x_{j}(\log x_{j})^{2^{\kappa}-1}$ for some $x_{1},x_{2}\gg 1$ and
$\eta>\max_{j}\\{(\log x_{j})^{-1/100}\\}$. We then show that there exists
$x\geq\min\\{x_{1},x_{2}\\}^{\xi^{2}}$ such that
$\left|S(x,f_{1}f_{2})\right|\geq\xi x(\log x)^{2^{2\kappa}-1}$, where
$\xi=C\eta^{1+2^{\kappa+3}}$ for some absolute constant $C>0$.

###### Key words and phrases:

Mean values, Multiplicative functions, Halász’s theorem, Modular forms, Sums
of Fourier coefficients

###### 2010 Mathematics Subject Classification:

Primary 11F30; secondary 11F11, 11F12, 11M41

The research of Claire Frechette is supported by NSF grant DMS-2203042, and
the research of Alia Hamieh is supported by NSERC Discovery grant
RGPIN-2018-06313.

##  1\. Introduction

A central theme in analytic number theory is studying mean values of
multiplicative functions. One enduring quest in this area, which has garnered
extensive research over the past century and continues to attract significant
interest to date, revolves around understanding the asymptotic behaviour of
character sums. Let $\chi$ be a primitive Dirichlet character modulo $q$ and
let $S(x,\chi)=\sum_{n\leq x}\chi(n)$ be the associated character sum.
Trivially, we have the bound

$|S(x,\chi)|\leq\min(x,q).$

An important result proven independently by Pólya and Vinogradov [1, pages
135–137] asserts that

$\max_{x}\left|S(x,\chi)\right|\ll\sqrt{q}\log q,$

which was further improved to

$\max_{x}\left|S(x,\chi)\right|\ll\sqrt{q}\log\log q,$

by assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet
$L$-functions thanks to the work of Montgomery and Vaughan [15]. It follows
that

(1.1) $S(x,\chi)=o(x)$

when $x\geq q^{\frac{1}{2}+\epsilon}$. In 1957, Burgess established (1.1) in
the range $x>q^{\frac{1}{4}+o(1)}$ for any quadratic character $\chi$ when $q$
is prime. This was generalized by Burgess himself to any non-principal
character provided that $q$ is cubefree, and in the smaller range
$x>q^{\frac{3}{8}+o(1)}$ otherwise. Burgess’s range has not been substantially
improved over the last five decades, although it is widely believed that (1.1)
should hold in the wider range $x\gg_{\epsilon}q^{\epsilon}$. In 2001,
Granville and Soundararajan [4] proved that, for a primitive character
$\chi\mod q$, (1.1) holds in the range $\log x/\log\log q\rightarrow\infty$
assuming the GRH for $L(s,\chi)$. In [6], they also showed that this
asymptotic holds under the weaker assumption that “100%” of the zeros of
$L(s,f)$ up to height $\frac{1}{4}$ lie on the critical line. This work of
Granville and Soundararajan is of particular interest to us because it makes
strong connections between character sums and zeros of the associated
$L$-functions using general results from multiplicative number theory
pertaining to mean values of 1-bounded multiplicative functions in the
pretentious framework. In [6], the authors record various other interesting
observations on large character sums applying variants of Halasz’s theorem and
Lipschitz estimate as developed in [5]. For example, they show that if
$\chi_{1}$ and $\chi_{2}$ have large character sums then so does their product
$\chi_{1}\chi_{2}$. This result follows as a direct application of [6, Theorem
6.2] which asserts that if the partial sums of two completely multiplicative
1-bounded functions are large, then the partial sum of their product is large
as well. In this note, we extend this theorem to multiplicative functions
valued outside of the unit disk, but whose magnitude is bounded by a power of
the divisor function.

######  Theorem 1.1.

Let $\kappa$ be a positive integer. Let $f_{1}$ and $f_{2}$ be multiplicative
functions with $|f_{j}(n)|\leq\tau(n)^{\kappa}$ for all $n\in\mathbb{N}$.
Suppose that, for $x_{1},x_{2}\gg 1$, there exists $\eta>\max_{j}\\{(\log
x_{j})^{-1/100}\\}$ such that

(1.2) $\left|\sum_{n\leq x_{j}}f_{j}(n)\right|\geq\eta x_{j}(\log
x_{j})^{2^{\kappa}-1}.$

Then, with $\xi=C\eta^{1+2^{\kappa+3}}$ for some absolute constant $C>0$,
there exists $x\geq\min\\{x_{1},x_{2}\\}^{\xi^{2}}$ such that

$\left|\sum_{n\leq x}f_{1}(n)f_{2}(n)\right|\geq\xi x(\log
x)^{2^{2\kappa}-1}.$

Theorem 1.1 is proved following the general structure of the argument of [6,
Theorem 6.1], but we extend it to divisor-bounded multiplicative functions.
The technical results employed in the proofs of the various theorems in [6]
have their roots in the work of Halász [7, 8] and the subsequent works of
Montgomery [14], Tenenbaum [16], Granville and Soundararajan [5], among other
mathematicians, on mean values of multiplicative functions that take their
values in the complex unit disc. These results have been extended to a broad
class of divisor-bounded multiplicative functions by Granville, Harper and
Soundararajan in [3], Mangerel in [11, 12], and Matthiesen in [13]. Our work
in the present paper and our previous work [2] hinges upon such
generalizations and is motivated by trying to study a
$\mathrm{GL}_{2}$-analogue of (1.1). In fact, for a primitive cusp form
$g(z)=\sum_{n\geq 1}\lambda_{g}(n)n^{\frac{k-1}{2}}e^{2\pi inz}$ in
$S_{k}(1)$, Lamzouri [9, Corollary 1.2] proved that

(1.3) $\sum_{n\leq x}\lambda_{g}(n)=o(x\log x)$

holds in the range $\log x/\log\log k\rightarrow\infty$ assuming the GRH for
$L(s,g)$. He also proved unconditionally that this range in $x$ is best
possible [9, Corollary 1.4]. In [2], we extend the methods of [6] to show that
(1.3) holds in the range $x\geq k^{\epsilon}$ under a weaker assumption than
GRH. In connection with the main result of this paper, we derive the following
immediate consequence for large sums of Fourier coefficients of primitive cusp
forms, which can be viewed as a ${\rm GL}_{2}$-analogue of [6, Corollary 1.7].

######  Corollary 1.2.

Let $g\in S_{k}(r)$ and $h\in S_{\ell}(q)$ be primitive cusp forms whose
Fourier coefficients are denoted by $\lambda_{g}(n)$ and $\lambda_{h}(n)$,
respecctively. Suppose that

$\left|\sum_{n\leq x_{1}}\lambda_{g}(n)\right|\geq\eta x_{1}\log
x_{1}\quad\text{and}\quad\left|\sum_{n\leq x_{2}}\lambda_{h}(n)\right|\geq\eta
x_{2}\log x_{2},$

for some $x_{1},x_{2}\gg 1$ and $\eta>\max_{j}\\{(\log x_{j})^{-1/100}\\}$.
Then there exists $x\geq(\min(x_{1},x_{2}))^{\xi^{2}}$ such that

$\left|\sum_{n\leq x_{1}}\lambda_{g}(n)\lambda_{h}(n)\right|\geq\xi x(\log
x)^{3},$

where $\xi=C\eta^{17}$ for some absolute positive constant $C$.

The paper is organized as follows. In Section 2, we record versions of
Halász’s Theorem and Lipschitz Formula that are suitable for our setup. We
also prove some consequences of these results that are crucial for the proof
of our main theorem. In Section 3, we introduce a distance function for
divisor-bounded multiplicative functions, for which we prove a triangle
inequality-type result. We also derive a lower bound on sums of multiplicative
functions in terms of the distance function. These results form the main
ingredients of the proof of Theorem 1.1, which appears in Section 4.

Conventions and Notation. In this work, we adopt the following conventions and
notation. Given two functions $f(x)$ and $g(x)$ we write $f(x)\ll g(x)$,
$g(x)\gg f(x)$ or $f(x)=O(g(x))$ to mean there exists some positive constant
$M$ such that $|f(x)|\leq M|g(x)|$ for $x$ large enough. The notation
$f(x)\asymp g(x)$ is used when both estimates $f(x)\ll g(x)$ and $f(x)\gg
g(x)$ hold simultaneously. We write $f(x)=o(g(x))$ when $g(x)\neq 0$ for
sufficiently large $x$ and
$\displaystyle{\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0}$. The letter $p$
will be exclusively used to represent a prime number.

##  2\. Halász’s Theorem and Consequences

Let $f(n)$ be a multiplicative function satisfying

(2.1) $|f(n)|\leq\tau(n)^{\kappa}$

for all $n\in\mathbb{N}$, and let $L(s,f)$ be its associated Dirichlet
$L$-series given by

$L(s,f)=\sum_{n\geq 1}\frac{f(n)}{n^{s}},\quad\text{Re}(s)>1.$

We begin with presenting slightly modified versions of Halász’s Theorem and
its corollary, following the ideas of [3, Theorem 1.1, Corollary 1.2].

######  Theorem 2.1.

Let $f$ be a multiplicative function such than $|f(n)|\leq\tau(n)^{\kappa}$
for all $n\in\mathbb{N}$. Let $x$ be sufficiently large. Then for any $1\ll
T\leq x^{\frac{9}{10}}$, we have

$\frac{1}{x}\sum_{n\leq x}f(n)\ll\frac{1}{\log x}\int_{\frac{1}{\log
x}}^{1}\left(\max_{|t|\leq
T}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|\right)\frac{d\sigma}{\sigma}+O\left(\frac{(\log
T)^{2^{\kappa}}}{\log x}+\frac{(\log x)^{2^{\kappa}}}{T}\right).$

###### Proof.

The result follows from [3, Eqs. 2.1, 2.2, & 2.9] with $y=T^{2}$. ∎

To further proceed, let $1\leq T_{0}\ll(\log x)^{2^{\kappa}}$, and let
$M_{f}(x,T_{0})$ be a real number satisfying

(2.2) $\max_{|t|\leq T_{0}}\left|\frac{L\left(1+\frac{1}{\log
x}+it,f\right)}{1+\frac{1}{\log x}+it}\right|=e^{-M_{f}(x,T_{0})}(\log
x)^{2^{\kappa}}.$

Then, we obtain the following corollary.

######  Corollary 2.2.

Suppose $f$ is a multiplicative function such that
$|f(n)|\leq\tau(n)^{\kappa}$ for all $n\in\mathbb{N}$. Let $(\log
x)^{2^{\kappa-1}}\ll T_{0}\ll(\log x)^{2^{\kappa}}$ for some sufficiently
large $x$, and let $M_{f}(x,T_{0})$ be the quantity defined in (2.2). Then, we
have

$\frac{1}{x}\sum_{n\leq x}f(n)\ll(M_{f}(x,T_{0})+1)e^{-M_{f}(x,T_{0})}(\log
x)^{2^{\kappa}-1}+\frac{(\log x)^{2^{\kappa}-1}}{T_{0}}+\frac{(\log\log
x)^{2^{\kappa}}}{\log x}.$

###### Proof.

We apply Theorem 2.1 with $T=T_{0}\log x$ to get

$\frac{1}{x}\sum_{n\leq x}f(n)\ll\frac{1}{\log x}\int_{\frac{1}{\log
x}}^{1}\left(\max_{|t|\leq
T}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|\right)\frac{d\sigma}{\sigma}+O\left(\frac{(\log
x)^{2^{\kappa}-1}}{T_{0}}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}\right).$

Noting that
$|f(n)|\leq\tau(n)^{\kappa}=\tau_{2}(n)^{\kappa}\leq\tau_{2^{\kappa}}(n)$,
where $\tau_{\ell}$ is the $\ell$-fold divisor function (i.e.,
$\tau_{\ell}(n)$ is the coefficient of $n^{-s}$ in the Dirichlet series
representation of $\zeta(s)^{\ell}$), and that $\sigma\leq 1$, we have

$L(1+\sigma+it,f)\ll\zeta(1+\sigma)^{2^{\kappa}}\ll\left(\frac{1}{\sigma}\right)^{2^{\kappa}},$

and therefore,

$\max_{|t|\leq
T}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|=\max_{|t|\leq
T_{0}}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|+O\left(\frac{1}{\sigma^{2^{\kappa}}T_{0}}\right).$

It follows that

$\frac{1}{\log x}\int_{\frac{1}{\log x}}^{1}\left(\max_{|t|\leq
T}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|\right)\frac{d\sigma}{\sigma}=\frac{1}{\log
x}\int_{\frac{1}{\log x}}^{1}\left(\max_{|t|\leq
T_{0}}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|\right)\frac{d\sigma}{\sigma}+O\left(\frac{(\log
x)^{2^{\kappa}-1}}{T_{0}}\right),$

and so

$\frac{1}{x}\sum_{n\leq x}f(n)\ll\frac{1}{\log x}\int_{\frac{1}{\log
x}}^{1}\left(\max_{|t|\leq
T_{0}}\left|\frac{L(1+\sigma+it,f)}{1+\sigma+it}\right|\right)\frac{d\sigma}{\sigma}+O\left(\frac{(\log
x)^{2^{\kappa}-1}}{T_{0}}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}\right).$

Using the maximum modulus principle on the region $\\{u+iv:1+\frac{1}{\log
x}\leq u\leq 2,\;|v|\leq T_{0}\\}$, the rest of the proof follows by applying
the same argument employed in the proof of [3, Corollary 1.2] with very minor
changes. We remark here that the lower bound $T_{0}\gg(\log x)^{2^{\kappa-1}}$
is imposed so that we could use [3, Lemma 2.7] directly. In practice, we will
apply this result with $T_{0}\asymp(\log x)^{2^{\kappa}}$. ∎

Let $\phi_{f}(x,T_{0})$ be a real number in the range $|t|\leq T_{0}$ for
which the function

$t\mapsto\left|L\left(1+\frac{1}{\log x}+it,f\right)\right|$

attains its maximum, and let $N_{f}(x,T_{0})$ be a real number satisfying

(2.3) $\left|L\left(1+\frac{1}{\log
x}+i\phi_{f}(x,T_{0}),f\right)\right|=e^{-N_{f}(x,T_{0})}(\log
x)^{2^{\kappa}}.$

As mentioned in the proof of Corollary 2.2, we will mostly be interested in
the case $T_{0}\asymp(\log x)^{2^{\kappa}}$. We also note that

$\displaystyle e^{-M_{f}(x,T_{0})}(\log x)^{2^{\kappa}}$
$\displaystyle=\max_{|t|\leq T_{0}}\left|\frac{L(1+\frac{1}{\log
x}+it,f)}{1+\frac{1}{\log x}+it}\right|$ $\displaystyle\leq\max_{|t|\leq
T_{0}}\left|L(1+\frac{1}{\log x}+it,f)\right|$
$\displaystyle=e^{-N_{f}(x,T_{0})}(\log x)^{2^{\kappa}}.$

It then follows that $M_{f}(x,T_{0})\geq N_{f}(x,T_{0})$, and so
$(M_{f}(x,T_{0})+1)e^{-M_{f}(x,T_{0})}\leq(N_{f}(x,T_{0})+1)e^{-N_{f}(x,T_{0})}$.
Thus, we arrive at the following result.

######  Corollary 2.3.

Suppose $f$ is a multiplicative function such that
$|f(n)|\leq\tau(n)^{\kappa}$ for all $n\in\mathbb{N}$. Let $(\log
x)^{2^{\kappa-1}}\ll T_{0}\ll(\log x)^{2^{\kappa}}$ for some sufficiently
large $x$, and let $N_{f}(x,T_{0})$ be the quantity defined in (2.3). Then, we
have

$\frac{1}{x}\sum_{n\leq x}f(n)\ll(N_{f}(x,T_{0})+1)e^{-N_{f}(x,T_{0})}(\log
x)^{2^{\kappa}-1}+\frac{(\log x)^{2^{\kappa}-1}}{T_{0}}+\frac{(\log\log
x)^{2^{\kappa}}}{\log x}.$

We now generalize [5, Theorem 2a], which will become useful in the proof of
Proposition 2.7 at the end of this section.

######  Proposition 2.4.

Suppose $f$ is a multiplicative function such that
$|f(n)|\leq\tau(n)^{\kappa}$ for all $n\in\mathbb{N}$. Let
$\phi=\phi_{f}(x,(\log x)^{2^{\kappa}})$ as in (2.3) and assume that
$|\phi|\gg(\log x)^{2^{\kappa-1}}$. Then

$\frac{1}{x}\sum_{n\leq x}f(n)\ll\frac{(\log
x)^{2^{\kappa}-1}}{|\phi|-2}+\frac{\left(\log\log
x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})+1}}{(\log
x)^{1-\frac{2^{\kappa+1}}{\pi}}}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}.$

###### Proof.

We apply Corollary 2.3 with $T_{0}=|\phi|-2$ to get

$\displaystyle\frac{1}{x}\sum_{n\leq x}f(n)$
$\displaystyle\ll(N_{f}(x,|\phi|-2)+1)e^{-N_{f}(x,|\phi|-2)}(\log
x)^{2^{\kappa}-1}+\frac{(\log x)^{2^{\kappa}-1}}{|\phi|-2}+\frac{(\log\log
x)^{2^{\kappa}}}{\log x}$ $\displaystyle\ll\frac{1}{\log
x}\max_{|t|\leq|\phi|-2}\left|L(1+\frac{1}{\log
x}+it,f)\right|\log\left(\frac{(\log
x)^{2^{\kappa}}}{\max_{|t|\leq|\phi|-2}\left|L(1+\frac{1}{\log
x}+it,f)\right|}\right)$ $\displaystyle\hskip 144.54pt+\frac{(\log
x)^{2^{\kappa}-1}}{|\phi|-2}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}.$

By [3, Lemma 4.1], we have

$\displaystyle L\left(1+\frac{1}{\log x}+it,f\right)$ $\displaystyle\ll(\log
x)^{\frac{2^{\kappa+1}}{\pi}}\left(\frac{\log x}{1+|t-\phi|\log x}+(\log\log
x)^{2}\right)^{2^{\kappa}(1-\frac{2}{\pi})}$ $\displaystyle\ll(\log
x)^{\frac{2^{\kappa+1}}{\pi}}\left(\log\log
x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})},$

for all $|t|\leq|\phi|-2$. It follows that

$\displaystyle\frac{1}{x}\sum_{n\leq x}f(n)$ $\displaystyle\ll\frac{1}{\log
x}(\log x)^{\frac{2^{\kappa+1}}{\pi}}\left(\log\log
x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})}\log\left(\frac{(\log
x)^{2^{\kappa}}}{\max_{|t|\leq|\phi|-2}\left|L(1+\frac{1}{\log
x}+it,f)\right|}\right)$ $\displaystyle\hskip 144.54pt+\frac{(\log
x)^{2^{\kappa}-1}}{|\phi|-2}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}$
$\displaystyle\ll\frac{\left(\log\log
x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})+1}}{(\log
x)^{1-\frac{2^{\kappa+1}}{\pi}}}+\frac{(\log
x)^{2^{\kappa}-1}}{|\phi|-2}+\frac{(\log\log x)^{2^{\kappa}}}{\log x},$

as required. ∎

Next, we require the following version of Lipschitz Theorem.

######  Theorem 2.5 (Lipschitz Theorem).

Suppose $f$ is a multiplicative function such that
$|f(n)|\leq\tau(n)^{\kappa}$. Let $\phi=\phi_{f}(x,(\log x)^{2^{\kappa}})$, as
given in (2.3). Then for all $1\leq w\leq x^{1/3}$ we have

$\displaystyle\left|\frac{1}{x}\sum_{n\leq
x}f(n)n^{-i\phi}-\frac{1}{x/w}\sum_{n\leq x/w}f(n)n^{-i\phi}\right|$
$\displaystyle\hskip 144.54pt\ll\log\left(\frac{\log x}{\log
ew}\right)\left(\frac{\log w+(\log\log x)^{2}}{\log
x}\right)^{\min\\{2^{\kappa}(1-\frac{2}{\pi}),1\\}}(\log x)^{2^{\kappa}-1}.$

###### Proof.

The proof is very similar to that of [3, Theorem 1.5], and the reader is
referred to their detailed exposition. ∎

######  Corollary 2.6.

Let $f$ and $\phi$ be as in Theorem 2.5. Then

$\frac{1}{x}\sum_{n\leq
x}f(n)=\frac{x^{i\phi}}{1+i\phi}\cdot\frac{1}{x}\sum_{n\leq
x}f(n)n^{-i\phi}+O(E_{\kappa}(x)),$

where

(2.4) $E_{\kappa}(x)=\begin{cases}(\log x)^{-1+4/\pi}(\log\log
x)^{5-8/\pi}\,\,\,\text{ if }\kappa=1\\\ (\log x)^{2^{\kappa}-2}(\log\log
x)^{3}\,\,\,\text{ if }\kappa>1.\end{cases}$

###### Proof.

We will show the equivalent statement that

$\frac{1}{x}\sum_{n\leq
x}f(n)n^{-i\phi}=\frac{1+i\phi}{x^{1+i\phi}}\sum_{n\leq
x}f(n)+O(|\phi|E_{\kappa}(x)).$

By partial summation, we have

$\displaystyle\frac{1}{x}\sum_{n\leq x}f(n)n^{-i\phi}$
$\displaystyle=\frac{1}{x}\int_{1}^{x}u^{-i\phi}d\left\\{\sum_{n\leq
x}f(n)\right\\}=\frac{1}{x^{1+i\phi}}\sum_{n\leq
x}f(n)+\frac{i\phi}{x}\int_{1}^{x}\frac{1}{u^{1+i\phi}}\sum_{n\leq u}f(n)du.$

We split the integral into two pieces as follows:

$\displaystyle\int_{1}^{x}\frac{1}{u^{1+i\phi}}\sum_{n\leq u}f(n)du$
$\displaystyle=\int_{1}^{x/(\log x)^{2}}\frac{1}{u^{1+i\phi}}\sum_{n\leq
u}f(n)du+\int_{x/(\log x)^{2}}^{x}\frac{1}{u^{1+i\phi}}\sum_{n\leq u}f(n)du.$

For the first integral, we apply the trivial bound

(2.5) $\sum_{n\leq x}|f(n)|\leq\sum_{n\leq x}\tau_{2^{\kappa}}(n)\ll x(\log
x)^{2^{\kappa}-1}$

(see [10] for example) to get

$\displaystyle\frac{i\phi}{x}\int_{1}^{x/(\log
x)^{2}}\frac{1}{u^{1+i\phi}}\sum_{n\leq u}f(n)du$
$\displaystyle\ll\frac{|\phi|}{x}\int_{1}^{x/(\log
x)^{2}}\frac{1}{u}\sum_{n\leq u}|f(n)|du$
$\displaystyle\leq\frac{|\phi|}{x}\int_{1}^{x/(\log x)^{2}}(\log
u)^{2^{\kappa}-1}\,du\ll|\phi|E_{\kappa}(x).$

Since $w=x/u$ is in the range of Theorem 2.5, the second integral is equal to

$\displaystyle\frac{i\phi}{x}\int_{x/(\log
x)^{2}}^{x}\left(\frac{1}{x^{1+i\phi}}\sum_{n\leq
x}f(n)+O\left(\log\left(\frac{\log x}{\log ew}\right)\left(\frac{\log
w+(\log\log x)^{2}}{\log
x}\right)^{\min\\{2^{\kappa}(1-\frac{2}{\pi}),1\\}}(\log
x)^{2^{\kappa}-1}\right)\right)du$ $\displaystyle=$
$\displaystyle\frac{i\phi}{x}\left(\frac{1}{x^{1+i\phi}}\sum_{n\leq
x}f(n)\right)\int_{x/(\log x)^{2}}^{x}du+\frac{i\phi}{x}\cdot O\left(\log\log
x\left(\frac{(\log\log x)^{2}}{\log
x}\right)^{\min\\{2^{\kappa}(1-\frac{2}{\pi}),1\\}}(\log
x)^{2^{\kappa}-1}\right)\int_{x/(\log x)^{2}}^{x}du$ $\displaystyle=$
$\displaystyle\frac{i\phi}{x^{1+i\phi}}\sum_{n\leq
x}f(n)+O(|\phi|E_{\kappa}(x)).$

Combining the two integrals gives the desired result. ∎

We now utilize Proposition 2.4 (when $|\phi|$ is large) and Corollary 2.6
(when $|\phi|$ is small) to get the proposition that will be used in our
argument.

######  Proposition 2.7.

Let $f$ be a multiplicative function such that $|f(n)|\leq\tau(n)^{\kappa}$,
and let $N=N_{f}(x,(\log x)^{2^{\kappa}})$ and $\phi=\phi_{f}(x,(\log
x)^{2^{\kappa}})$ for sufficiently large $x$. Then,

$\frac{1}{x}\sum_{n\leq x}f(n)\ll(\log
x)^{2^{\kappa}-1}\left(\frac{(N+1)e^{-N}}{1+|\phi|}+\frac{E_{\kappa}(x)}{(\log
x)^{2^{\kappa}-1}}\right),$

where $E_{\kappa}(x)$ is given in (2.4).

###### Proof.

Suppose that $|\phi|\geq\frac{1}{2}(\log x)^{2^{\kappa}}$. By Proposition 2.4,
we have

$\frac{1}{x}\sum_{n\leq x}f(n)\ll\frac{(\log x)^{2^{\kappa}-1}}{(\log
x)^{2^{\kappa}}}+\frac{\left(\log\log
x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})+1}}{(\log
x)^{1-\frac{2^{\kappa+1}}{\pi}}}+\frac{(\log\log x)^{2^{\kappa}}}{\log
x}\ll\frac{\left(\log\log x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})+1}}{(\log
x)^{1-\frac{2^{\kappa+1}}{\pi}}}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}.$

It follows that

$\displaystyle\frac{1}{x}\sum_{n\leq x}f(n)-(\log
x)^{2^{\kappa}-1}\frac{(N+1)e^{-N}}{1+|\phi|}$
$\displaystyle\ll\frac{\left(\log\log
x\right)^{2^{\kappa+1}(1-\frac{2}{\pi})+1}}{(\log
x)^{1-\frac{2^{\kappa+1}}{\pi}}}+\frac{(\log\log x)^{2^{\kappa}}}{\log x}\ll
E_{\kappa}(x).$

Hence, we may assume that $|\phi|\leq\frac{1}{2}(\log x)^{2^{\kappa}}$.
Applying Corollary 2.3 to the function $g(n)=f(n)n^{-i\phi}$ with
$T_{0}=\frac{1}{2}(\log x)^{2^{\kappa}}$ gives

$\displaystyle\frac{1}{x}\sum_{n\leq x}f(n)n^{-i\phi}=\frac{1}{x}\sum_{n\leq
x}g(n)$ $\displaystyle\ll\left(N_{g}\left(x,\frac{(\log
x)^{2^{\kappa}}}{2}\right)+1\right)e^{-N_{g}(x,\frac{1}{2}(\log
x)^{2^{\kappa}})}(\log x)^{2^{\kappa}-1}+\frac{(\log\log x)^{2^{\kappa}}}{\log
x}$ (2.6) $\displaystyle=(N+1)e^{-N}(\log x)^{2^{\kappa}-1}+\frac{(\log\log
x)^{2^{\kappa}}}{\log x}.$

The last line follows from the fact that $N_{g}(x,\frac{1}{2}(\log
x)^{2^{\kappa}})=N_{f}(x,(\log x)^{2^{\kappa}})=N$. Indeed, this can be
observed as following: Since $L(s,g)=L(s+i\phi,f)$, we have

$\displaystyle\max_{|t|<\frac{1}{2}(\log
x)^{2^{\kappa}}}\left|L\left(1+\frac{1}{\log x}+it,g\right)\right|$
$\displaystyle=\max_{|t|<\frac{1}{2}(\log
x)^{2^{\kappa}}}\left|L\left(1+\frac{1}{\log x}+i(t+\phi),f\right)\right|$
$\displaystyle=\max_{|t|<(\log x)^{2^{\kappa}}}\left|L\left(1+\frac{1}{\log
x}+it,f\right)\right|.$

Applying Corollary 2.6 and (2), we get

$\displaystyle\frac{1}{x}\sum_{n\leq x}f(n)$
$\displaystyle\ll\left|\frac{x^{i\phi}}{1+i\phi}\right|\left((N+1)e^{-N}(\log
x)^{2^{\kappa}-1}+\frac{(\log\log x)^{2^{\kappa}}}{\log
x}\right)+O(E_{\kappa}(x))$ $\displaystyle\ll\frac{(N+1)e^{-N}}{1+|\phi|}(\log
x)^{2^{\kappa}-1}+O\left(\frac{(\log\log x)^{2^{\kappa}}}{\log
x}\right)+O\left(E_{\kappa}(x)\right).$

The first error term is subsumed into the second one since it is smaller. ∎

Before concluding this section, we establish the following lemma, which will
play a pivotal role in Section 4.

######  Lemma 2.8.

Let $f$ be a multiplicative function such that $|f(n)|\leq\tau(n)^{\kappa}$
for all $n\in\mathbb{N}$. Let $N=N_{f}(x,(\log x)^{2^{\kappa}})$ and
$\phi=\phi_{f}(x,(\log x)^{2^{\kappa}})$ as in (2.3). Suppose that there
exists $x\gg 1$ and $\eta>(\log x)^{-1/100}$ such that

(2.7) $\left|\sum_{n\leq x}f(n)\right|\geq\eta x(\log x)^{2^{\kappa}-1}.$

Then,

$|\phi|\ll\frac{1}{\eta}\quad\text{and}\quad N\leq
2\log\left(\frac{1}{\eta}\right)+O(1).$

###### Proof.

By Proposition 2.7, we know that

$\left|\sum_{n\leq x}f(n)\right|\ll x(\log
x)^{2^{\kappa}-1}\left(\frac{(N+1)e^{-N}}{1+|\phi|}+\frac{E_{\kappa}(x)}{(\log
x)^{2^{\kappa}-1}}\right).$

Therefore, together with our assumption (2.7), we have

(2.8) $\eta\leq\frac{C(N+1)e^{-N}}{1+|\phi|}+\frac{CE_{\kappa}(x)}{(\log
x)^{2^{\kappa}-1}},$

for some absolute constant $C>0$. Writing the second term on the right-hand
side as $A_{\kappa}(x)$, the inequality (2.8) can be written as

$(1+|\phi|)(\eta-A_{\kappa}(x))\leq C(N+1)e^{-N}.$

Since $\eta\geq 2A_{\kappa}(x)$ for sufficiently large $x$, we see that

(2.9) $1+|\phi|\leq\frac{C(N+1)e^{-N}}{\eta-
A_{\kappa}(x)}\leq\frac{2C(N+1)e^{-N}}{\eta}.$

In particular, $|\phi|\ll\frac{1}{\eta}$ and $\eta\ll e^{-\frac{N}{2}}$. It
follows that $N\leq 2\log\left(\frac{1}{\eta}\right)+O(1)$ as desired. ∎

##  3\. Technical Results

While the notion of a distance between multiplicative functions makes most
sense in the context of functions taking values in the complex unit disc, we
adapt the standard definition (see for example [6, Section 2]) and set the
following notation for a distance function associated with multiplicative
functions $f$ satisfying (2.1):

(3.1) $\displaystyle\mathbb{D}^{2}(f,n^{it};x)=\sum_{p\leq
x}\frac{2^{\kappa}-\text{Re}(f(p)p^{-it})}{p}.$

The distance function $\mathbb{D}^{2}(f,n^{it};x)$ is related to the Dirichlet
series $L(s,f)$ via

$\left|L\left(1+\frac{1}{\log x}+it,f\right)\right|\asymp(\log
x)^{2^{\kappa}}\exp(-\mathbb{D}^{2}(f,n^{it};x)),$

which can be observed in the following way. We have

(3.2) $\displaystyle\left|L\left(1+\frac{1}{\log x}+it,f\right)\right|$
$\displaystyle=\exp\left(\sum_{p\leq
x}\frac{\text{Re}(f(p)p^{-it})}{p}+O(1)\right)$
$\displaystyle=\exp\left(\sum_{p\leq x}\frac{2^{\kappa}}{p}-\sum_{p\leq
x}\frac{2^{\kappa}-\text{Re}(f(p)p^{-it})}{p}+O(1)\right)$
$\displaystyle\asymp(\log x)^{2^{\kappa}}\exp\left(-\sum_{p\leq
x}\frac{2^{\kappa}-\text{Re}(f(p)p^{-it})}{p}\right)$ $\displaystyle=(\log
x)^{2^{\kappa}}\exp(-\mathbb{D}^{2}(f,n^{it};x)),$

where the first equality follows from [11, Lemma 2.2.15].

We note that this distance function is related to $N=N_{f}(x,(\log
x)^{2^{\kappa}})$, defined in (2.3), in the following way:

(3.3) $e^{-N}=\frac{1}{(\log x)^{2^{\kappa}}}\left|L(1+\frac{1}{\log
x}+i\phi,f)\right|\asymp\exp(-\mathbb{D}^{2}(f,n^{i\phi};x)).$

It is known that the standard distance function between 1-bounded
multiplicative functions satisfies a triangle inequality. While such a result
does not hold in our setting, we next establish a triangle-type inequality
satisfied by the function $\mathbb{D}(f,n^{it};x)$ we introduced in (3.1).

######  Lemma 3.1.

For $j=1,2$, let $f_{j}$ be a multiplicative function such that
$|f_{j}(n)|\leq\tau(n)^{\kappa}$, and let $t_{j}$ be a real number. We have

$\mathbb{D}(f_{1},n^{it_{1}};x_{1})+\mathbb{D}(f_{2},n^{it_{2}};x_{2})\geq\frac{1}{\sqrt{2^{\kappa}}}\mathbb{D}(f_{1}f_{2},n^{i(t_{1}+t_{2})};\min\\{x_{1},x_{2}\\}),$

where the distance function on the right-hand side is defined with respect to
the condition $|f_{1}(n)f_{2}(n)|\leq\tau(n)^{2\kappa}$.

###### Proof.

For simplicity, let us denote
$\mathbb{D}_{j}=\mathbb{D}(f_{j},n^{it_{j}};x_{j})$ for $j=1,2$,
$x=\min\\{x_{1},x_{2}\\}$, and
$\mathbb{D}=\mathbb{D}(f_{1}f_{2},n^{i(t_{1}+t_{2})};x)$. Then, we have

$\displaystyle(\mathbb{D}_{1}+\mathbb{D}_{2})^{2}$ $\displaystyle=\sum_{p\leq
x_{1}}\frac{2^{\kappa}-\text{Re}(f_{1}(p)p^{-it_{1}})}{p}+\sum_{p\leq
x_{2}}\frac{2^{\kappa}-\text{Re}(f_{2}(p)p^{-it_{2}})}{p}$
$\displaystyle\hskip 72.26999pt+2\left(\sum_{p\leq
x_{1}}\frac{2^{\kappa}-\text{Re}(f_{1}(p)p^{-it_{1}})}{p}\right)^{1/2}\left(\sum_{p\leq
x_{2}}\frac{2^{\kappa}-\text{Re}(f_{2}(p)p^{-it_{2}})}{p}\right)^{1/2}$
$\displaystyle\geq\sum_{p\leq
x}\frac{2^{\kappa+1}-\text{Re}(f_{1}(p)p^{-it_{1}})-\text{Re}(f_{2}(p)p^{-it_{2}})}{p}$
$\displaystyle\hskip 72.26999pt+2\left(\sum_{p\leq
x}\frac{2^{\kappa}-\text{Re}(f_{1}(p)p^{-it_{1}})}{p}\right)^{1/2}\left(\sum_{p\leq
x}\frac{2^{\kappa}-\text{Re}(f_{2}(p)p^{-it_{2}})}{p}\right)^{1/2}$
$\displaystyle\geq\sum_{p\leq
x}\frac{2^{\kappa}}{p}\left[2-\text{Re}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)-\text{Re}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)\right]$
$\displaystyle\hskip 72.26999pt+\sum_{p\leq
x}\frac{2^{\kappa}}{p}\sqrt{2\left(1-\text{Re}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)\right)}\sqrt{2\left(1-\text{Re}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)\right)}$
$\displaystyle\geq\sum_{p\leq
x}\frac{2^{\kappa}}{p}\left[2-\text{Re}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)-\text{Re}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)+\left|\text{Im}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)\text{Im}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)\right|\right],$

where the last inequality follows from
$\left|\frac{f_{j}(p)}{2^{\kappa}}p^{-it_{j}}\right|\leq 1$ for $j=1,2$. This
fact also gives us

$\displaystyle
2-\text{Re}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)-\text{Re}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)+\left|\text{Im}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)\text{Im}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-i\phi_{2}}\right)\right|$
$\displaystyle\hskip 72.26999pt\geq
1-\text{Re}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)\text{Re}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)+\text{Im}\left(\frac{f_{1}(p)}{2^{\kappa}}p^{-it_{1}}\right)\text{Im}\left(\frac{f_{2}(p)}{2^{\kappa}}p^{-it_{2}}\right)$
$\displaystyle\hskip
72.26999pt=1-\text{Re}\left(\frac{f_{1}(p)f_{2}(p))}{2^{2\kappa}}p^{-i(t_{1}+t_{2}})\right).$

It then follows that

$\displaystyle(\mathbb{D}_{1}+\mathbb{D}_{2})^{2}$
$\displaystyle\geq\sum_{p\leq
x}\frac{2^{\kappa}}{p}\left(1-\text{Re}\left(\frac{f_{1}(p)f_{2}(p))}{2^{2\kappa}}p^{-i(\phi_{1}+\phi_{2}})\right)\right)$
$\displaystyle=\frac{1}{2^{\kappa}}\sum_{p\leq
x}\frac{2^{2\kappa}-\text{Re}(f_{1}(p)f_{2}(p)p^{-i(\phi_{1}+\phi_{2})})}{p}=\frac{1}{2^{\kappa}}\mathbb{D}^{2},$

as desired. ∎

To conclude this section, we exhibit lower bounds of sums of multiplicative
functions in terms of the distance function. More precisely, we show that if
the distance $\mathbb{D}(f,n^{i\psi};x)$ is small then the partial sums of $f$
get large in suitable ranges.

######  Proposition 3.2.

Let $f$ be a multiplicative function satisfying $|f(n)|\leq\tau(n)^{\kappa}$
for all $n\in\mathbb{N}$. Let $\psi=\psi_{f}(x,(\log x)^{2^{\kappa}})$ be a
number in the range $|t|\leq(\log x)^{2^{\kappa}}$ where the maximum in (2.2)
is attained. Let $x\gg 1$ and set
$\lambda=\mathbb{D}^{2}(f,n^{i\psi};x)+\log(1+|\psi|)+c$ where $c$ is a
suitably large constant. Then, there exists $y\in[x^{\gamma},x]$, with
$\gamma=1/(\lambda e^{\lambda})$, such that

(3.4) $\displaystyle\left|\sum_{n\leq
y}f(n)\right|\gg\frac{\exp(-\mathbb{D}^{2}(f,n^{i\psi};x))}{1+|\psi|}y(\log
y)^{2^{\kappa}-1}.$

###### Proof.

The strategy of the proof is to produce contradictory upper and lower bounds
on the quantity

$\left|L\left(1+\frac{2\lambda}{\log x}+i\psi,f\right)\right|,$

under the assumption that (3.4) fails.

For the lower bound, we observe that

(3.5) $\displaystyle\begin{split}\left|L\left(1+\frac{2\lambda}{\log
x}+i\psi,f\right)\right|&=\exp\left(\sum_{p\leq
x^{\frac{1}{2\lambda}}}\frac{\text{Re}(f(p)p^{-i\psi})}{p}+O(1)\right)\\\
&\geq\exp\left(\sum_{p\leq
x^{\frac{1}{2\lambda}}}\frac{\text{Re}(f(p)p^{-i\psi})}{p}-\sum_{x^{\frac{1}{2\lambda}}<p\leq
x}\frac{2^{\kappa}-\text{Re}(f(p)p^{-i\psi})}{p}+O(1)\right)\\\
&\geq\exp\left(\sum_{p\leq
x}\frac{\text{Re}(f(p)p^{-i\psi})}{p}-\sum_{x^{\frac{1}{2\lambda}}\leq p\leq
x}\frac{2^{\kappa}}{p}+O(1)\right)\\\
&=\left(2\lambda\right)^{-2^{\kappa}}\exp\left(\sum_{p\leq
x}\frac{\text{Re}(f(p)p^{-i\psi})}{p}+O(1)\right)\\\
&=\left(2\lambda\right)^{-2^{\kappa}}(\log
x)^{2^{\kappa}}\exp(-\mathbb{D}^{2}(f,n^{i\psi};x)).\end{split}$

We now move to the upper bound. Set $\delta:=\frac{2\lambda}{\log x}$. From
the Mellin transform representation of Dirichlet series, we get

(3.6)
$\left|L\left(1+\delta+i\psi,f\right)\right|\leq|1+\delta+i\psi|\left(\int_{1}^{\infty}\frac{1}{y^{2+\delta}}\left|\sum_{n\leq
y}f(n)\right|dy\right).$

Applying (2.5), the integral on the right-hand side of (3.6) is bounded by a
constant multiple of

$\int_{1}^{\infty}y^{-1-\delta}(\log y)^{2^{\kappa}-1}dy.$

Observe that

$\int y^{-1-\delta}(\log y)^{2^{\kappa}-1}\;dy=-\frac{P(\delta\log
y)}{\delta^{2^{\kappa}}y^{\delta}},$

where

$P(X):=(2^{\kappa}-1)!\sum_{k=0}^{2^{\kappa}-1}\frac{X^{k}}{k!}.$

For contradiction, we now assume that the upper bound

(3.7) $\left|\sum_{n\leq y}f(n)\right|\leq e^{-\lambda}y(\log y)^{2^{\kappa}}$

holds for all $y\in[x^{\gamma},x]$, and split the integral into three parts,
so that

$\int_{1}^{\infty}y^{-1-\delta}(\log
y)^{2^{\kappa}-1}dy=\left[-\frac{P(\delta\log
y)}{\delta^{2^{\kappa}}y^{\delta}}\right]^{x^{\gamma}}_{1}+\left[-e^{-\lambda}\frac{P(\delta\log
y)}{\delta^{2^{\kappa}}y^{\delta}}\right]^{x}_{x^{\gamma}}+\left[-\frac{P(\delta\log
y)}{\delta^{2^{\kappa}}y^{\delta}}\right]^{\infty}_{x}.$

Noting that $P(0)=(2^{\kappa}-1)!$ and recalling that
$\delta:=\frac{2\lambda}{\log x}$, we get

$\displaystyle\int_{1}^{\infty}y^{-1-\delta}(\log y)^{2^{\kappa}-1}dy$
$\displaystyle=\frac{(2^{\kappa}-1)!}{\delta^{2^{\kappa}}}-\frac{P(\delta\gamma\log
x)}{\delta^{2^{\kappa}}x^{\delta\gamma}}+e^{-\lambda}\frac{P(\delta\gamma\log
x)}{\delta^{2^{\kappa}}x^{\delta\gamma}}-e^{-\lambda}\frac{P(\delta\log
x)}{\delta^{2^{\kappa}}x^{\delta}}+\frac{P(\delta\log
x)}{\delta^{2^{\kappa}}x^{\delta}}$ $\displaystyle=$
$\displaystyle\frac{(2^{\kappa}-1)!(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}}-\frac{(\log
x)^{2^{\kappa}}P(2\gamma\lambda)}{(2\lambda)^{2^{\kappa}}e^{2\lambda\gamma}}+e^{-\lambda}\frac{(\log
x)^{2^{\kappa}}P(2\gamma\lambda)}{(2\lambda)^{2^{\kappa}}e^{2\lambda\gamma}}$
$\displaystyle\hskip 10.00002pt-e^{-\lambda}\frac{(\log
x)^{2^{\kappa}}P(2\lambda)}{(2\lambda)^{2^{\kappa}}e^{2\lambda}}+\frac{(\log
x)^{2^{\kappa}}P(2\lambda)}{(2\lambda)^{2^{\kappa}}e^{2\lambda}}$
$\displaystyle=$ $\displaystyle\frac{e^{-\lambda}(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}}\left((2^{\kappa}-1)!e^{\lambda}-\frac{e^{\lambda}P(2\gamma\lambda)}{e^{2\lambda\gamma}}+\frac{P(2\gamma\lambda)}{e^{2\lambda\gamma}}-\frac{P(2\lambda)}{e^{2\lambda}}+\frac{e^{\lambda}P(2\lambda)}{e^{2\lambda}}\right).$

We want to verify that the inside of the parenthesis is bounded as
$\lambda\rightarrow\infty$. Recall that $\gamma=\frac{1}{\lambda e^{\lambda}}$
and $\kappa\geq 1$, so that

$\frac{P(2\gamma\lambda)}{e^{2\lambda\gamma}}-\frac{P(2\lambda)}{e^{2\lambda}}+\frac{e^{\lambda}P(2\lambda)}{e^{2\lambda}}\ll
1.$

For the first two terms in the parenthesis, let $Q(X):=P(X)-(2^{\kappa}-1)!$.
We have

$(2^{\kappa}-1)!e^{\lambda}-\frac{e^{\lambda}P(2\gamma\lambda)}{e^{2\lambda\gamma}}=(2^{\kappa}-1)!e^{\lambda}(1-e^{-2\lambda\gamma})+\frac{e^{\lambda}Q(2e^{-\lambda})}{e^{2e^{-\lambda}}}=(2^{\kappa}-1)!e^{\lambda}(1-e^{-2\lambda\gamma})+O(1),$

since $Q(X)$ is a polynomial with vanishing constant term. It remains to bound
$(2^{\kappa}-1)!e^{\lambda}(1-e^{-2\lambda\gamma})$. The Taylor expansion of
$e^{X}$ for $X=o(1)$ gives

$(2^{\kappa}-1)!e^{\lambda}(1-e^{-2\lambda\gamma})=(2^{\kappa}-1)!e^{\lambda}(2\lambda\gamma+O((2\lambda\gamma)^{2}))=(2^{\kappa}-1)!e^{\lambda}(2e^{-\lambda}+O(e^{-2\lambda}))=O(1).$

We conclude that

$\left|L\left(1+\frac{2\lambda}{\log
x}+i\psi,f\right)\right|\ll\left|1+\frac{2\lambda}{\log
x}+i\psi\right|\frac{e^{-\lambda}(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}}.$

From the definition of $\lambda$, we have

$\lambda=\mathbb{D}^{2}(f,n^{i\psi};x)+\log(1+|\psi|)+c=O(\log\log x),$

so that

$\displaystyle\left|L\left(1+\frac{2\lambda}{\log x}+i\psi,f\right)\right|$
$\displaystyle\ll\left|1+i\psi\right|\frac{\exp(-\mathbb{D}^{2}(f,n^{i\psi};x))\exp(-\log\left|1+i\psi\right|-c)(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}}$
$\displaystyle=\frac{\exp(-\mathbb{D}^{2}(f,n^{i\psi};x))\exp(-c)(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}}.$

Comparing the upper bound from this last line with the conclusion of (3.5), we
get

$\frac{\exp(-\mathbb{D}^{2}(f,n^{i\psi};x))(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}}\ll\frac{\exp(-\mathbb{D}^{2}(f,n^{i\psi};x))e^{-c}(\log
x)^{2^{\kappa}}}{(2\lambda)^{2^{\kappa}}},$

which can be made false for $c$ large enough. It follows that the assumption
(3.7) is false, and that the result holds. ∎

##  4\. Proof of Theorem 1.1

Now, we will prove the main theorem. For $j=1,2$, we set
$\phi_{j}=\phi_{f_{j}}(x_{j},(\log x_{j})^{2^{\kappa}})$ and
$N_{j}=N_{f_{j}}(x_{j},(\log x_{j})^{2^{\kappa}})$ as in (2.3). We also set
$f=f_{1}f_{2}$ and $\phi=\phi_{f}(x,(\log x)^{2^{\kappa}})$ with
$x=\min\\{x_{1},x_{2}\\}$. By Lemma 3.1, we have

$\displaystyle\mathbb{D}^{2}(f_{1}f_{2},n^{i(\phi_{1}+\phi_{2})};x)$
$\displaystyle\leq
2^{\kappa}\left[\mathbb{D}(f_{1},n^{i\phi_{1}};x_{1})+\mathbb{D}(f_{2},n^{i\phi_{2}};x_{2})\right]^{2}$
$\displaystyle\leq
2^{\kappa+2}\max_{j}\left\\{\mathbb{D}^{2}(f_{j},n^{i\phi_{j}};x_{j})\right\\}.$

Together with (3.3), it follows that

$\exp\left(-\mathbb{D}^{2}(f_{1}f_{2},n^{i(\phi_{1}+\phi_{2})};x)\right)\geq\min_{j}\left\\{e^{-2^{\kappa+2}N_{j}}\right\\}.$

Applying Lemma 2.8 yields

(4.1)
$\exp\left(-\mathbb{D}^{2}(f_{1}f_{2},n^{i(\phi_{1}+\phi_{2})};x)\right)\geq\left(\eta^{2}\right)^{2^{\kappa+2}}=\eta^{2^{\kappa+3}}.$

On the other hand, by Proposition 3.2, there exists $y\in[x^{1/(\lambda
e^{\lambda})},x]$ such that

$\displaystyle\left|\sum_{n\leq y}f_{1}(n)f_{2}(n)\right|$
$\displaystyle\gg\frac{\exp(-\mathbb{D}^{2}(f,n^{i\psi};x))y(\log
y)^{2^{\kappa}-1}}{1+|\psi|}$ (4.2)
$\displaystyle\gg\frac{\exp(-\mathbb{D}^{2}(f,n^{i(\phi_{1}+\phi_{2})};x))y(\log
y)^{2^{\kappa}-1}}{1+|\phi_{1}+\phi_{2}|},$

where we recall that $\lambda=\mathbb{D}^{2}(f,n^{i\psi};x)+\log(1+|\psi|)+c$
for some sufficiently large constant $c$. Observe that the last inequality in
(4) follows from

$\displaystyle\frac{(\log
x)^{2^{\kappa}}\exp(-\mathbb{D}^{2}(f,n^{i\psi_{f}};x))}{\left|1+\frac{1}{\log
x}+i\psi_{f}\right|}$ $\displaystyle\asymp\frac{\left|L\left(1+\frac{1}{\log
x}+i\psi_{f},f\right)\right|}{\left|1+\frac{1}{\log
x}+i\psi_{f}\right|}=\max_{|t|\leq(\log
x)^{2^{\kappa}}}\frac{\left|L\left(1+\frac{1}{\log
x}+it,f\right)\right|}{\left|1+\frac{1}{\log x}+it\right|}$
$\displaystyle\geq\frac{\left|L\left(1+\frac{1}{\log
x}+i(\phi_{1}+\phi_{2}),f\right)\right|}{\left|1+\frac{1}{\log
x}+i(\phi_{1}+\phi_{2})\right|}$ $\displaystyle\asymp\frac{(\log
x)^{2^{\kappa}}\exp(-\mathbb{D}^{2}(f,n^{i(\phi_{1}+\phi_{2})};x))}{\left|1+\frac{1}{\log
x}+i(\phi_{1}+\phi_{2})\right|}.$

Now, putting (4.1) and (4) together gives

$\displaystyle\left|\sum_{n\leq y}f_{1}(n)f_{2}(n)\right|$
$\displaystyle\gg\frac{\eta^{2^{\kappa+3}}y(\log
y)^{2^{2\kappa}-1}}{1+|\phi_{1}+\phi_{2}|}.$

Finally, since $1+|\phi_{1}+\phi_{2}|\leq 1+|\phi_{1}|+|\phi_{2}|\ll 1/\eta$
by Lemma 2.8, we conclude that

$\displaystyle\left|\sum_{n\leq
y}f_{1}(n)f_{2}(n)\right|\gg\eta^{1+2^{\kappa+3}}y(\log y)^{2^{2\kappa}-1},$

which completes the proof by observing that

$\displaystyle\frac{1}{\lambda e^{\lambda}}$ $\displaystyle\geq e^{-2\lambda}$
$\displaystyle=\left(\exp(-\mathbb{D}^{2}(f,n^{i\psi};x)-\log(1+|\psi|)-c)\right)^{2}$
$\displaystyle\gg\left(\frac{\exp(-\mathbb{D}^{2}(f,n^{i(\phi_{1}+\phi_{2})};x))}{\left|1+\frac{1}{\log
x}+i(\phi_{1}+\phi_{2})\right|}\right)^{2}$
$\displaystyle\gg\left(\eta^{1+2^{\kappa+3}}\right)^{2},$

and setting $\xi=C\eta^{1+2^{\kappa+3}}$, for some absolute positive constant
$C$. $\square$

## Acknowledgements

This project originated from Women In Numbers 6 Research Workshop that took
place at Banff International Research Station in March 2023. The authors
express their utmost gratitude to the organizers for the invaluable
opportunity provided by the workshop. The third author would also like to
thank Oleksiy Klurman for answering a question related to this work.

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