# Weyl sums with multiplicative coefficients and joint equidistribution

Matteo Bordignon  KTH Royal Institute of Technology, Stockholm
and
Charles University, Faculty of Mathematics and Physics, Department of Algebra,
Sokolovská 83, 186 00 Praha 8, Czech Republic Department of Mathematics
<EMAIL_ADDRESS>,  Cynthia Bortolotto  Department of Mathematics,
ETH Zürich, Switzerland<EMAIL_ADDRESS>and  Bryce Kerr  UNSW
Canberra, School of Science<EMAIL_ADDRESS>
###### Abstract.

In this paper we generalize a result of Montgomery and Vaughan regarding
exponential sums with multiplicative coefficients to the setting of Weyl sums.
As applications, we establish a joint equidistribution result for roots of
polynomial congruences and polynomial values and obtain some new results for
mixed character sums.

###### Contents

  1. 1 Introduction
  2. 2 Main results
  3. 3 Applications
    1. 3.1 Joint equidistribution
    2. 3.2 Mixed character sums
    3. 3.3 Acknowledgements
    4. 3.4 Funding
  4. 4 Preliminary results
    1. 4.1 Reduction to bilinear forms
  5. 5 Sums over bilinear forms
    1. 5.1 The Vinogradov mean value theorem
    2. 5.2 Bounding bilinear forms
  6. 6 Proof of Theorem 1
  7. 7 Proof of Corollary 1
  8. 8 Short mixed character sums
  9. 9 On the correlation between roots of polynomial congruences and polynomial values
    1. 9.1 First reductions
    2. 9.2 Decomposing $D_{r}(s,\chi)$
    3. 9.3 Bounds on the Artin L-functions near $s=1$
    4. 9.4 Finishing the proof
  10. 10 Sharpness of Theorem 1

##  1\. Introduction

Let $A\geqslant 1$ and $f$ a multiplicative function satisfying
$|f(p)|\leqslant A$ for any prime $p$ and $\sum_{n\leq N}|f(n)|^{2}\leqslant
A^{2}N$ for all natural numbers $N$. For $\alpha\in\mathbb{R}$ set

$S(\alpha):=\sum_{1\leqslant n\leqslant N}f(n)e(\alpha n),$

where $e(x)=\exp(2\pi ix)$.

These sums first appear to be considered by Daboussi [11], who showed that if
$|\alpha-a/q|\leqslant 1/q^{2}$, $(a,q)=1$ and $3\leqslant q\leqslant(N/\log
N)^{\frac{1}{2}}$, then

$S(\alpha)\ll\frac{N}{(\log\log q)^{\frac{1}{2}}},$

and implied constant depending only on $A$.

This result was improved by Montgomery and Vaughan [32, Corollary 1] who show
that assuming $|\alpha-a/q|\leqslant 1/q^{2}$, $(a,q)=1$ and $2\leqslant
R\leqslant q\leqslant N/R$, we have

(1) $S(\alpha)\ll\frac{N}{\log N}+\frac{N(\log
R)^{\frac{3}{2}}}{R^{\frac{1}{2}}}.$

We refer the reader to [32, Section 7] for a demonstration that the term
$N/\log{N}$ is sharp.

The optimal dependence on $R$ in (1) is an open problem and has been the
subject of a number of works, see for example [1, 4], and it is expected the
estimate (1) may be improved to

(2) $S(\alpha)\ll\frac{N}{\log N}+\frac{N}{R^{\frac{1}{2}}}.$

Recently, Bretéche and Granville [5] have studied in detail the sums
$S(\alpha)$ on minor arcs. Their estimates suggest the following conjecture
(see [5, Equation 1.4])

$S(\alpha)\ll\frac{N}{\log
N}+\frac{N}{q^{\frac{1}{2}}(1+|\beta|x)},\quad\alpha=\frac{a}{q}+\beta,\quad(a,q)=1.$

We also note that [5] contains some nice applications to circle-method type
problems.

The estimate (1) has important applications to Dirichlet L-functions. For
example, Montgomery and Vaughan [32] have shown how it may be combined with
the Generalized Riemann Hypothesis (GRH) to obtain a sharp upper bound for
Dirichlet L-functions at the point $s=1$. One may also combine (1) with
progress around the Burgess bound to obtain unconditional variants of
Montgomery and Vaughan’s result and we refer the reader to [14, 18, 19] for
progress in this direction.

Since the work of Montgomery and Vaughan, exponential sums with multiplicative
coefficients have appeared in a number of different contexts and a variety of
techniques have been developed to facilitate the reduction to exponential sums
over bilinear forms. Some examples include Karatsuba’s work on short
Kloosterman sums [24], which has been refined by Korolev, see for example
[26]. Bourgain, Sarnak and Ziegler [2] have established a finite version of
Vinogradov’s bilinear sum inequality. Gong and Jia [15] have considered
shifted character sums with multiplicative coefficients and Korolev and
Shparlinski [27] dealt with sums over trace functions with multiplicative
coefficients.

In this paper, we revisit the approach of Montgomery and Vaughan and
generalise into the setting of sums of the form

(3) $\displaystyle\sum_{n\leqslant N}f(n)e(g(n)),$

where $g$ is a polynomial with real coefficients and $f$ is a multiplicative
function satisfying

$\displaystyle f(p)=O(1),\quad\sum_{n\leqslant
N}|f(n)|=O(N),\quad\sum_{n\leqslant N}|f(n)|^{2}=O(N(\log{N})^{A}),$

for any $A\geq 0$.

Problems of this sort have previously been considered by Jiang, Lü and Wang
[23], who showed that one may replace an assumption on the $\ell_{2}$ norm

$\sum_{n\leqslant N}|f(n)|^{2}\ll N,$

with an assumption of the form

$\sum_{\begin{subarray}{c}p\leqslant N\\\ p\
\text{prime}\end{subarray}}|f(p)||f(p+h)|\ll\frac{h}{\phi(h)}\frac{N}{(\log{N})^{2}}.$

Such a relaxation is significant in the context of $\text{GL}_{m}$
$L$-functions in the absence of progress towards the Ramanujan conjectures.

Matthiesen [29] has considered sums of the form (3) over polynomial
nilsequences with slightly weaker conditions on $f$ than Montgomery and
Vaughan. These results were later applied to linear correlations of
multiplicative functions [30]. We also mention the recent work [28] which
considers exponential sums with multiplicative functions over nilsequences on
average over short intervals.

##  2\. Main results

Our first result is as follows.

######  Theorem 1.

Let $A>0$ and $c>0$ be real numbers and $f$ be a multiplicative function
satisfying

(4) $\displaystyle|f(p)|\leqslant C\quad\text{for each prime $p$},$ (5)
$\displaystyle\sum_{n\leqslant N}|f(n)|=O(N),$ (6)
$\displaystyle\sum_{n\leqslant N}|f(n)|^{2}=O(N(\log{N})^{A}).$

Let $F$ be a polynomial of degree $d\geqslant 1$ with real coefficients given
by

$F(x)=\alpha_{d}x^{d}+\dots+\alpha_{1}x.$

Let $R\geqslant 1$ and suppose there exist integers $l,a,q$ with $1\leqslant
q\leqslant R$ and $1\leqslant\ell\leqslant d$, $(a,q)=1$ and

$\displaystyle\left|\alpha_{\ell}-\frac{a}{q}\right|\leqslant\frac{1}{Rq},$

Denote $C=\frac{A}{2r}$. Then for any $r>d(d+1)$, we have

$\displaystyle\sum_{1\leqslant n\leqslant N}f(n)e(F(n))$ $\displaystyle\ll
N\left(\frac{1}{(\log{N})^{1-C}}+(\log{N})^{C}\left(\frac{q}{N^{\ell}}+\frac{1}{q}\right)^{1/4r^{2}}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+(NR^{1/\ell})^{1/2},$

where the implied constant depends on $A$ and $r$. In particular, if we
suppose that

$(\log{N})^{4r^{2}}\leqslant q\leqslant\frac{N^{\ell}}{(\log{N})^{4r^{2}}},$

then

(7) $\displaystyle\sum_{1\leqslant n\leqslant
N}f(n)e(F(n))\ll\frac{N}{(\log{N})^{1-C}}.$

To demonstrate the precision of the above estimate, in Section 10 we prove
that for any polynomial $F$ and for all $N$, there exists $f=f_{F,N}$ such
that

$\displaystyle\left|\sum_{1\leqslant n\leqslant
N}f(n)e(F(n))\right|\geqslant\frac{1}{10}\frac{N}{\log{N}}.$

The proof of Theorem 1 follows the outline of Montgomery and
Vaughan [32] and starts with a combinatorial decomposition of multiplicative
functions based on Möbius inversion and reduces the problem to estimating
bilinear forms over polynomials with summation restricted to points under the
hyperbola. We then use Montgomery and Vaughan’s partition of the parabola into
disjoint rectangles to which techniques related to the Vinogradov Mean Value
theorem may be applied. It will be fundamental to develop a version of
Vinogradov Mean Value theorem for primes in large translated intervals, this
is Lemma 5. We should note that we introduced the general condition (6) with
the aiming of using Theorem 1 in the proof of Theorem 2.

##  3\. Applications

###  3.1. Joint equidistribution

As an application of Theorem 1, we prove a joint distribution result.
Throughout this section, we let $p\in{\mathbb{Z}}[x]$ be irreducible over
${\mathbb{Q}}$ of degree $e\geqslant 2$ and we consider the ratios $v/n$,
where $v$ are the roots of the polynomial $p$ modulo $n$

$\displaystyle p(v)\equiv 0\bmod{n}.$

Consider the sequence $(g_{k})_{k\geqslant 1}$ of these ratios so that the
corresponding denominators are in ascending order. Hooley proved, in [22,
Theorem 2], that this sequence is equidistributed in
${\mathbb{R}}/{\mathbb{Z}}$.

We now let $F(x)=\alpha_{1}x+\ldots+\alpha_{d}x^{d}\in{\mathbb{R}}[x]$ with
$d\geqslant 1$ and with an irrational coefficient and define

$\displaystyle A(F,p)_{k}=(g_{k},F(k^{\prime}))_{k\geqslant 1},$

where $g_{k}$ is as above and for $g_{k}=v/n$, $p(v)\equiv 0\bmod{n}$, we take
$k^{\prime}=n$. We prove the following

######  Theorem 2.

The sequence $(A(F,p)_{k})_{k\geqslant 1}$ is equidistributed in
$({\mathbb{R}}/{\mathbb{Z}})^{2}$.

This indicates that the sequence $(F(n))_{n\in\geqslant 1}$ is somehow not
correlated with the sequence $(g_{n})_{n\geqslant 1}.$

###  3.2. Mixed character sums

We next explain how Theorem 1 is related to sums considered by Enflo [12],
Chang [7] and Heath-Brown and Pierce [17]. Theorem 1 implies large exponential
sums must correspond to pretentious multiplicative functions.

######  Corollary 1.

With notation and conditions as in Theorem 1, let $r\geqslant d(d+1)$ and
suppose that

$\displaystyle\left|\sum_{1\leqslant n\leqslant
N}f(n)e(F(n))\right|\gg\frac{N}{(\log{N})^{1-A/2r}}.$

There exists an integer

$k\leqslant(\log{N})^{d(4r^{2}+4rA)},$

and a multiplicative character $\psi\mod{k}$ such that

$\displaystyle\sum_{1\leqslant n\leqslant
N}f(n)e(F(n))\ll(\log{N})^{4r^{2}+4rA}\max_{u\leqslant
N}\left|\sum_{n\leqslant u}\psi(n)f(n)\right|.$

Corollary 1 implies one may bound character sums mixed by polynomials by
reducing to pure character sums, we refer the reader to Section 8 for more
precise results.

######  Corollary 2.

Let $F(n)$ a polynomial of degree $d$ with real coefficients and $\chi$ a
primitive characters modulo $q$. Suppose that $\delta,\varepsilon$ satisfy

$\max_{\begin{subarray}{c}k\ll(\log{q})^{100d^{3}}\\\ \psi\mod{k}\\\
u\leqslant N\end{subarray}}\left|\sum_{n\leqslant
u}\psi(n)\chi(n)\right|\leqslant Nq^{-\varepsilon}\quad\text{provided}\quad
N\geqslant q^{\delta}.$

Then we have

(8) $\sum_{n\leqslant N}\chi(n)e(F(n))\ll\frac{N}{(\log N)^{1-1/d(d+1)}}.$

In particular, the estimate (8) holds under the following conditions:

  * •

For arbitrarily small $\delta$ assuming the Generalised Riemann hypothesis

  * •

For $\delta=1/3$ and arbitrary integer $q$, which follows from the Burgess
bound, see for example [16].

Let $\varepsilon>0$ be small. Enflo [12] has previously established that

$\sum_{n\leqslant N}\chi(n)e(F(n))\ll N^{1-\delta},\quad\text{provided}\quad
N\geqslant q^{1/4+\varepsilon},$

and we refer the reader to [7, 17] for quantitative improvements on Enflo’s
result. Chang [8] has shown

$\sum_{n\leqslant N}\chi(n)e(F(n))\ll N^{1-\delta},\quad\text{provided}\quad
N\geqslant q^{\varepsilon},$

provided $q$ is suitably smooth/powerful.

Corollary 2 provides some new instances where one may bound mixed character
sums nontrivially. We refer the reader to Section 8 for more details.

###  3.3. Acknowledgements

We would like to thank Emmanuel Kowalski for his helpful comments and
suggestions. We would also like to thank Alberto Perelli for a useful
conversation and his helpful comments.

###  3.4. Funding

Part of the research done by Matteo Bordignon was supported by OP RDE project
No. CZ.$02.2.69/0.0/0.0/18\\_053/0016976$ International mobility of research,
technical and administrative staff at the Charles University, part by an
Australian Mathematical Society Lift-off Fellowship and part was done while
the author was a PhD student of the University of New South Wales Canberra.

The third author would like to acknowledge the support of the Max Planck
Institute for Mathematics and the Australian Research Council (DE220100859).

##  4\. Preliminary results

###  4.1. Reduction to bilinear forms

We proceed in a similar fashion to [32, Section 2], which reduces the proof of
Theorem 1 to bounding bilinear forms under the hyperbola.

######  Lemma 1.

Let $f$ be a multiplicative function satisfying the conditions of Theorem 1,
$g$ be any real valued function and let $\epsilon>0$. Then for any integer $N$
we have

(9) $\displaystyle\begin{split}&\left|\sum_{1\leqslant n\leqslant
N}f(n)e(g(n))\right|\ll\\\
&\frac{N}{\log{N}^{1-\epsilon}}+\frac{1}{\log{N}}\left|\sum_{1\leqslant
np\leqslant N}f(n)f(p)(\log{p})e(g(np))\right|.\end{split}$

###### Proof.

We follow the argument from [32, Section 2] with some modifications to deal
with the condition

$\sum_{n\leqslant N}|f(n)|^{2}\ll N(\log{N})^{A}.$

Consider

$S=\sum_{n\leqslant N}f(n)e(g(n))\log N/n.$

Since

$S=\log{N}\sum_{n\leqslant N}f(n)e(g(n))-\sum_{n\leqslant
N}f(n)(\log{n})e(g(n)),$

it is sufficient to show

(10) $\displaystyle S\ll_{\epsilon}N(\log{N})^{\epsilon},$

and

(11) $\displaystyle\sum_{n\leqslant N}f(n)$
$\displaystyle(\log{n})e(g(n))\ll_{\epsilon}$
$\displaystyle\left|\sum_{1\leqslant np\leqslant
N}f(n)f(p)(\log{p})e(g(np))\right|+N(\log{N})^{\epsilon}.$

Let $r$ be a large real number and apply Hölder’s inequality, (5) and (6) to
get

$\displaystyle|S|^{2r}$ $\displaystyle\ll\left(\sum_{n\leqslant
N}(\log{N/n})^{2r}\right)\left(\sum_{n\leqslant
N}|f(n)|^{2}\right)\left(\sum_{n\leqslant N}|f(n)|\right)^{2r-2}$
$\displaystyle\ll\left(\sum_{n\leqslant
N}(\log{N/n})^{2r}\right)N^{2r-1}(\log{N})^{A}.$

Since

$\displaystyle\sum_{n\leqslant N}(\log{N/n})^{2r}$
$\displaystyle\ll\sum_{\begin{subarray}{c}j\\\ 1\leqslant 2^{j}\leqslant
N\end{subarray}}j^{2r}\sum_{N/2^{j+1}\leqslant n\leqslant N/2^{j-1}}1$
$\displaystyle\ll N\sum_{j\geqslant 1}\frac{j^{2r}}{2^{j}}\ll N,$

we obtain (10) after taking $r$ sufficiently large. Since

$\log n=\sum_{d|n}\Lambda(d),$

we have

$\displaystyle\sum_{n\leqslant N}f(n)(\log{n})e(g(n))$
$\displaystyle=\sum_{nm\leqslant N}\Lambda(m)f(nm)e(g(nm))$
$\displaystyle=\sum_{mn\leqslant N}\Lambda(m)f(n)f(m)e(g(nm))$
$\displaystyle+O\left(\sum_{mn\leqslant N}\Lambda(m)|f(nm)-f(n)f(m)|\right).$

Using (4) and (5)

$\displaystyle\sum_{mn\leqslant N}\Lambda(m)f(n)f(m)e(g(nm))$
$\displaystyle=\sum_{pn\leqslant N}(\log{p})f(p)f(n)e(g(nm))$
$\displaystyle+O\left(N\sum_{k\geqslant 2}\sum_{p^{k}\leqslant
N}\frac{|f(p^{k})|}{p^{k}}.\right)$ $\displaystyle=\sum_{pn\leqslant
N}(\log{p})f(p)f(n)e(g(nm))$ $\displaystyle+O(N),$

since the Cauchy-Schwarz inequality, (6)

$\displaystyle\sum_{k\geqslant 2}\sum_{p^{k}\leqslant
N}\frac{|f(p^{k})|}{p^{k}}\ll\left(\sum_{k\geqslant
2}\sum_{p}\frac{1}{p^{0.8k}}\right)\left(\sum_{n}\frac{|f(n)|^{2}}{n^{1.2}}\right)=O(1).$

Hence it remains to show

(12) $\displaystyle\sum_{mn\leqslant
N}\Lambda(m)|f(nm)-f(n)f(m)|=O(N(\log{N})^{\epsilon}).$

From (5)

$\displaystyle\sum_{mn\leqslant N}\Lambda(m)|f(nm)-f(n)f(m)|$
$\displaystyle\ll\sum_{k\geqslant 1}\sum_{p^{k}\leqslant
N}\sum_{\begin{subarray}{c}n\leqslant N/p^{k}\\\
p|n\end{subarray}}|f(p^{k})||f(n)|+|f(p^{k}n)|$
$\displaystyle\ll\sum_{k,j\geqslant 1}\sum_{p^{k+j}\leqslant
N}\sum_{\begin{subarray}{c}n\leqslant N/p^{k+j}\\\
(n,p)=1\end{subarray}}(|f(p^{k+j})|+|f(p^{k})||f(p^{j})|)|f(n)|$
$\displaystyle\ll N\sum_{k,j\geqslant
1}\sum_{p}\frac{|f(p^{k})||f(p^{j})|+|f(p^{k+j})|}{p^{k+j}}.$

Applying the Cauchy-Schwarz inequality, (6) and partial summation to summation
over $p,k,j$ as above, we establish (12) which completes the proof. ∎

We require a generalisation of [32, Section 3] for multiplicative functions
$f$ satisfying (6).

######  Lemma 2.

Let notation and conditions be as in Theorem 1. Suppose $s$ is a parameter and
for each $0\leqslant i\leqslant\log_{2}{N}$ write

(13) $\displaystyle
J_{i}=\min\left(i+1,\lfloor\log_{2}{N}\rfloor-i+1,\lfloor\log_{2}(64N/s)/2\rfloor\right).$

Define rectangles

(14)
$\displaystyle{\mathcal{R}}_{i}=(0,2^{i}]\times\left(\frac{N}{2^{i+1}},\frac{N}{2^{i}}\right],\quad
0\leqslant i\leqslant\log_{2}{N},$ (15)
$\displaystyle\begin{split}{\mathcal{R}}_{i,j,k}=&\left(\frac{2^{i+j}}{k},\frac{2^{i+j+1}}{2k-1}\right]\times\left(\frac{(k-1)N}{2^{i+j}},\frac{(2k-1)N}{2^{i+j+1}}\right],\\\
&0\leqslant i\leqslant\log_{2}{N},\quad 1\leqslant j\leqslant J_{i},\quad
2^{j-1}\leqslant k\leqslant 2^{j}.\end{split}$

Then each ${\mathcal{R}}_{i,j,k}$ is a rectangle of the form
$(P,P^{\prime}]\times(N,N^{\prime}],$ with $P,P^{\prime},N,N^{\prime}$
satisfying

$\displaystyle P^{\prime}-P\geqslant\frac{1}{4},\quad
N^{\prime}-N\geqslant\frac{1}{4},\quad(P^{\prime}-P)(N^{\prime}-N)\gg s.$

Let ${\mathcal{E}}$ denote the set of points $(p,n)$ satisfying $1\leqslant
pn\leqslant N$ which do not lie in any of the rectangles (14) or (15). Then,
for any fixed $\epsilon>0$, we have

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}}f(p)(\log{p})$ $\displaystyle
f(n)e(g(pn))\ll$ $\displaystyle\ll(\log
N)^{\epsilon}(N+(Ns)^{1/2}\log({2N/s})\log{s}).$

###### Proof.

Our proof is similar to that of [32, Section 3] with some minor modifications.
We first partition

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}}f(p)(\log{p})f(n)e(g(pn))\ll\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(p)||f(n)|(\log{p})$
$\displaystyle+\sum_{(p,n)\in{\mathcal{E}}_{2}}|f(p)||f(n)|(\log{p})+\sum_{(p,n)\in{\mathcal{E}}_{3}}|f(p)||f(n)|(\log{p}),$

where

$\displaystyle{\mathcal{E}}_{1}=\\{(p,n)\in{\mathcal{E}}:\leavevmode\nobreak\
pn\leqslant N,\leavevmode\nobreak\
\frac{N}{2^{i+1}}<n\leqslant\frac{N}{2^{i}},\leavevmode\nobreak\
J_{i}>i+1\\},$
$\displaystyle{\mathcal{E}}_{2}=\\{(p,n)\in{\mathcal{E}}:\leavevmode\nobreak\
pn\leqslant N,\leavevmode\nobreak\
\frac{N}{2^{i+1}}<n\leqslant\frac{N}{2^{i}},\leavevmode\nobreak\
J_{i}=\lfloor\log_{2}{N}\rfloor-i+1\\},$
$\displaystyle{\mathcal{E}}_{3}=\\{(p,n)\in{\mathcal{E}}:\leavevmode\nobreak\
pn\leqslant N,\leavevmode\nobreak\
\frac{N}{2^{i+1}}<n\leqslant\frac{N}{2^{i}},\leavevmode\nobreak\
J_{i}=\lfloor\log_{2}(64N/s)/2\rfloor\\}.$

Consider first ${\mathcal{E}}_{1}$. By Hölder’s inequality and (4)

$\displaystyle\left(\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(p)||f(n)|(\log{p})\right)^{2r}\ll$
$\displaystyle\left(\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(n)|\right)^{2r-2}\left(\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(n)|^{2}\right)\left(\sum_{(p,n)\in{\mathcal{E}}_{1}}(\log{p})^{2r}\right).$

For each prime $p$,

$\displaystyle\\#\\{n\ :\ (p,n)\in{\mathcal{E}}_{1}\\}\ll\frac{N}{p^{2}},$

so that

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{1}}(\log{p})^{2r}\ll
N\sum_{p}\frac{(\log{p})^{2r}}{p^{2}}\ll N,$

and for each $n$

$\\#\\{p\ :\ (p,n)\in{\mathcal{E}}_{1}\\}\ll 1.$

Hence by (5) and (6)

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(n)|$
$\displaystyle\ll\sum_{n\leqslant N}|f(n)|\ll N,$
$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(n)|^{2}$
$\displaystyle\ll\sum_{n\leqslant N}|f(n)|^{2}\ll N(\log{N})^{A}.$

Taking $r$ sufficiently large, the above estimates combine to give

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{1}}|f(p)||f(n)|(\log{p})\ll_{\epsilon}N(\log{N})^{\epsilon}.$

Consider next ${\mathcal{E}}_{2}$. By the Cauchy-Schwarz inequality

$\displaystyle\left(\sum_{(p,n)\in{\mathcal{E}}_{2}}|f(p)||f(n)|(\log{p})\right)^{2}\ll\sum_{(p,n)\in{\mathcal{E}}_{2}}|f(n)|^{2}\sum_{(p,n)\in{\mathcal{E}}_{2}}(\log{p})^{2}.$

If $(p,n)\in{\mathcal{E}}_{2}$ then $n\ll N^{1/2}$ and for fixed $n$, there
exists some $H$ such that

$\\{p\ ;\ (p,n)\in{\mathcal{E}}_{2}\\}\subseteq[H,H+N/n^{2}].$

Hence by the Brun-Titchmarsh theorem

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{2}}|f(p)||f(n)|(\log{p})\ll
N\sum_{n\leqslant N^{1/2}}\frac{|f(n)|^{2}}{n^{2}\log{(4N/n^{2})}},$

which combined with (6) and partial summation

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{2}}|f(n)|^{2}\ll\frac{N}{\log{N}}.$

For each $p$

$\\#\\{n\ :(p,n)\in{\mathcal{E}}_{2}\\}\ll 1,$

so that

$\sum_{(p,n)\in{\mathcal{E}}_{2}}(\log{p})^{2}\ll\sum_{p\ll N}(\log{p})^{2}\ll
N\log{N}.$

The above estimates combine to give

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{2}}|f(p)||f(n)|(\log{p})\ll N.$

Finally consider ${\mathcal{E}}_{3}$. By Hölder’s inequality

$\displaystyle\left(\sum_{(p,n)\in{\mathcal{E}}_{3}}|f(p)||f(n)|(\log{p})\right)^{2r}\ll$
$\displaystyle\left(\sum_{(N/q)^{1/2}\leqslant
n\leqslant(Nq)^{1/2}}|f(n)|^{2r/(2r+1)}(\log{p})\right)^{2r-1}$
$\displaystyle\times\left(\sum_{(N/q)^{1/2}\leqslant
p\leqslant(Nq)^{1/2}}\frac{\log{p}}{p}\right).$

If $(p,n)\in{\mathcal{E}}_{3}$ then

$\displaystyle\left(\frac{N}{s}\right)^{1/2}\leqslant p\leqslant(Ns)^{1/2},$

and for each $p$, there exists some $H$ such that

$\\#\\{n\ :\ (p,n)\in{\mathcal{E}}_{3}\\}\subseteq[H,H+O((Ns)^{1/2}/p)],$

and for each $n$ there exists some $H$ such that

$\\#\\{p\ :\ (p,n)\in{\mathcal{E}}_{3}\\}\subseteq[H,H+O((Ns)^{1/2}/n)].$

Using the Brun-Titchmarsh inequality, we see that

$\displaystyle\left(\sum_{(p,n)\in{\mathcal{E}}_{3}}|f(p)||f(n)|(\log{p})\right)^{2r}\ll(Ns)^{1/2}$
$\displaystyle\times\left(\sum_{(N/s)^{1/2}\leqslant
n\leqslant(Ns)^{1/2}}|f(n)|^{2r/(2r+1)}\frac{\log{2N/n}}{n\log{2Ns/n^{2}}}\right)^{2r-1}$
$\displaystyle\times\left(\sum_{(N/s)^{1/2}\leqslant
p\leqslant(Ns)^{1/2}}\frac{\log{p}}{p}\right).$

We have

$\displaystyle\sum_{(N/s)^{1/2}\leqslant
p\leqslant(Ns)^{1/2}}\frac{\log{p}}{p}\ll\log{s},$

and Hölder’s inequality combined with (5) and (6) give

$\displaystyle\left(\sum_{(N/s)^{1/2}\leqslant
n\leqslant(Ns)^{1/2}}|f(n)|^{2r/(2r+1)}\frac{\log{2N/n}}{n\log{2Ns/n^{2}}}\right)^{2r-1}$
$\displaystyle\ll(\log(2N/s))^{2r-1}\left(\sum_{(N/s)^{1/2}\leqslant
n\leqslant(Ns)^{1/2}}\frac{|f(n)|}{n}\right)^{2r-2}$
$\displaystyle\times\left(\sum_{(N/s)^{1/2}\leqslant
n\leqslant(Ns)^{1/2}}\frac{|f(n)|^{2}}{n}\right)$
$\displaystyle\ll(\log(2N/s))^{2r-1}(\log{N})^{A}(\log{s})^{2r-1}.$

Which after taking $r$ sufficiently large gives

$\displaystyle\sum_{(p,n)\in{\mathcal{E}}_{3}}|f(p)||f(n)|(\log{p})\ll(Ns)^{1/2}(\log{N})^{\varepsilon}(\log{s})\log{(2N/s)}.$

from which the result follows. ∎

##  5\. Sums over bilinear forms

###  5.1. The Vinogradov mean value theorem

Given integers $r,d,V$ we let $J_{r,d}(V)$ count the number of solutions to
the system of equations

$\displaystyle v_{1}^{j}+\dots-v_{2r}^{j}=0,\quad 1\leqslant j\leqslant d,$

with variables satisfying

$1\leqslant v_{1},\dots,v_{2r}\leqslant V.$

We will use a consequence of Bourgain, Demeter and Guth’s work on the
Vinogradov mean value theorem, see [3, Section 5].

######  Lemma 3.

Assume $d\geqslant 2$ and $r>d(d+1)$. Then we have

$\displaystyle J_{r,d}(V)\ll_{k}V^{2r-d(d+1)/2}.$

Combining Lemma 3 with the fact that the Vinogradov system is translation
invariant, we obtain the following.

######  Corollary 3.

Let $d\geqslant 2$, $r>d(d+1)$ be integers and ${\mathcal{M}}(k),1\leqslant
k\leqslant K$ disjoint intervals satisfying

$\displaystyle{\mathcal{M}}(k)=(M^{\prime}(k),M^{\prime\prime}(k)],\quad
M^{\prime\prime}(k)-M^{\prime}(k)\leqslant Y.$

Then the number of solutions to the system of equations

(16) $\displaystyle n_{1}^{j}+\dots-n_{2r}^{j}=0,\quad
n_{i}\in{\mathcal{M}}(k),\quad 1\leqslant k\leqslant K,$

is bounded by

$O(KY^{2r-d(d+1)/2}).$

We will also require an estimate for the number of solutions to the Vinogradov
system with prime variables in translated intervals, here it will be
fundamental that the intervals will not be ’too short’. To obtain such a
result we need the following intermediary lemma that follows directly from
[21, Theorem 10], here appears clear why it is important that the intervals we
work with are quite large compared to their starting point. We use $L=\log P$.

######  Lemma 4.

Let $0<Q\leqslant c_{1}(k)L^{\sigma_{1}}$, $X\gg 1$ and

$S(X,P)=\sum_{\begin{subarray}{c}X<p\leqslant X+P\\\ p\equiv
t\pmod{Q}\end{subarray}}e(f(p))$

in which

$P\gg\frac{X}{(\log X)^{M}},$

for any $M\gg 1$ and

$f(x)=\frac{h}{q}x^{k}+\alpha_{1}x^{k-1}+\cdots+\alpha_{k},\quad(h,q)=1,$

the number $\alpha$ being real. Suppose that $L^{\sigma}<q\leqslant
P^{k}L^{-\sigma}$. For arbitrary $\sigma_{0}>0$, when $\sigma\leqslant
2^{6k}(\sigma_{0}+\sigma_{1}+1)$, we always have

$|S(X,P)|\leqslant c_{2}(k)\frac{P}{QL^{\sigma_{0}-M}}.$

This lemma allows us to prove the following estimate for the number of
solutions to the Vinogradov system with prime variables in large translated
intervals.

######  Lemma 5.

Let $d\geqslant 2$ be an integer and $X,Y\gg 1$. If $r>d(d+1)$, the number of
solutions to the equation

$\displaystyle p_{1}^{j}+\dots-p_{2r}^{j}=0,\quad 1\leqslant j\leqslant
d,\quad Y\leqslant p_{i}\leqslant Y+X,$ $\displaystyle X\gg\frac{Y}{(\log
Y)^{M}},\quad p_{i}\quad\text{prime},$

for any $M\gg 1$, is bounded by

$\displaystyle O\left(\frac{X^{2r-d(d+1)/2}}{(\log{X})^{2r}}\right).$

###### Proof.

The result follows in a straightforward way from [21, Theorem 16] re-defining
in part 1)

$S(\alpha_{k},\cdots,\alpha_{1})=\sum_{Y\leqslant p\leqslant Y+X}e(f(p)),\quad
f(x)=\alpha_{k}x^{k}+\cdots+\alpha_{1}x,$

and introducing the the two following changes which account for the
translation in the set of primes and optimize the range of $r$ in Hua’s
result. In part 3) of the proof we use our Lemma 4 instead of [21, Theorem
10], this is possible as $s_{1}$ in [21, Lemma 10.8] is of arbitrary size.
While in part 5) of the proof we need to substitute [21, Theorem 15] with [3,
Theorem 1]. Doing this we need to be careful to isolate, in [21, pag. 144],
$\left|S(\alpha_{k},\cdots,\alpha_{1})\right|^{d(d+1)-\epsilon}$ instead of
$\left|S(\alpha_{k},\cdots,\alpha_{1})\right|^{s_{0}-1}$ and, in [21, pag.
145], $\left|S(\alpha_{k},\cdots,\alpha_{1})\right|^{s-\epsilon}$ instead of
$\left|S(\alpha_{k},\cdots,\alpha_{1})\right|^{s-1}$. Here $\epsilon>0$ it is
such that $s=d(d+1)+2\epsilon$. ∎

###  5.2. Bounding bilinear forms

It is well known that one may use Lemma 3 to estimate bilinear forms with Weyl
sums over rectangles. We next show how one may obtain sharper results by
averaging bilinear forms over a family of disjoint rectangles.

######  Lemma 6.

Let $F\in{\mathbb{R}}[X]$ be a polynomial of degree $d\geqslant 2$ of the form

$F(x)=\alpha_{d}x^{d}+\dots+\alpha_{1}x.$

Let $M,N$ be integers and $\alpha(n),\beta(m)$ two sequences of complex
numbers satisfying

$\displaystyle|\beta(m)|\leqslant 1,$

and

(17) $\displaystyle\sum_{n\leqslant N}|\alpha(n)|\ll N,\quad\sum_{n\leqslant
N}|\alpha(n)|^{2}\ll N(\log{N})^{A},$

for $A\geqslant 0$. For $1\leqslant k\leqslant K$ let

(18) $\displaystyle{\mathcal{R}}(k)={\mathcal{L}}(k)\times{\mathcal{M}}(k),$

be a rectangle of the form

${\mathcal{L}}(k)=(Q^{\prime}(k),Q^{\prime\prime}(k)],\quad{\mathcal{M}}(k)=(M^{\prime}(k)\times
M^{\prime\prime}(k)],$

such that ${\mathcal{L}}(k)\subseteq(0,Q)$, with $Q\gg 1$, are disjoint and
satisfy

(19) $Q^{\prime\prime}(k)-Q^{\prime}(k)\leqslant X,$

or

(20) $Q^{\prime\prime}(k)-Q^{\prime}(k)=X\gg\frac{Q^{\prime}(k)}{(\log
Q^{\prime}(k))^{M}}$

and ${\mathcal{M}}(k)\subseteq(0,M]$ are disjoint and satisfy

$M^{\prime\prime}(k)-M^{\prime}(k)\leqslant Y,\quad
M^{\prime\prime}(k)\leqslant 2M^{\prime}(k),$

and $K\ll M$. Let $I$ denote the sum

(21) $\displaystyle
I=\sum_{k=1}^{K}\sum_{\begin{subarray}{c}(p,n)\in{\mathcal{R}}(k)\end{subarray}}\alpha(n)\beta(p)e(F(np)).$

Let $R\geqslant 1$ and suppose there exists $q\leqslant R$ and
$1\leqslant\ell\leqslant d$ such that

(22)
$\displaystyle\left|\alpha_{\ell}-\frac{a}{q}\right|\leqslant\frac{1}{qR},$

for some $(a,q)=1$. For $r>d(d+1)$, we have

$\displaystyle I^{4r^{2}}$ $\displaystyle\ll
m(X)(\log{M})^{2rA}M^{4r^{2}}\left(\frac{X}{\log{X}}\right)^{4r^{2}}\left(\frac{M}{Y}\right)^{d(d-1)/2}$
$\displaystyle\left(\frac{Q}{X}\right)^{d(d-1)/2}\left(\frac{q}{XQ^{\ell-1}YM^{\ell-1}}+\frac{1}{XQ^{\ell-1}}+\frac{1}{YM^{\ell-1}}+\frac{1}{q}\right),$

where

$m(x)=\begin{cases*}(\log{x})^{4r}&{when} \eqref{eq:cond1} {holds},\\\
1&{when} \eqref{eq:cond2} {holds}.\end{cases*}$

###### Proof.

Recall that

$F(x)=\alpha_{d}x^{d}+\dots+\alpha_{1}x,$

and define

(23) $\displaystyle
S_{k}=\sum_{\begin{subarray}{c}(p,n)\in{\mathcal{R}}(k)\end{subarray}}\alpha(n)\beta(p)e(F(np)),$

so that

(24) $\displaystyle I=\sum_{k=1}^{K}S_{k}.$

Fix some $1\leqslant k\leqslant K$ and consider (23). Recalling (18), by
Hölder’s inequality, for any integer $r\geqslant 1$

$\displaystyle S_{k}\leqslant$
$\displaystyle\left(\sum_{n\in{\mathcal{M}}(k)}|\alpha(n)|^{2r/(2r-1)}\right)^{1-1/2r}\left(\sum_{n\in{\mathcal{M}}(k)}\left|\sum_{p\in{\mathcal{L}}(k)}\beta(p)e(F(np))\right|^{2r}\right)^{1/2r}.$

After another application of Hölder’s inequality

$\displaystyle I^{2r}\leqslant$ $\displaystyle\left(\sum_{1\leqslant
k\leqslant
K}\sum_{n\in{\mathcal{M}}(k)}|\alpha(n)|^{2r/(2r-1)}\right)^{2r-1}\sum_{1\leqslant
k\leqslant
K}\sum_{n\in{\mathcal{M}}(k)}\left|\sum_{p\in{\mathcal{L}}(k)}\beta(p)e(F(np))\right|^{2r}.$

Hence from assumptions on the ${\mathcal{M}}(k)$

$\displaystyle I^{2r}\ll\left(\sum_{n\leqslant
M}|\alpha(n)|^{2r/(2r-1)}\right)^{2r-1}\sum_{1\leqslant k\leqslant
K}\sum_{n\in{\mathcal{M}}(k)}\left|\sum_{p\in{\mathcal{L}}(k)}\beta(p)e(F(np))\right|^{2r},$

which combined with (17) implies that

$\displaystyle I^{2r}\ll M^{2r-1}(\log{M})^{A}\sum_{1\leqslant k\leqslant
K}\sum_{n\in{\mathcal{M}}(k)}\left|\sum_{p\in{\mathcal{L}}(k)}\beta(p)e(F(np))\right|^{2r}.$

Write $\lambda=(\lambda_{1},\dots,\lambda_{d})$ and let $J_{k}(\lambda)$ count
the number of solutions to

$p_{1}^{j}+\dots-p_{2r}^{j}=\lambda_{j},\quad 1\leqslant j\leqslant d,\quad
p_{i}\in{\mathcal{L}}(k),\quad p_{i}\ \ \text{prime}.$

Note by assumptions on ${\mathcal{L}}(k)$, if
$p_{1},\dots,p_{2r}\in{\mathcal{L}}(k)$ satisfy

$p_{1}^{j}+\dots-p_{2r}^{j}=\lambda_{j},$

then

$\lambda_{j}\ll XQ^{j-1}.$

Expanding the $2r$-th power and interchanging summation gives

$\displaystyle I^{2r}\ll$ $\displaystyle M^{2r-1}(\log{M})^{A}$
$\displaystyle\times\sum_{1\leqslant k\leqslant
K}\sum_{\begin{subarray}{c}\lambda\\\ \lambda_{j}\ll
XQ^{j-1}\end{subarray}}J_{k}(\lambda)\left|\sum_{n\in{\mathcal{M}}(k)}e(\alpha_{d}\lambda_{d}n^{d}+\dots+\alpha_{1}\lambda_{1}n)\right|.$

After another application of Hölder’s inequality, we get

$\displaystyle I^{4r^{2}}\ll
M^{4r^{2}-2r}(\log{M})^{2rA}\left(\sum_{1\leqslant k\leqslant
K}\sum_{\lambda}J_{k}(\lambda)\right)^{2r-2}\left(\sum_{1\leqslant k\leqslant
K}\sum_{\lambda}J_{k}(\lambda)^{2}\right)$
$\displaystyle\quad\times\sum_{1\leqslant k\leqslant
K}\sum_{\begin{subarray}{c}\lambda_{j}\ll
XQ^{j-1}\end{subarray}}\left|\sum_{n\in{\mathcal{M}}(k)}e(\alpha_{d}\lambda_{d}n^{d}+\dots+\alpha_{1}\lambda_{1}n)\right|^{2r}.$

We have

$\displaystyle\sum_{1\leqslant k\leqslant
K}\sum_{\lambda}J_{k}(\lambda)\leqslant\sum_{1\leqslant k\leqslant
K}|\\{p_{1},\dots,p_{2r}\in{\mathcal{L}}(k)\\}|.$

Since for each $1\leqslant k\leqslant K$, ${\mathcal{L}}(k)$ is an interval of
length at most $X$, the Brun-Titshmarsh theorem implies

$\displaystyle\sum_{1\leqslant k\leqslant
K}\sum_{\lambda}J_{k}(\lambda)\ll\frac{KX^{2r}}{(\log{X})^{2r}}.$

The term

$\displaystyle\sum_{1\leqslant k\leqslant K}\sum_{\lambda}J_{k}(\lambda)^{2},$

counts the number of solutions to the system of equations

(25) $\displaystyle p_{1}^{j}+\dots-p_{4r}^{j}=0,\quad 1\leqslant j\leqslant
d,\quad p_{i}\in{\mathcal{L}}(k),\ \ \text{ $p_{i}$ prime},\quad 1\leqslant
k\leqslant K.$

Ignoring the condition that $p_{i}$ is prime, by Corollary 3 and Lemma 5,

$\displaystyle\sum_{1\leqslant k\leqslant
K}\sum_{\lambda}J_{k}(\lambda)^{2}\ll Km_{1}(X),$

where

$m_{1}(x)=\begin{cases*}X^{4r-d(d+1)/2}&{when} \eqref{eq:cond1} {holds},\\\
\frac{X^{4r-d(d+1)/2}}{(\log X)^{4r}}&{when} \eqref{eq:cond2}
{holds}.\end{cases*}$

since $r>d(d+1)$. Combined with the observations above, this implies

(26) $\displaystyle I^{4r^{2}}\ll
m(X)M^{4r^{2}-2r}(\log{M})^{2rA}K^{2r-1}\left(\frac{X}{\log{X}}\right)^{4r^{2}}\frac{I_{0}}{X^{d(d+1)/2}},$

where

$I_{0}=\sum_{1\leqslant k\leqslant K}\sum_{\begin{subarray}{c}\lambda_{j}\ll
XQ^{j-1}\end{subarray}}\left|\sum_{n\in{\mathcal{M}}(k)}e(\alpha_{d}\lambda_{d}n^{d}+\dots+\alpha_{1}\lambda_{1}n)\right|^{2r}.$

Let

$S(x)=\left(\frac{\sin{\pi x}}{\pi x}\right)^{2},$

so that

(27) $\displaystyle\widehat{S}(x)=\max\\{0,1-|x|\\},$

and

$S(x)\gg 1\quad\text{if}\quad|x|\leqslant\frac{1}{4}.$

There exists a constant $c_{0}$ such that

$\displaystyle I_{0}\ll$ $\displaystyle\sum_{1\leqslant k\leqslant
K}\sum_{\begin{subarray}{c}\lambda_{j}\ll XQ^{j-1}\\\
j\neq\ell\end{subarray}}\sum_{\lambda_{\ell}\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda_{\ell}}{XQ^{\ell-1}}\right)\left|\sum_{n\in{\mathcal{M}}(k)}e(\alpha_{d}\lambda_{d}n^{d}+\dots+\alpha_{1}\lambda_{1}n)\right|^{2r}.$

Expanding the $2r$-th power and interchanging summation

$\displaystyle I_{0}\ll$ $\displaystyle\sum_{\begin{subarray}{c}1\leqslant
k\leqslant K\\\ \mu_{j}\ll
YN^{j-1}\end{subarray}}L_{k}(\mu)\prod_{\begin{subarray}{c}j=1\\\
j\neq\ell\end{subarray}}^{d}\left|\sum_{\lambda\ll
XQ^{j-1}}e(\alpha_{j}\lambda\mu_{j})\right|\left|\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda_{\ell}}{XQ^{\ell-1}}\right)e(\alpha_{\ell}\lambda\mu_{\ell})\right|,$

with $\mu=(\mu_{1},\dots,\mu_{d})$ and $L_{k}(\mu)$ counts the number of
solutions to

$\displaystyle n^{j}_{1}+\dots-n_{2r}^{j}=\mu_{j},\quad
n_{i}\in{\mathcal{M}}(k),\quad 1\leqslant k\leqslant K.$

Using that

$L_{k}(\mu)\leqslant L_{k}(0),$

and applying Corollary 3, we obtain

$\displaystyle I_{0}\ll KY^{2r-d(d+1)/2}$
$\displaystyle\sum_{\begin{subarray}{c}\mu\\\ \mu_{j}\ll
YM^{j-1}\end{subarray}}\prod_{\begin{subarray}{c}j=1\\\
j\neq\ell\end{subarray}}^{d}\left|\sum_{\lambda\ll
XQ^{j-1}}e(\alpha_{j}\lambda\mu_{j})\right|\left|\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda}{Xq^{\ell-1}}\right)e(\alpha_{\ell}\lambda\mu_{\ell})\right|.$

Combining the inequality above with (26) we have

(28) $\displaystyle I^{4r^{2}}\ll
m(X)(\log{M})^{2rA}M^{4r^{2}}\left(\frac{X}{\log{X}}\right)^{4r^{2}}\frac{I_{1}}{(XY)^{d(d+1)/2}},$

where

$\displaystyle I_{1}=\sum_{\begin{subarray}{c}\mu_{j}\ll
YM^{j-1}\end{subarray}}\prod_{\begin{subarray}{c}j=1\\\
j\neq\ell\end{subarray}}^{d}\left|\sum_{\lambda\ll
XQ^{j-1}}e(\alpha_{j}\lambda\mu_{j})\right|\left|\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda}{Xq^{\ell-1}}\right)e(\alpha_{\ell}\lambda\mu_{\ell})\right|$

In $I_{1}$, we bound every term trivially except the one with index $\ell$ to
obtain

(29) $\displaystyle I_{1}$
$\displaystyle\ll\left(\frac{Y}{M}\right)^{d}\left(\frac{X}{Q}\right)^{d}(QM)^{d(d+1)/2}\frac{1}{XQ^{\ell-1}}\frac{1}{YM^{\ell-1}}$
(30) $\displaystyle\sum_{\mu\ll
YM^{\ell-1}}\left|\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda}{Xq^{\ell-1}}\right)e(\alpha_{\ell}\lambda\mu_{\ell})\right|.$

By Poisson summation

$\displaystyle\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda_{\ell}}{XQ^{\ell-1}}\right)e(\alpha_{\ell}\mu\lambda)=\frac{XQ^{\ell-1}}{c_{0}}\sum_{\lambda\in{\mathbb{Z}}}\widehat{S}\left(\frac{XQ^{\ell-1}(\lambda-\alpha_{\ell}\mu)}{c_{0}}\right),$

and hence from (27)

$\displaystyle\sum_{\mu\ll
YM^{\ell-1}}\left|\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda_{\ell}}{XQ^{\ell-1}}\right)e(\alpha_{\ell}\mu\lambda)\right|$
$\displaystyle\quad\quad\quad\quad\quad\ll XQ^{\ell-1}\sum_{\mu\ll
YM^{\ell-1}}\sum_{\lambda\in{\mathbb{Z}}}\max\left\\{0,1-\left|\frac{XQ^{\ell-1}(\lambda-\alpha_{\ell}\mu)}{c_{0}}\right|\right\\}$
$\displaystyle\quad\quad\quad\quad\quad\ll XQ^{\ell-1}\left|\left\\{\mu\ll
YM^{\ell-1}\ :\
\|\alpha_{\ell}\mu\|\leqslant\frac{100c_{0}}{XQ^{\ell-1}}\right\\}\right|.$

There exists some real number $\beta$ such that

$\displaystyle\left|\left\\{\mu\ll YM^{\ell-1}\ :\
\|\alpha_{\ell}\mu\|\leqslant\frac{100c_{0}}{XQ^{\ell-1}}\right\\}\right|$
$\displaystyle\quad\quad\quad\ll\left(1+\frac{YM^{\ell-1}}{q}\right)\left|\left\\{0\leqslant\mu\leqslant
q\ :\
\|\alpha_{\ell}\mu+\beta\|\leqslant\frac{100c_{0}}{XQ^{\ell-1}}\right\\}\right|.$

Recalling (22)

$\displaystyle\left|\left\\{\mu\ll YM^{\ell-1}\ :\
\|\alpha_{\ell}\mu\|\leqslant\frac{100c_{0}}{XQ^{\ell-1}}\right\\}\right|$
$\displaystyle\quad\quad\ll\left(1+\frac{YM^{\ell-1}}{q}\right)\left|\left\\{0\leqslant\mu\leqslant
q\ :\
\|\frac{a\mu}{q}+\beta\|\leqslant\frac{100c_{0}}{XQ^{\ell-1}}+\frac{1}{R}\right\\}\right|,$

which implies that

$\displaystyle\left|\left\\{\mu\ll YM^{\ell-1}\ :\
\|\alpha_{\ell}\mu\|\leqslant\frac{100c_{0}}{XQ^{\ell-1}}\right\\}\right|$
$\displaystyle\quad\quad\quad\quad\quad\quad\ll\left(1+\frac{YM^{\ell-1}}{q}\right)\left(1+\frac{q}{XQ^{\ell-1}}\right),$

and hence

$\displaystyle\sum_{\mu\ll
YM^{\ell-1}}\left|\sum_{\lambda\in{\mathbb{Z}}}S\left(\frac{c_{0}\lambda}{Xq^{\ell-1}}\right)e(\alpha_{\ell}\lambda\mu_{\ell})\right|\ll$
$\displaystyle
XQ^{\ell-1}YM^{\ell-1}\left(\frac{1}{YM^{\ell-1}}+\frac{1}{q}\right)\left(1+\frac{q}{XQ^{\ell-1}}\right).$

Combined with (28) and (29) gives

$\displaystyle I^{4r^{2}}\ll$ $\displaystyle
m(X)(\log{M})^{2rA}M^{4r^{2}}\left(\frac{X}{\log{X}}\right)^{4r^{2}}\left(\frac{M}{Y}\right)^{d(d-1)/2}\left(\frac{Q}{X}\right)^{d(d-1)/2}$
$\displaystyle\left(\frac{q}{XQ^{\ell-1}YM^{\ell-1}}+\frac{1}{XQ^{\ell-1}}+\frac{1}{YM^{\ell-1}}+\frac{1}{q}\right),$

which completes the proof. ∎

##  6\. Proof of Theorem 1

We apply Lemma 1 and Lemma 2 with

$s=\frac{q^{1/\ell}}{(\log{N})^{4}},$

to get

$\displaystyle\sum_{1\leqslant n\leqslant N}f(n)e(F(n))\ll$
$\displaystyle\frac{N}{\log{N}^{1+o(1)}}+(Nq^{1/\ell})^{1/2}+\sum_{0\leqslant
i\leqslant\log_{2}{N}}\frac{i}{\log{N}}S_{i}$ $\displaystyle+\sum_{0\leqslant
i\leqslant\log_{2}{N}}\frac{i}{\log{N}}\sum_{1\leqslant j\leqslant
J_{i}}S_{i,j},$

where

$\displaystyle
S_{i}=\sum_{(p,n)\in{\mathcal{R}}_{i}}\frac{\log{p}}{i}f(p)f(n)e(F(pn)),$

and

$\displaystyle S_{i,j}=\sum_{2^{j-1}\leqslant k\leqslant
2^{j}}\sum_{(p,n)\in{\mathcal{R}}_{i,j,k}}\frac{\log{p}}{i}f(p)f(n)e(F(pn)).$

Note if $(p,n)\in{\mathcal{R}}_{i}$ or $(p,n)\in{\mathcal{R}}_{i,j,k}$ then

$\frac{\log{p}}{i}\ll 1.$

By Lemma 6, observing that in this case condition (20) holds,

$\displaystyle S_{i}$
$\displaystyle\ll(\log{N})^{A/2r}\frac{N}{i}\left(\frac{q}{N^{\ell}}+\frac{1}{2^{\ell
i}}+\frac{2^{\ell i}}{N}+\frac{1}{q}\right)^{1/4r^{2}},$

which implies that

(31) $\displaystyle\sum_{0\leqslant
i\leqslant\log_{2}{N}}\frac{i}{\log{N}}S_{i}\ll\frac{N}{(\log{N})^{1-A/2r}}\left(1+\log{N}\left(\frac{q}{N^{\ell}}+\frac{1}{q}\right)^{1/4r^{2}}\right).$

Consider next the sums $S_{i,j}$. We apply Lemma 6 with parameters

$\displaystyle K=2^{j-1},\quad M=\frac{N}{2^{i}},\quad Q=2^{i+1},\quad
X=2^{i-j+1},\quad Y=32N2^{-i-j}.$

We first focus on the case when $j\geqslant\log i^{c}$, for a $c\gg 1$. In
this case we use Lemma 6 with condition (19), which easily gives

$\displaystyle
S_{i,j}\ll\frac{N}{i^{c_{1}}2^{j/8}}\left(\frac{q}{N^{\ell}}+\frac{1}{2^{\ell
i}}+\left(\frac{2^{i}}{N}\right)^{\ell}+\frac{1}{q}\right)^{1/4r^{2}},$

for $c_{1}\gg 1$. For fixed $i\leqslant\log_{2}{N}$, recalling that $J_{i}$ is
given by (13), we have

$\displaystyle\sum_{\log i^{c}\leqslant j\leqslant
J_{i}}S_{i,j}\ll\frac{N}{i^{c_{1}}}\left(\frac{q}{N^{\ell}}+\frac{1}{2^{\ell
i}}+\left(\frac{2^{i}}{N}\right)^{\ell}+\frac{1}{q}\right)^{1/4r^{2}},$

and hence

$\displaystyle\sum_{0\leqslant
i\leqslant\log_{2}{N}}\frac{i}{\log{N}}\sum_{\log i^{c}\leqslant j\leqslant
J_{i}}S_{i,j}$
$\displaystyle\ll\frac{N}{\log{N}}\sum_{i\leqslant\log_{2}{N}}\frac{1}{i^{c_{1}-1}}\left(\frac{q}{N^{\ell}}+\frac{1}{2^{\ell
i}}+\left(\frac{2^{i}}{N}\right)^{\ell}+\frac{1}{q}\right)^{1/4r^{2}}$
$\displaystyle\ll
N\left(\frac{1}{\log{N}}+\frac{q}{N^{\ell}}+\frac{1}{q}\right).$

Denote $C=\frac{A}{2r}$. We then focus on the case when $j\leqslant\log
i^{c}$. In this case we use can use Lemma 6 with condition (20), which easily
gives

$\displaystyle S_{i,j}$
$\displaystyle\ll\frac{1}{(i-j+1)^{1-C}}\frac{N}{2^{j(1-d(d-1)/4r^{2})}}$
$\displaystyle\left(\frac{2^{2j}q}{N^{\ell}}+\frac{2^{j}}{2^{\ell
i}}+2^{j}\left(\frac{2^{i}}{N}\right)^{\ell}+\frac{1}{q}\right)^{1/4r^{2}}.$

For fixed $i\leqslant\log_{2}{N}$, we have

$\displaystyle\sum_{1\leqslant j\leqslant\log
i^{c}}S_{i,j}\ll\frac{N}{i^{1-C}}\left(\frac{q}{N^{\ell}}+\frac{1}{2^{\ell
i}}+\left(\frac{2^{i}}{N}\right)^{\ell}+\frac{1}{q}\right)^{1/4r^{2}},$

and hence

$\displaystyle\sum_{0\leqslant
i\leqslant\log_{2}{N}}\frac{i}{\log{N}}\sum_{1\leqslant j\leqslant\log
i^{c}}S_{i,j}$
$\displaystyle\ll\frac{N}{\log{N}}\sum_{i\leqslant\log_{2}{N}}i^{C}\left(\frac{q}{N^{\ell}}+\frac{1}{2^{\ell
i}}+\left(\frac{2^{i}}{N}\right)^{\ell}+\frac{1}{q}\right)^{1/4r^{2}}$
$\displaystyle\ll
N\left(\frac{1}{(\log{N})^{1-C}}+\frac{q(\log{N})^{C}}{N^{\ell}}+\frac{(\log{N})^{C}}{q}\right).$

Combining the above estimates we complete the proof.

##  7\. Proof of Corollary 1

Suppose

$F(x)=\alpha_{1}x+\dots+\alpha_{d}x^{d}.$

By Dirichlet’s theorem, for each $1\leqslant\ell\leqslant d$, there exists
integers $r_{\ell},s_{\ell}$ with $(r_{\ell},s_{\ell})=1$ and

$\displaystyle r_{\ell}\leqslant\frac{N^{\ell}}{(\log{N})^{4r^{2}+4rA}},$

such that

$\displaystyle\left|\alpha_{\ell}-\frac{r_{\ell}}{s_{\ell}}\right|\leqslant\frac{(\log{N})^{4r^{2}+4rA}}{q_{\ell}N^{\ell}}.$

By Theorem 1, we may suppose for each $1\leqslant\ell\leqslant d$

$\displaystyle s_{\ell}\leqslant(\log{N})^{4r^{2}+4rA}.$

By partial summation, we have

$\left|\sum_{n\leqslant
N}f(n)e(F(n))\right|\ll(\log{N})^{4r^{2}+4rA}\max_{u\leqslant N}T(u),$

with

$T(u):=\sum_{n\leqslant u}f(n)e(F_{1}(n)),$

and

$F_{1}(x):=\frac{r_{d}}{s_{d}}x^{d}+\dots+\frac{r_{1}}{s_{1}}x.$

Defining $k:=lcm(s_{d},\cdots,s_{1})$, we have

$T(u)=\sum_{a\leqslant k}e(F_{1}(a))S(a),$

with

$S(a):=\sum_{\begin{subarray}{c}n\leqslant u\\\ n\equiv
a\bmod{k}\end{subarray}}f(n).$

Let $d=(a,k)$ and write

$a^{\prime}=\frac{a}{d},\quad k^{\prime}=\frac{k}{d},$

so that

$S(a)=\sum_{\begin{subarray}{c}n\leqslant u\\\ n\equiv
a^{\prime}\bmod{k^{\prime}}\\\ n\equiv
0\bmod{d}\end{subarray}}f(n)=\frac{1}{\phi(k^{\prime})}\sum_{\psi\mod
k^{\prime}}\overline{\psi}(a^{\prime})\sum_{\begin{subarray}{c}n\leqslant u\\\
n\equiv 0\bmod{d}\end{subarray}}\psi(n)f(n).$

If $(d,q)\neq 1$ then $S(a)=0$. If $(d,q)=1$, then

$|S(a)|\leqslant\frac{1}{\phi(k^{\prime})}\sum_{\psi\mod
k^{\prime}}\left|\sum_{n\leqslant u/d}\psi(n)f(n)\right|.$

Thus, observing that

$\operatorname{lcm}(s_{d},\cdots,s_{1})\leqslant\prod_{1\leqslant i\leqslant
d}s_{i}\leqslant(\log{N})^{d(4r^{2}+4rA)}$

we conclude the proof.

##  8\. Short mixed character sums

We next use Corollary 1 to show how one may estimate short mixed character
sums assuming GRH.

######  Corollary 4.

Let $F(n)$ a polynomial of degree $d$ with real coefficients given by

$F(x)=\alpha_{d}x^{d}+\dots+\alpha_{1}x$

and taken $\chi$ a primitive characters modulo $q$. Then assuming GRH,
uniformly for all primitive characters, polynomials $F$ of degree $d$ and

(32) $\log{N}\geqslant B\log\log{q},$

for a suitable fixed constant $B$, we have

$\left|\sum_{n\leqslant N}\chi(n)e(F(n))\right|\ll\frac{N}{(\log
N)^{1-\varepsilon}},$

for arbitrary $\varepsilon>0$.

We need the following lemma on convolution of Dirichlet characters, that
follows from Theorem 2 in [13] and Corollary 1.3 in [20], remembering that
there they have $u=\log x/\log y$.

######  Lemma 7.

Let $\chi$ be a non-principal character modulo $q$. Assuming the Generalised
Riemann hypothesis, for any $x$ such that

$\frac{\log x}{\log\log q}\geqslant B,$

for a sufficiently large constant $B$, we have

$\left|\sum_{n\leqslant x}\chi(n)\right|\ll x(\log q)^{-(1+o(1))\frac{\log
x\log\left(\frac{\log x}{2\log\log q}\right)}{2(\log\log q)^{2}}},$

We can thus prove Corollary 4 in a similar fashion to [18, Lemma 3] and [25,
Lemma 7].

###### Proof of Corollary 4.

Clearly we may suppose

$\left|\sum_{n\leqslant N}\chi(n)e(F(n))\right|\gg\frac{N}{(\log
N)^{1-\varepsilon}}.$

For any $\psi$ with modulus smaller than or equal to
$k:=lcm(s_{d},\cdots,s_{1})$, by Lemma 7, we have

${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\max_{u\leqslant
N}}\left|\sum_{n\leqslant u}\psi(n)\chi(n)\right|\ll N(\log
q)^{-(1+o(1))\frac{\log N\log\left(\frac{\log N}{2\log\log
q}\right)}{2(\log\log q)^{2}}}.$

Thus, by Corollary 1 and (32), we have

$\displaystyle\left|\sum_{n\leqslant N}\chi(n)e(F(n))\right|$
$\displaystyle\ll N(\log{x})^{d^{5}}(\log q)^{-(1+o(1))\frac{\log
N\log\left(\frac{\log N}{2\log\log q}\right)}{2(\log\log q)^{2}}}$
$\displaystyle\ll\frac{N}{(\log N)^{1-\varepsilon}},$

and this concludes the proof. ∎

##  9\. On the correlation between roots of polynomial congruences and
polynomial values

We now prove Theorem 2. Let $p\in{\mathbb{Z}}[x]$ be irreducible over
${\mathbb{Q}}$ of degree $e\geqslant 2$ and consider the ratios $v/n$, where
$v$ are the roots of the polynomial $p$ modulo $n$

$\displaystyle p(v)\equiv 0\bmod{n}.$

Define the sequence $(g_{k})_{k\geqslant 1}$ of these ratios so that the
corresponding denominators are in ascending order.

Hooley [22, Theorem 1] proved that $(g_{k})_{k\geqslant 1}$ is equidistributed
in ${\mathbb{R}}/{\mathbb{Z}}$. Even stronger than that, he showed

(33) $\displaystyle\lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leqslant
x}\left|\sum_{\begin{subarray}{c}v\in{\mathbb{Z}}/n{\mathbb{Z}}\\\ p(v)\equiv
0\bmod{n}\end{subarray}}e\left(\frac{hv}{n}\right)\right|=0,$

for every $h\in{\mathbb{Z}}\backslash\\{0\\}$. To prove Theorem 2, by the Weyl
equidistribution criteria, it suffices to prove that for any
$(h_{1},h_{2})\neq 0$, we have

$\displaystyle\sum_{n\leqslant
N}e\left(h_{1}F(n)\right)\sum_{\begin{subarray}{c}v\in{\mathbb{Z}}/n{\mathbb{Z}}\\\
{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}p}(v)\equiv
0\bmod{n}\end{subarray}}e\left(\frac{h_{2}v}{n}\right)=o(N).$

If $h_{2}\neq 0$ this is true by equation (33). Thus, the only cases that need
to be considered are $h_{2}=0$ and $h_{1}\neq 0$. If we let

(34)
$\displaystyle\varrho(n)=|\\{v\in{\mathbb{Z}}/n{\mathbb{Z}};{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}p}(v)\equiv{0}\bmod{n}\\}|,$

then the problem is therefore to show that

$\displaystyle\sum_{n\leqslant N}\varrho(n)e\left(F(n)\right)=o(N).$

We first prove some properties of $\varrho$.

######  Lemma 8.

The function $\varrho$ is multiplicative and satisfies:

  1. (1)

For all $\epsilon>0$, $\varrho(n)\leqslant_{\epsilon}n^{\epsilon}$, for
$n\rightarrow\infty$.

  2. (2)

There exists a constant $\lambda>0$ such that

$\sum_{n=1}^{N}\varrho(n)\sim\lambda N.$

  3. (3)

There exists a constant $A\geqslant 0$ such that

$\sum_{n=1}^{N}\varrho(n)^{2}\ll N(\log{N})^{A},$

for $N\geqslant 2$.

  4. (4)

There exists a constant $D\geqslant 1$ such that

$\varrho(mn)\leqslant D^{\operatorname{disc}(f)}\varrho(m)\varrho(n),$

for all $m,n\in{\mathbb{Z}}_{>0}$.

###### Proof.

The Chinese Remainder Theorem implies that $\varrho$ is a multiplicative
function. From Wirsing’s theorem [34, Satz 1] we conclude the second item. The
proof of the third one is standard (see, for e.g [10, Lemma 2.7]).

To conclude the fourth item, we observe that $\varrho(p^{\alpha})=\varrho(p)$
for all $p$ primes, $p\nmid\operatorname{disc}(f)$ and
$\alpha\in{\mathbb{Z}}_{\geqslant 1}$ and $\varrho(p^{\alpha})\leqslant D$ for
all $p$ primes and a constant $D>0$ (for a proof, see, for e.g. [10, Lemma
2.4]). We also note that if $p|\operatorname{disc}(f)$ and
$\varrho(p^{\alpha})\neq 0$ then $\varrho(p^{\beta})\neq 0$ for all
$\beta\leqslant\alpha$ and we can conclude the result by factoring $m$ and $n$
in primes. Note this also implies the first item, since if
$n=p_{1}^{\alpha_{1}}\dots p_{k}^{\alpha_{k}}$ then

$\varrho(n)\ll D^{k}\ll D^{\log{n}/{\log\log{n}}}.$

∎

From now on, we fix $r^{\ast}=2\max(d(d+1),A)$ and $B=4{r^{\ast}}^{2}+4rA+1$
and the constants $\lambda$, $D$ and $A$ as in Lemma 8.

From Dirichlet’s Theorem, we can find $a_{i}$ and $1\leqslant
q_{i}\leqslant\frac{N^{i}}{(\log{N})^{B}}$ coprime integers satisfying

(35)
$\left|\alpha_{i}-\frac{a_{i}}{q_{i}}\right|\leqslant\frac{(\log{N})^{B}}{q_{i}N^{i}},$

for $1\leqslant i\leqslant d$. We also let

$q=\operatorname{lcm}(q_{1},\ldots,q_{d}).$

######  Proposition 1.

If there exists an $l\in\\{1,\ldots,d\\}$ such that $q_{l}$ satisfies
$q_{l}\geqslant(\log{N})^{B}$ then

$\sum_{n=1}^{N}\varrho(n)e(F(n))\ll\frac{N}{(\log{N})^{1/2}}.$

###### Proof.

This follows directly from Theorem 1 applied to $f=\rho$, $r=r^{\ast}$ and
$R=N^{\ell}/(\log{N})^{B}$. ∎

We next focus on establishing the following result.

######  Proposition 2.

Suppose that $q_{i}<(\log{N})^{B}$, for all $1\leqslant i\leqslant d$. Then it
holds that

$\sum_{n=1}^{N}\varrho(n)e(F(n))\ll\frac{N}{q^{\frac{2}{d(d+1)}-\epsilon}}.$

Combining Proposition 2 with Proposition 1 one may deduce Theorem 2. Indeed,
since at least one of the $\alpha_{i}$ is irrational, as $N\rightarrow\infty$,
the integers $q_{i}$ in (35) must satisfy $q_{i}\rightarrow\infty$ which
implies

$\sum_{n=1}\varrho(n)e(F(n))=o(N).$

To prove Proposition 2 we will proceed with an algebraic approach. We split
the task in four subsections: first we make some reductions and provide a
proof of the proposition, conditional to further analysis of a different sum.
This new sum will be analysed using a Dirichlet series. In the next
subsection, we write the Dirichlet series that we want to analyse in terms of
better understood L-functions and in the third subsection we state some of its
properties. At last, we gather all the results and conclude the proof.

###  9.1. First reductions

We assume $q_{i}<(\log{N})^{B}$, for all $1\leqslant i\leqslant d$ and we set,
for $r|q$, and $Q\in{\mathbb{Z}}[x]$

$\displaystyle S_{r}(N,Q)=\sum_{\begin{subarray}{c}n=1\\\
(n,q)=r\end{subarray}}^{N}\varrho(n)e(Q(n)),$

and we omit $r$ from the notation when the sum is over all integers
$1\leqslant n\leqslant N$.

We can split $S(N,F)$ in two sums as follows

(36) $\displaystyle
S(N,F)=\sum_{r|q}S_{r}(N,F)=\sum_{\begin{subarray}{c}r|q\\\
r<q^{1/d(d+1)}\end{subarray}}S_{r}(N,F)+\sum_{\begin{subarray}{c}r|q\\\
r\geqslant q^{1/d(d+1)}\end{subarray}}S_{r}(N,F).$

We can bound trivially the second sum of the right hand side of equation (36)
using Lemma 8. Indeed,

$\displaystyle\left|\sum_{\begin{subarray}{c}r|q\\\ r\geqslant
q^{1/d(d+1)}\end{subarray}}S_{r}(N,F)\right|\leqslant\sum_{\begin{subarray}{c}r|q\\\
r\geqslant q^{1/d(d+1)}\end{subarray}}\sum_{\begin{subarray}{c}n=1\\\
(n,q)=r\end{subarray}}^{N}\varrho(n)$
$\displaystyle=\sum_{\begin{subarray}{c}r|q\\\ r\geqslant
q^{1/d(d+1)}\end{subarray}}\sum_{\begin{subarray}{c}n=1\\\
(n,q/r)=1\end{subarray}}^{N/r}\varrho(rn)$ (37)
$\displaystyle\ll_{\epsilon}\frac{N}{q^{1/d(d+1)-\epsilon}}.$

We turn our attention to $S_{r}(N,F)$ when $r\leqslant 1/d(d+1)$. Let
$b_{j}=\frac{a_{j}\cdot q}{q_{j}},$ for $1\leqslant j\leqslant d$,
$\tilde{F}(x)=\sum_{j=1}^{d}\frac{b_{j}}{q_{j}}x^{j}$ and
$Q(x)=F(x)-\tilde{F}(x)$. Observe that integration by parts yields

(38) $\displaystyle\begin{split}S_{r}(N,F)&=\
e\left(Q(N)\right)S_{r}(N,\tilde{F})\\\ &-2\pi
i\sum_{j=1}^{d}j\beta_{j}\int_{1}^{N}u^{j-1}e\left(Q(u)\right)S_{r}(u,\tilde{F})du,\end{split}$

where $\beta_{j}$ is defined by

(39) $\displaystyle\alpha_{j}=\frac{a_{j}}{q_{j}}+\beta_{j}.$

In particular, from (35)

(40) $\displaystyle|\beta_{j}|\leqslant\frac{(\log{N})^{B}}{q_{j}N^{j}}.$

The above reduces to analysing $S_{r}(N,\tilde{F})$ instead of $S_{r}(N,F)$.

Observe that since we are summing over multiples of $r$, we can rewrite
$e(\tilde{F}(n))$ as $e(\tilde{F_{r}}(n))$, where
$\tilde{F_{r}}(n):=\tilde{F}(rn)$, so that the latter is a periodic function
modulo $q/r$. We let $q/r=r^{\prime}$ and we decompose
$S_{r}(u,\tilde{F_{r}})$ using Dirichlet characters modulo $r^{\prime}$ as
follows:

(41)
$\begin{split}S_{r}(N,\tilde{F})&=\sum_{x\bmod{r^{\prime}}}e(\tilde{F}_{r}(x))\sum_{\begin{subarray}{c}m\leqslant
N/r\\\ (m,r^{\prime})=1\\\ m\equiv
x\bmod{r^{\prime}}\end{subarray}}\varrho(rm)\\\
&=\frac{1}{\varphi(r^{\prime})}\sum_{x\bmod{r^{\prime}}}e(\tilde{F}_{r}(x))\sum_{\chi\bmod{r^{\prime}}}\overline{\chi(x)}\sum_{m\leqslant
N/r}\chi(m)\varrho(rm).\end{split}$

We will prove:

######  Proposition 3.

Let $\chi$ be a Dirichlet character $\bmod{r^{\prime}}$. There exist constants
$c>0$, $0\leqslant\delta(\chi)\leqslant\lambda r$ such that

$\displaystyle\sum_{m\leqslant
N/r}\chi(m)\varrho(rm)=\delta(\chi)\frac{N}{r}+O(Ne^{-\frac{1}{c}\sqrt{\log{N}}}).$

Moreover, if $\delta(\chi)\neq 0$ then $\chi$ has conductor
$h|\operatorname{disc}(p)$.

We next show that Proposition 3 implies we can conclude the proof of
Proposition 2. Indeed, we obtain

$\displaystyle
S_{r}(N,\tilde{F})=\frac{N}{r\varphi(r^{\prime})}\sum_{x\bmod{r^{\prime}}}e(\tilde{F}_{r}(x))\sum_{\chi\bmod{r^{\prime}}}\overline{\chi(x)}\delta(\chi)+O(Ne^{-\frac{1}{c^{\prime}}\sqrt{\log{N}}}),$

which substituted into (38) and using (40) implies

$\displaystyle S_{r}(N,F)$ $\displaystyle=\
e\left(Q(N)\right)S_{r}(N,\tilde{F})$ $\displaystyle-C\int_{1}^{N}u\left(2\pi
i\sum_{j=1}^{d}j\beta_{j}u^{j-1}\right)e\left(Q(u)\right)du+O(Ne^{-\frac{1}{c}\sqrt{\log{N}}}),$

with

$\displaystyle
C=\frac{1}{r\varphi(r^{\prime})}\sum_{x\bmod{r^{\prime}}}e(\tilde{F}_{r}(x))\sum_{\chi\bmod{r^{\prime}}}\overline{\chi(x)}\delta(\chi).$

Note that integrating by parts

$\displaystyle\int_{1}^{N}u\left(2\pi
i\sum_{j=1}^{d}j\beta_{j}u^{j-1}\right)e\left(Q(u)\right)du\ll N,$

and since if $\delta(\chi)\neq 0$ then $\chi$ has conductor
$h|\operatorname{disc}(p)$, the above implies

(42) $\displaystyle
S_{r}(N,F)\ll\frac{N}{r}\frac{1}{\phi(r^{\prime})}\sum_{x\mod{r^{\prime}}}\chi(x)e(\tilde{F}_{r}(x))+O(Ne^{-\frac{1}{c}\sqrt{\log{N}}}),$

for some $\chi$ with conductor $h|\operatorname{disc}(p)$.

Recall that

$\displaystyle\tilde{F}_{r}(x)=\tilde{F}(rx)=\sum_{j=1}^{d}\frac{b_{j}r^{j}}{q_{j}}x^{j}=\frac{\sum_{j=1}^{d}(qb_{j}r^{j}/q_{j})x^{j}}{q}.$

Let $q^{\prime}$ denote the smallest divisor of $q$ such that the polynomial

$P(x)=\sum_{j=1}^{d}(qb_{j}r^{j}/q_{j})x^{j},$

is constant mod $q/q^{\prime}$. Note that for each $1\leqslant j\leqslant d$

$\frac{qb_{j}r^{j}}{q_{j}}\equiv 0\mod{q/q^{\prime}}.$

Since $(b_{j},q_{j})=1$, this implies

$q^{\prime}>\frac{q_{j}}{r^{j}}\geqslant\left(\prod_{j=1}^{d}\frac{q_{j}}{r^{j}}\right)^{1/d}\geqslant\frac{q^{1/d}}{r^{(d+1)/2}}\geqslant
q^{1/2d},$

provided $r\leqslant q^{1/d(d+1)}.$

It follows from work of Cochrane and Zheng [9] and the Chinese remainder
theorem that

$\displaystyle\sum_{x\mod{r^{\prime}}}\chi(x)e(\tilde{F}_{r}(x))\ll\frac{r^{\prime}}{q^{\prime}}(q^{\prime})^{1-1/d}\ll\frac{r^{\prime}}{q^{1/2d(d+1)}}.$

Combining the above with (42) gives

$S_{r}(N,F)\leqslant\frac{N}{q^{1/2d(d+1)}}\quad\text{provided}\quad
r\leqslant q^{1/d(d+1)}.$

Summing over $r<q^{1/d(d+1)}$ we get

$\displaystyle\sum_{\begin{subarray}{c}r|q\\\
r<q^{1/d(d+1)}\end{subarray}}S_{r}(N,\tilde{F})$
$\displaystyle\leqslant\frac{N}{q^{1/2d(d+1)+o(1)}}.$

Together with (36) and (9.1) this concludes the proof of Proposition 2. We
will now focus on the proof of Proposition 3. In order to understand the sum

(43) $\sum_{n\leqslant N/r}\varrho(rn)\chi(n),$

we study the Dirichlet series for $\chi\bmod{r^{\prime}}$

$D_{r}(s,\chi)=\sum_{n=1}^{\infty}\varrho(rn)\chi(n)n^{-s}.$

###  9.2. Decomposing $D_{r}(s,\chi)$

Obverse that $D_{r}(s,\chi)$ is absolutely convergent for
$\operatorname{Re}(s)>1$. Our next goal is to extend $D_{r}(s,\chi)$ to the
left of $\operatorname{Re}(s)>1$ so we can use contour integration to estimate
the sum (43). We will do this by decomposing $D_{r}$ in terms of well-known
Artin L-functions. With this in mind, we fix some notation.

Denote by $K_{f}$ the splitting field of $f$ in ${\mathbb{C}}$, by $G$ the
Galois group of $K_{f}$ over ${\mathbb{Q}}$ and by
${\mathbb{Q}}(e(1/r^{\prime}))$ the cyclotomic field generated by
$r^{\prime}$-roots of unity. Observe that the latter is a Galois extension of
${\mathbb{Q}}$, with Galois group $C_{r^{\prime}}$ isomorphic to
$({\mathbb{Z}}/r^{\prime}{\mathbb{Z}})^{\times}$.

We also consider the compositum of $K_{f}$ and ${\mathbb{Q}}(e(1/r^{\prime}))$
and denote it by $K_{f,r^{\prime}}$ and its Galois group over ${\mathbb{Q}}$
by $G_{r^{\prime}}$. Observe that there is a natural injection

$\displaystyle G_{r^{\prime}}\rightarrow G\times C_{r^{\prime}}\simeq
G\times({\mathbb{Z}}/r^{\prime}{\mathbb{Z}})^{\times}.$

Moreover, observe that from the extension theorem for field automoprhism, it
follows that the projections $p_{1}:G_{r^{\prime}}\rightarrow G$ and
$p_{2}:G_{r^{\prime}}\rightarrow({\mathbb{Z}}/r^{\prime}{\mathbb{Z}})^{\times}$
are surjective.

We denote by $\hat{G}$ the finite set of isomorphism classes of complex
irreducible representations of $G$ and for $\pi\in\hat{G}$ we write
$\chi_{\pi}$ for the character of $\pi$.

Note that $\chi$ can be viewed as a character of
$({\mathbb{Z}}/r^{\prime}{\mathbb{Z}})^{\times}$, and consequently as a
character $\eta:C_{r^{\prime}}\rightarrow{\mathbb{C}}^{\times}$. It is known
that $\eta(\sigma_{p})=\chi(p)$ for $p\nmid r^{\prime}$ and $\sigma_{p}$ the
Frobenius automorphism at $p$.

So, we consider the representations $p_{1}^{\ast}\pi=\pi\circ p_{1}$ and
$p^{\ast}_{2}\eta=\eta\circ p_{2}$ of $G_{r^{\prime}}$ and observe that their
tensor product satisfies

(44) $\displaystyle\operatorname{tr}(p_{1}^{\ast}\pi\otimes
p_{2}^{\ast}\eta(\sigma_{p}))=\chi_{\pi}(\sigma_{p})\chi(p),$

for $p\nmid\operatorname{disc}(f)$ and $p\nmid r^{\prime}$.

######  Proposition 4.

$D_{r}(s,\chi)$ has an expression

$\displaystyle
D_{r}(s,\chi)=F_{r}(s,\chi)\prod_{\pi\in\hat{G}}E_{r,\pi}(s,\chi)(L(s,K_{f,r^{\prime}}/{\mathbb{Q}},p_{1}^{\ast}\pi\otimes
p_{2}^{\ast}\eta))^{m_{\pi}},$

for $m_{\pi}\geqslant 0$ an integer and, for any $\epsilon>0$,
$F_{r}\ll_{\epsilon}q^{\epsilon}$ and $E_{r,\pi}\ll q^{\epsilon}$ for
$\operatorname{Re}(s)\geqslant 3/4$.

To prove the proposition we will need the following lemma:

######  Lemma 9.

Denote by $\pi_{f}$ the permutation representation of $G$ acting on the set of
roots of $f$ in ${\mathbb{C}}$ and let

$\displaystyle\pi_{f}=\bigoplus_{\pi\in\hat{G}}m_{\pi}\pi$

be its decomposition in irreducible representations, where $m_{\pi}\geqslant
0$ are integers. For all $p\nmid\operatorname{disc}(f)$ it holds

(45) $\displaystyle\varrho(p)=\sum_{\pi}m_{\pi}\cdot\chi_{\pi}(\sigma_{p}),$

for $\sigma_{p}\in G$ the Frobenius automorphism at $p$.

###### Proof.

The proof follows by noting that $\varrho(p)$ is the number of fixed points of
the Frobenius automorphism $\sigma_{p}$ at $p$, which is also the character at
$\sigma_{p}$ of the permutation representation.

∎

###### Proof of Proposition 4.

First we observe that $D_{r}$ can be decomposed the following way

$\displaystyle D_{r}(s,\chi)$
$\displaystyle=\sum_{e|r^{\infty}}\sum_{(k,r)=1}\chi(ek)\varrho(erk)(ek)^{-s}$
$\displaystyle=\sum_{e|r^{\infty}}\varrho(er)\chi(e)e^{-s}\sum_{(k,r)=1}\chi(k)\varrho(k)k^{-s},$

where the notation $e|r^{\infty}$ means that $e$ runs over all possible
products of powers of the primes that divide $r$.

We denote by

$\displaystyle E_{r}(s,\chi)$
$\displaystyle=\sum_{e|r^{\infty}}\varrho(er)\chi(e)e^{-s}$
$\displaystyle\tilde{D}_{r}(s,\chi)$
$\displaystyle=\sum_{(k,r)=1}\chi(k)\varrho(k)k^{-s},$

and using Lemma 8 and the property $\varrho(p^{n})\leqslant C,$ for all
$n\geqslant 1$ and $p$ prime, we observe that $E_{r}(s,\chi)\ll r^{\epsilon}$
uniformly for $\operatorname{Re}(s)\geqslant 3/4$ and that
$\tilde{D}_{r}(s,\chi)$ converges absolutely for $\operatorname{Re}(s)>1$.

To be able to explore the properties of $\varrho$, we factor the primes that
divide the discriminant of $f$ in $\tilde{D}_{r}(s,\chi)$ and, since $r|q$, we
also factor the remaining primes that divide $q$. This yields

$\displaystyle\tilde{D}_{r}(s,\chi)=E_{1,r}(s,\chi)\prod_{\begin{subarray}{c}p\nmid\operatorname{disc}(f)\\\
p\nmid q\end{subarray}}\sum_{k\geqslant 0}\varrho(p^{k})\chi(p)^{k}p^{-ks},$

where $E_{1,r}(s,\chi)$ is entire and bounded for
$\operatorname{Re}(s)\geqslant 3/4$ by $C(\epsilon)q^{\epsilon}$.

We proceed in a similar manner and factor out the prime powers with exponent
bigger than $2$, obtaining

(46)
$\displaystyle\tilde{D}_{r}(s,\chi)=E_{1,r}(s,\chi)E_{2}(s,\chi)\prod_{\begin{subarray}{c}p\nmid\operatorname{disc}(f)\\\
p\nmid q\end{subarray}}(1+\varrho(p)\chi(p)p^{-s}),$

for $E_{2}(s,\chi)$ holomorphic and bounded in vertical strips by
$C(\epsilon)q^{\epsilon}$ for $\operatorname{Re}(s)\geqslant 3/4$.

Now we can use Lemma 9 and a further decomposition to write

$\displaystyle\tilde{D}_{r}(s,\chi)=E_{1,r}(s,\chi)E_{2}(s,\chi)E_{3,r}(s,\chi)\prod_{\pi\in\hat{G}}\prod_{p}(1+\chi_{\pi}(\sigma_{p})\chi(p)p^{-s})^{m_{\pi}},$

for $E_{3,r}$ holomorphic and uniformly bounded in vertical strips by
$C(\epsilon,f)q^{\epsilon}$ for $\operatorname{Re}(s)\geqslant 3/4$. Set
$F_{r}=E_{1,r}E_{2}E_{3,r}$. To conclude, we use equality (44), and yet
another similar decomposition as before and obtain

$\displaystyle\prod_{p}(1+\chi_{\pi}(\sigma_{p})\chi(p)p^{-s})=E_{r,\pi}(s,\chi)L(s,K_{f,q}/{\mathbb{Q}},p_{1}^{\ast}\pi\otimes
p_{2}^{\ast}\eta).$

∎

Observe that now we are dealing with Artin L-functions, which are better
understood than the original Dirichlet series. Since $p_{1}$ is surjective and
$\pi$ is irreducible it follows that $p_{1}^{\ast}\pi$ is an irreducible
representation. Furthermore, since $p_{2}^{\ast}\eta$ is of dimension $1$ we
can conclude that $p_{1}^{\ast}\pi\otimes p_{2}^{\ast}\eta$ is also an
irreducible representation. Artin’s conjecture states that this L-function is
entire except if the representation is trivial - which can occur if
$p_{1}^{\ast}\pi$ is the inverse of $p_{2}^{\ast}\eta$.

To avoid assuming Artin’s conjecture we will use the Brauer induction theorem
(see, e.g, [33, Theorem 19]) instead. It states that for every subgroup $H$ of
$G$ and 1-dimensional character $\beta_{H}:H\rightarrow{\mathbb{C}}^{\times}$,
there exist integers $n_{\pi,\beta_{H}}$ such that

$\displaystyle\chi_{\pi}=\sum_{H}\sum_{\beta_{H}}n_{\pi,\beta_{H}}\cdot{\operatorname{Ind}_{H}^{G}}\beta_{H}.$

Observe that this implies that $L(s,K_{f,q}/{\mathbb{Q}},p_{1}^{\ast}\pi)$ can
be represented as follows

$\displaystyle
L(s,K_{f,q}/{\mathbb{Q}},p_{1}^{\ast}\pi)=L(s,K_{f}/{\mathbb{Q}},\pi)$
$\displaystyle=\prod_{H}\prod_{\beta_{H}}L(s,K_{f}/{\mathbb{Q}},\operatorname{Ind}_{H}^{G}\beta_{H})^{n_{\pi,\beta_{H}}}$
$\displaystyle=\prod_{H}\prod_{\beta_{H}}L(s,K_{f}/K_{H},\beta_{H})^{n_{\pi,\beta_{H}}},$

where $K_{H}\subset K_{f}$ is the subfield fixed by $H$ \- which implies that
$H$ is the Galois gorup of $K_{f}$ over $K_{H}$.

We can now introduce the twist by the Dirichlet character. We denote by
$H^{\prime}=p_{1}^{-1}(H)$ and observe that we can write

$\displaystyle
p_{1}^{\ast}\pi=\bigoplus_{H}n_{\pi,\beta_{H}}\operatorname{Ind}_{H^{\prime}}^{G_{r^{\prime}}}(p_{1}^{\ast}\beta_{H}),$

and thus

$\displaystyle p_{1}^{\ast}\pi\otimes
p_{2}^{\ast}\eta=\bigoplus_{H}\bigoplus_{\beta_{H}}n_{\pi,\beta_{H}}\operatorname{Ind}_{H^{\prime}}^{G_{r^{\prime}}}(p_{1}^{\ast}\beta_{H}\otimes\operatorname{Res}_{H^{\prime}}^{G_{r^{\prime}}}p_{2}^{\ast}\eta).$

So denoting
$\theta_{\beta_{H}}=p_{1}^{\ast}\beta_{H}\otimes\operatorname{Res}_{H^{\prime}}^{G_{r^{\prime}}}p_{2}^{\ast}\eta$,
we can finally write

$\displaystyle L(s,K_{f,q}/{\mathbb{Q}},p_{1}^{\ast}\chi\otimes
p_{2}^{\ast}\eta)=\prod_{H}\prod_{\beta_{H}}L(s,K_{f,q}/K_{H},\theta_{\beta_{H}})^{n_{\pi,\beta_{H}}},$

thus $D_{r}$ has the following representation

(47) $\displaystyle
D_{r}(s,\chi)=F_{r}(s,\chi)\prod_{\pi\in\hat{G}}E_{r,\pi}(s,\chi)\prod_{H}\prod_{\beta_{H}}L(s,K_{f,q}/K_{H},\theta_{\beta_{H}})^{n_{\pi,\beta_{H}}}.$

Now $L(s,K_{f,q}/K_{H},\theta_{\beta_{H}})^{n_{\pi,\beta_{H}}}$ are Artin
L-functions of dimension one, which we know that are entire except for a pole
in $s=1$ if $\theta_{\beta_{H}}$ is the trivial one dimensional character.

###  9.3. Bounds on the Artin L-functions near $s=1$

We denote by $q_{\pi,i}$ the conductor of
$L(s,K_{f,q}/K_{H},\theta_{\beta_{H}})$ and observe that we can bound it as
follows, (see, e.g. [6]),

$q_{\pi,\beta_{H}}\leqslant\operatorname{cond}(p_{1}^{\ast}\beta_{H})\cdot\operatorname{cond}(\operatorname{Res}_{H^{\prime}}^{G_{q}}p_{2}^{\ast}\eta)\leqslant
Mq,$

where $M=\sup_{\pi,\beta_{H}}\operatorname{cond}(p_{1}^{\ast}\beta_{H})$. It
is well known that $L(s,K_{f,q}/K_{H},\theta_{\beta_{H}})$ can have at most
one real zero $\beta$ in the region

(48) $\displaystyle\sigma\geqslant
1-\frac{c}{\log({q_{\pi,\beta_{H}}}(|t|+3))},$

for $s=\sigma+it\in{\mathbb{C}}$ and $c$ a universal constant, (see, e.g. [16,
Theorem 5.35]). Note that this zero can only exist if $\theta_{i}$ is a
quadratic character.

To simplify the notation, in what follows we let $g=\theta_{\beta_{H}}$ and we
omit the field $K_{f,q}/K_{H}$. Denote by $\beta$ the possible exception in
region (48) and let $\bm{q}$ be the conductor of $g$. Also, we set $r_{1}=1$
if $g$ has an exceptional zero $\beta$ and $r_{1}=0$ otherwise. Analogously,
$r_{2}=1$ is $g$ is the trivial character, and consequently has a pole in
$s=1$, and $r_{2}=0$ if not.

######  Proposition 5.

Let $s=\sigma+it\in{\mathbb{C}}$ satisfy the conditions:

(49) $\displaystyle\sigma\geqslant 1$
$\displaystyle-\frac{c}{10\log(M{q}(|t|+3))}$ (50) $\displaystyle|s-\beta|$
$\displaystyle\geqslant\frac{1}{20\log{(3M{q})}}$ (51) $\displaystyle|s-1|$
$\displaystyle\geqslant\frac{1}{20\log(3M{q})}.$

It holds that

$\begin{split}L(s,g)&\ll\log(Mq(|t|+3))\\\
\frac{1}{L(s,g)}&\ll\log(Mq(|t|+3)),\\\ \end{split}$

where the implicit constant does not depend on the parameters.

If $g$ is not trivial condition (51) can be dropped. Condition (50) can be
dropped if $g$ has no exceptional zero.

###### Proof.

We set $\gamma(s,g)=\pi^{-s/2}\gamma\left(\frac{s+\kappa_{g}}{2}\right)$,
where $\kappa_{g}=0$ or $1$, and let
$\Lambda(s,g)=\bm{q}^{s/2}\gamma(s,g)L(s,g)$ be the extended L-function. It is
a known fact that $\Lambda(s,g)$ is a meromorphic function of order at most 1.
Thus, we can proceed as in [31, Theorem 11.4] to obtain the result of the
theorem.

∎

###  9.4. Finishing the proof

###### Proof of Proposition 2.

To bound the sum $\sum_{n\leqslant N/r}\varrho(rn)\chi(n)$ we first smooth it

$\displaystyle\sum_{\begin{subarray}{c}n=1\\\
(n,q)=r\end{subarray}}^{N/r}\varrho(n)\chi(n)=\sum_{\begin{subarray}{c}n=1\\\
(n,q)=r\end{subarray}}^{\infty}\varrho(n)\chi(n)\phi(n)+O(\tilde{N}),$

where $\phi$ is defined as

$\phi(x)=\min\left(x,1,1+\frac{N/r-x}{\tilde{N}}\right)$

for $0\leqslant x\leqslant N/r+\tilde{N}$ and $\phi(x)=0$ for
$x>N/r+\tilde{N}$, $\tilde{N}>0$ to be chosen later. Observe that Mellin
inversion implies that

(52) $\begin{split}&\sum_{\begin{subarray}{c}n=1\\\
(n,q)=r\end{subarray}}^{\infty}\varrho(n)\chi(n)\phi(n)=\frac{1}{2\pi
i}\int_{(3)}D_{r}(s,\chi)\hat{\phi}(s)ds\\\ &=\frac{1}{2\pi
i}\int_{(3)}F_{r}(s,\chi)\prod_{\pi\in\hat{G}}E_{r,\pi}(s,\chi)(L(s,K_{f,r^{\prime}}/{\mathbb{Q}},p_{1}^{\ast}\pi\otimes
p_{2}^{\ast}\eta))^{m_{\pi}}\hat{\phi}(s)ds\\\ &=\frac{1}{2\pi
i}\int_{(3)}F_{r}(s,\chi)\prod_{\pi\in\hat{G}}E_{r,\pi}(s,\chi)\prod_{H}\prod_{\beta_{H}}L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})^{n_{\pi,\beta_{H}}}\hat{\phi}(s)ds.\end{split}$

Our goal is to shift the contour of integration and use the information on the
L-functions $L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})$ to obtain good
bounds for $S_{r}(s,\tilde{F}_{r})$. Observe that we can find $c/2\leqslant
Q\leqslant c/10$ such that all the conditions of Proposition 5 are satisfied
for

$s\in\mathcal{Z}=\\{s=\sigma+it,\sigma=1-\frac{Q}{\log{(M{q}(|t|+3))}}\\}.$

Moving the contour of integration in equation (LABEL:eq:mellin) to
$\mathcal{Z}$ we get

(53) $\begin{split}\sum_{\begin{subarray}{c}n=1\\\
(n,q)=r\end{subarray}}^{\infty}\varrho(n)\chi(n)\phi(n)&=\frac{1}{2\pi
i}\int_{\mathcal{Z}}D_{r}(s,\chi)\widehat{\phi}(s)ds+\alpha\operatorname{Res}(D_{r}(s,\chi),1)\cdot\frac{N}{r}\\\
&+\tilde{\alpha}\operatorname{Res}(D_{r}(s,\chi),\beta)\cdot N^{\beta},\\\
\end{split}$

where $\alpha=1$ if there is $\pi\in\hat{G}$, for which
$p_{1}^{\ast}\pi\otimes p_{2}^{\ast}\eta$ is the principal character in
$K_{f,r^{\prime}}/{\mathbb{Q}}$ with $m_{\pi}=1$ and $\alpha=0$ otherwise.
Likewise, $\tilde{\alpha}=1$ if
$L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})$ has an exceptional zero
$\beta$ and $n_{\pi,\beta_{H}}=-1$, and $\tilde{\alpha}=0$ otherwise. Note
from Proposition 4 that there is at most one trivial character
$p_{1}^{\ast}\pi\otimes p_{2}^{\ast}\eta$ and one quadratic character
$\theta_{\beta_{H}}$. Moreover, if $p_{1}^{\ast}\pi\otimes p_{2}^{\ast}\eta$
is the trivial character, then we observe that $\chi$ must have a conductor
with modulus $h$ that divides $\operatorname{disc}(f)$.

We bound each term of the right hand side of equation (53) separately. To
bound the first term, we note that the Mellin transform of $\phi$ satisfies

$\displaystyle\widehat{\phi}(s)=\int_{0}^{N+\tilde{N}}\phi(z)z^{s-1}dz\ll\frac{N^{\sigma}}{|s|}\min\left(1,\frac{N}{|s|\tilde{N}}\right),$

(see [16, Theorem 5.12]). Thus,

$\displaystyle\int_{\mathcal{Z}}D_{r}(s,\chi)\hat{\phi}(s)ds\ll\int_{\mathcal{Z}}|D_{r}(s,\chi)|\frac{N^{\sigma}}{|s|}\min\left(1,\frac{N}{|s|\tilde{N}}\right)|ds|.$

We let $T=N/r\tilde{N}$ and $\sigma(T)=1-Q/\log(Mq(T+3))$ and note that the
observation above, Proposition 4 and Proposition 5 imply that

$\displaystyle\int_{\mathcal{Z}}D_{r}(s,\chi)\hat{\phi}(s)ds\ll
N^{\sigma(T)}q^{Y\epsilon},$

for a constant $Y>0$ that only depends on $f$.

We now deal with the last term
$\tilde{\alpha}\operatorname{Res}(D_{r}(s,\chi),\beta)\cdot N^{\beta}$. Siegel
proved (see, e.g. [16, Theorem 5.28]) that if the exceptional zero $\beta$
exists then for every $\epsilon>0$ there exists a constant $C(\epsilon)>0$
such that

(54) $\displaystyle\beta\leqslant
1-\frac{c(\epsilon)}{\operatorname{cond}(\theta_{i})^{\epsilon}}\leqslant
1-\frac{c(\epsilon)}{Mq^{\epsilon}}.$

To bound the residue at $\beta$ we notice that if $\theta_{\pi,\beta_{H}}$ is
not trivial we can use Proposition 5 and obtain

$\displaystyle|L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})|^{n_{\pi,i}}\ll|\log{3Mq}|^{|n_{\pi,\beta_{H}}|}.$

If $\theta_{\beta_{H}}$ is trivial then

$\displaystyle|L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})|^{n_{\pi,\beta_{H}}}\ll_{\epsilon}q^{\epsilon|n_{\pi,\beta_{H}}|},$

and for the residue given by the L-function with quadratic character we have

$\displaystyle\operatorname{Res}\left(\frac{1}{L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})},\beta\right)=\frac{1}{L(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})}.$

We can deduce that $L^{\prime}(s,K_{f,q}/K^{\prime}_{H},\theta_{\beta_{H}})\gg
1$ from the last part of Theorem 11.4 in [31]. Putting everything together we
obtain

$\displaystyle\operatorname{Res}(D_{r}(s,\chi),\beta)\cdot N^{\beta}\leqslant
q^{\epsilon}N^{1-C(\epsilon)/q^{\epsilon}}.$

Pick $T=\exp\left(\frac{1}{3}\sqrt{\log{N}}\right)$. Since
$q\leqslant(\log{N})^{B}$, the considerations above imply that

(55) $\displaystyle
S_{r}(N,\tilde{F}_{r})=\alpha\operatorname{Res}(D_{r}(s,\chi),1)\cdot\frac{N}{r}+O(Ne^{-\frac{1}{c(\epsilon)}\sqrt{\log{N}}}),$

for a constant $c(\epsilon)$ and $\epsilon>0$ sufficiently small. Since

$\displaystyle\sum_{n\leqslant N}\varrho(n)\leqslant\lambda N,$

and the error term in equation (55) is $o(N)$ we conclude that

$\operatorname{Res}(D_{r}(s,\chi),1)\leqslant\lambda r$

. ∎

##  10\. Sharpness of Theorem 1

Following the approach of [32], we will construct a completely multiplicative
function $f=f_{N,F}$, with $|f(n)|\leqslant 1$, such that

(56) $\displaystyle\left|\sum_{n\leqslant
N}f(n)e\left(F(n)\right)\right|\geqslant\frac{1}{10}\frac{N}{\log{N}}.$

We first observe that the function $G:{\mathbb{C}}\rightarrow{\mathbb{C}}$
given by

$\displaystyle G(z)=\sum_{n\leqslant
N}z^{\Omega(n)}e(F(n))+\sum_{\frac{N}{2}<p\leqslant N}(1-ze(F(p)))$

is entire and its value at zero satifies

$\displaystyle G(0)=\sum_{\frac{N}{2}<p\leqslant
N}1\geqslant\frac{1}{10}\frac{N}{\log{N}},$

for $N$ sufficiently large. Thus, by the maximum modulus principle, there
exists a $z_{0}\in{\mathbb{C}}$, $|z_{0}|=1$, such that
$|G(z_{0})|\geqslant|G(0)|$.

Define the completely multiplicative function $f$ by $f(p)=z_{0}$, for
$p\leqslant N/2$, and $f(p)=e(-F(p))$, for $p>N/2$. To conclude that equation
(56) is satisfied, we just observe that

$\displaystyle\sum_{n\leqslant N}f(n)e\left(F(n)\right)$
$\displaystyle=\sum_{n\leqslant
N}z_{0}^{\Omega(n)}e(F(n))+\sum_{\frac{N}{2}<p\leqslant N}(1-z_{0}e(F(p)))$
$\displaystyle=G(z_{0})\gg\frac{N}{\log{N}}.$

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