1 Introduction
Denote by
$\unicode[STIX]{x1D70F}_{k}(n)$
the number of positive integral solutions to the equation
$x_{1}x_{2}\cdots x_{k}=n$
(and write
$\unicode[STIX]{x1D70F}_{2}=\unicode[STIX]{x1D70F}$
). A basic problem in analytic number theory is to study the asymptotic behavior of the average
$$\begin{eqnarray}\displaystyle \mathscr{D}_{k}(X;q,a):=\mathop{\sum }_{\substack{ n\leqslant X \\ n\equiv a\,(\text{mod}\,q)}}\unicode[STIX]{x1D70F}_{k}(n) & & \displaystyle\end{eqnarray}$$
for
$q\geqslant 1$
and
$(a,q)=1$
.
Define
$$\begin{eqnarray}\displaystyle \mathscr{M}_{k}(X;q,a):=\frac{1}{\unicode[STIX]{x1D711}(q)}\mathop{\sum }_{\substack{ n\leqslant X \\ (n,q)=1}}\unicode[STIX]{x1D70F}_{k}(n), & & \displaystyle \nonumber\end{eqnarray}$$
which is expected to be the main term in the asymptotic formula for
$\mathscr{D}_{k}(X;q,a)$
. Note that
$\mathscr{M}_{k}(X;q,a)\sim c_{k,q}X(\log X)^{k-1}$
with some positive constant
$c_{k,q}\gg (\log \log q)^{-k}.$
For each fixed
$k\geqslant 1$
, we are interested in the admissible value of
$\unicode[STIX]{x1D703}_{k}$
being as large as possible such that for any
$\unicode[STIX]{x1D700}>0,$
the bound
$$\begin{eqnarray}\displaystyle \mathscr{D}_{k}(X;q,a)-\mathscr{M}_{k}(X;q,a)\ll \frac{X}{q}(\log X)^{-A} & & \displaystyle\end{eqnarray}$$
holds for all
$q<X^{\unicode[STIX]{x1D703}_{k}-\unicode[STIX]{x1D700}}$
and
$A>0$
. It is widely believed that one can take
$\unicode[STIX]{x1D703}_{k}=1$
for each
$k\geqslant 1;$
this is, however, only known for the trivial case
$k=1$
. Relying on Weil’s bound for Kloosterman sums, Selberg [Reference Selberg19] and Hooley [Reference Hooley12] obtained independently that
$\unicode[STIX]{x1D703}_{2}=\frac{2}{3}$
is admissible. For
$k\geqslant 3$
, the determination of
$\unicode[STIX]{x1D703}_{k}$
becomes difficult. Anyway, it was already known to Linnik [Reference Linnik16] that
$\unicode[STIX]{x1D703}_{3}=\unicode[STIX]{x1D703}_{4}=\frac{1}{2}$
is acceptable. It would be much more difficult to go beyond
$\frac{1}{2}$
in general. However, Friedlander and Iwaniec [Reference Friedlander and Iwaniec9] were able to prove that one can take
$\unicode[STIX]{x1D703}_{3}=\frac{1}{2}+\frac{1}{230}$
, and this was later increased to
$\frac{1}{2}+\frac{1}{82}$
by Heath-Brown [Reference Heath-Brown11]. In the case of prime moduli, Fouvry, Kowalski and Michel [Reference Fouvry, Kowalski and Michel6] can prove that
$\unicode[STIX]{x1D703}_{3}=\frac{1}{2}+\frac{1}{46}$
is admissible appealing to their bilinear forms of Kloosterman sums with smooth coefficients (see [Reference Fouvry, Kowalski and Michel5, Theorem 1.16]).
The problem becomes much easier if conditions are imposed on the prime factorization of
$q$
. For instance, Irving [Reference Irving13] can beat the record of Selberg–Hooley if
$q$
has no prime factors exceeding
$q^{1/18.1}$
. In particular, if
$\unicode[STIX]{x1D702}>0$
is sufficiently small and
$q$
has no prime factors exceeding
$q^{\unicode[STIX]{x1D702}}$
, then the admissible value of
$\unicode[STIX]{x1D703}_{2}$
can be enlarged to
$\frac{2}{3}+\frac{1}{246}+O(\unicode[STIX]{x1D702}).$
As a consequence of arithmetic exponent pairs developed in [Reference Wu and Xi22], we may obtain a slightly larger value
$\frac{2}{3}+\frac{1}{232}$
.
In this paper, we focus on the estimation of
$\unicode[STIX]{x1D703}_{3}$
under the assumptions that
$q$
is squarefree and has only small prime factors.
Theorem 1.1. Let
$$\begin{eqnarray}\unicode[STIX]{x1D703}<{\textstyle \frac{9}{17}}={\textstyle \frac{1}{2}}+{\textstyle \frac{1}{34}}.\end{eqnarray}$$
There exists some constant
$\unicode[STIX]{x1D702}>0$
such that if
$q\asymp X^{\unicode[STIX]{x1D703}}$
is squarefree and has all its prime factors not exceeding
$q^{\unicode[STIX]{x1D702}},$
then (1.2) with
$k=3$
holds uniformly for
$(a,q)=1$
and the implied constant therein may depend on
$\unicode[STIX]{x1D703}$
and
$A$
.
As in [Reference Wu and Xi22], we benefit a lot from the
$q$
-van der Corput method and arithmetic exponent pairs. In addition, we will develop in Lemma 2.4 a kind of double
$q$
-van der Corput method to estimate smooth bilinear forms of exponential sums; in particular, Lemma 2.4 is non-trivial if
$M>q^{\unicode[STIX]{x1D700}},N>q^{\unicode[STIX]{x1D700}}$
, and
$\mathit{MN}>q^{2/3+\unicode[STIX]{x1D700}}$
(see Remark 2 for detailed arguments). On the other hand, Fouvry, Kowalski and Michel [Reference Fouvry, Kowalski and Michel5, Theorem 1.16] have to assume
$\mathit{MN}>q^{3/4+\unicode[STIX]{x1D700}}$
in the case of prime moduli
$q$
.
In fact, only two exponent pairs
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706})=(\frac{1}{2},\frac{1}{2}),(\frac{1}{6},\frac{2}{3})$
will be essentially used, the former of which comes from the completion method and does not require conditions on the factorization of
$q$
. In fact, we can restrict
$q$
to be a product of two distinct primes of certain sizes; the aim of this paper, however, is to see how far one can go if assuming the factorization of
$q$
being as good as possible such that the arithmetic exponent pairs apply freely.
Notation
As usual, we write
$\text{e}(x)=\text{e}^{2\unicode[STIX]{x1D70B}\text{i}x}$
;
$\unicode[STIX]{x1D707}$
and
$\unicode[STIX]{x1D711}$
denote the Möbius and Euler functions, respectively. The multiplicative inverse
$\overline{x}$
of
$x\,(\text{mod}\,q)$
is defined by
$\overline{x}x\equiv 1\,(\text{mod}\,q).$
For a coefficient
$\boldsymbol{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{m})$
, denote by
$\Vert \boldsymbol{\unicode[STIX]{x1D6FC}}\Vert$
the
$\ell _{2}$
-norm
$(\sum _{m}|\unicode[STIX]{x1D6FC}_{m}|^{2})^{1/2}.$
For
$g\in L^{1}(\mathbf{R})$
, define the Fourier transform
$$\begin{eqnarray}\widehat{g}(\unicode[STIX]{x1D706})=\int _{\mathbf{R}}g(x)\text{e}(-\unicode[STIX]{x1D706}x)\,dx.\end{eqnarray}$$
We use
$\unicode[STIX]{x1D700}$
to denote a very small positive number, which might be different at each occurrence; we also write
$X^{\unicode[STIX]{x1D700}}\log X\ll X^{\unicode[STIX]{x1D700}}.$
The notation
$n\sim N$
means
$N<n\leqslant 2N.$
We write
$A\ll B$
or
$A=O(B)$
to mean
$|A|\leqslant cB$
with some positive constant
$c$
, which is independent from the involved variables. Also,
$A\asymp B$
is defined to be
$B\ll A\ll B$
.
2 Averages of trace functions
This section is devoted to several estimates on certain averages of trace functions over finite fields. Since the main concern of this paper is to study the distribution of
$\unicode[STIX]{x1D70F}_{3}$
, we will not spend too much time on the background of trace functions, which we will leave in a black box. Some necessary terminology on trace functions will be mentioned and complete details can be found, for instance, in [Reference Fouvry, Kowalski and Michel4, Reference Fouvry, Kowalski and Michel7, Reference Katz14, Reference Wu and Xi22], etc.
2.1 Kloosterman sums as trace functions
For an odd prime
$p$
, we may consider the middle-extension sheaves on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
. Recall that a sheaf
${\mathcal{F}}$
on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
is said to be middle-extension, if
${\mathcal{F}}\simeq j_{\ast }(j^{\ast }{\mathcal{F}})$
holds for each
$j:U\rightarrow \mathbf{A}^{1}$
as the open immersion of a non-empty open set on which
${\mathcal{F}}$
is lisse. Denote by
$\mathfrak{c}({\mathcal{F}})$
the conductor of
${\mathcal{F}}$
. The trace function
$K$
associated to
${\mathcal{F}}$
is given by
$K(x)=\unicode[STIX]{x1D704}((\text{tr}{\mathcal{F}})(\mathbf{F}_{\!p},x))$
for some isomorphism
$\unicode[STIX]{x1D704}:\overline{\mathbf{Q}}_{\ell }\rightarrow \mathbf{C}$
with
$\ell \neq p$
. A trace function
$K$
can be associated to different sheaves, and the minimum of the conductors of these sheaves is defined to be the conductor of
$K$
, denoted by
$\mathfrak{c}(K)$
.
In terms of trace functions
$\,(\text{mod}\,p),$
we may construct the composite trace functions following the manners in [Reference Polymath18] and [Reference Wu and Xi22]. Let
$q\geqslant 3$
be a squarefree number and
$K=K_{q}(\cdot )$
a composite trace function (formally) defined by the product
$$\begin{eqnarray}\displaystyle K_{q}(n)=\mathop{\prod }_{p\mid q}K_{p}(n), & & \displaystyle\end{eqnarray}$$
where
$K_{p}$
is the trace function of certain
$\ell$
-adic middle-extension sheaf
${\mathcal{F}}_{p}$
on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
for each
$p\mid q$
. In addition, we may define the conductor of
$K$
by
$$\begin{eqnarray}\mathfrak{c}(K)=\mathop{\prod }_{p\mid q}\mathfrak{c}(K_{p}).\end{eqnarray}$$
Note that
$\mathfrak{c}(K)\ll q^{\unicode[STIX]{x1D700}}$
for any
$\unicode[STIX]{x1D700}>0$
if
$\mathfrak{c}(K_{p})$
is bounded for each
$p\mid q$
. Originally,
$K_{p}$
is a function living in
$\mathbf{Z}/p\mathbf{Z}$
; one can extend its domain to
$\mathbf{Z}$
by periodicity, and one can do the same to
$K=K_{q}$
.
Some of the subsequent arguments are applicable to general trace functions, however we only focus on Kloosterman sums, which are the main tools in this paper and usually serve as typical examples of trace functions.
There are many ways to define Kloosterman sums. For instance, for a positive integer
$\unicode[STIX]{x1D708}\geqslant 1$
, we may define the exponential sum
for each positive integer
$q$
and vector
$\mathbf{n}=(n_{1},\ldots ,n_{\unicode[STIX]{x1D708}})\in \mathbf{Z}^{\unicode[STIX]{x1D708}}.$
One may get slightly more general sums if the congruence condition is replaced by
$a_{1}\cdots a_{\unicode[STIX]{x1D708}}\equiv a\,(\text{mod}\,q)$
for some
$a\,(\text{mod}\,q),$
which can be expressed in terms of
$S_{\unicode[STIX]{x1D708}}(\mathbf{n};q)$
by virtue of Möbius inversion.
For
$\unicode[STIX]{x1D708}=2$
,
$S_{2}(\mathbf{n};q)$
is the classical Kloosterman sum
$$\begin{eqnarray}\displaystyle S_{2}(m,n;q)=\frac{S(m,n;q)}{\sqrt{q}}=\frac{1}{\sqrt{q}}~\sum ^{\ast }_{a\,(\text{mod}\,q)}\text{e}\biggl(\frac{ma+n\overline{a}}{q}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
The Weil bound for
$S_{2}(\mathbf{n};q)$
reads
$$\begin{eqnarray}\displaystyle |S_{2}(m,n;q)|\leqslant \unicode[STIX]{x1D70F}(q)(m,n,q)^{1/2}. & & \displaystyle\end{eqnarray}$$
For
$\unicode[STIX]{x1D708}\geqslant 3$
, one may view
$S_{\unicode[STIX]{x1D708}}(\mathbf{n};q)$
as the hyper-Kloosterman sum introduced by Mordell (slightly different from that of Deligne in notation). What will be used in this paper is the case
$\unicode[STIX]{x1D708}=3$
, for which Smith [Reference Smith20] deduced from Deligne’s bound that
$$\begin{eqnarray}\displaystyle |S_{3}(m,n,l;q)|\leqslant \unicode[STIX]{x1D70F}_{3}(q)(m,n,l,q). & & \displaystyle\end{eqnarray}$$
We should remark that
$S_{\unicode[STIX]{x1D708}}(\mathbf{n};q)$
appears naturally in the study of
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D708}}$
in arithmetic progressions (as well as analytic theory of GL
$_{\unicode[STIX]{x1D708}}$
automorphic forms). However, there are more algebraic interpretations with the slightly more general sum
Note that
$S_{\unicode[STIX]{x1D708}}(\mathbf{n};q)=\text{Kl}_{\unicode[STIX]{x1D708}}(n_{1}n_{2}\cdots n_{\unicode[STIX]{x1D708}},q)$
if
$(n_{1}n_{2}\cdots n_{\unicode[STIX]{x1D708}},q)=1.$
According to Deligne, there exists an
$\ell$
-adic middle-extension sheaf
${\mathcal{K}}\ell _{\unicode[STIX]{x1D708}}$
modulo
$p$
, called a Kloosterman sheaf with the corresponding trace function
$$\begin{eqnarray}\displaystyle K_{{\mathcal{K}}\ell _{\unicode[STIX]{x1D708}}}(x)=\text{Kl}_{\unicode[STIX]{x1D708}}(x,p),\quad x\in \mathbf{F}_{\!p}^{\times }. & & \displaystyle \nonumber\end{eqnarray}$$
Such a sheaf was constructed by Deligne [Reference Deligne2]. According to him,
${\mathcal{K}}\ell _{\unicode[STIX]{x1D708}}$
is geometrically irreducible, of rank
$\unicode[STIX]{x1D708}$
, and with the conductor bounded by
$\unicode[STIX]{x1D708}+3$
. Therefore, one needs to transform
$S_{\unicode[STIX]{x1D708}}(\mathbf{n};q)$
to the shape
$\text{Kl}_{\unicode[STIX]{x1D708}}(n,q)$
, where
$n$
can be defined explicitly by certain products of each coordinate of
$\mathbf{n}.$
In this way, the average of
$S_{\unicode[STIX]{x1D708}}(\mathbf{n};q)$
over
$\mathbf{n}$
can be interpreted by multilinear sums of
$\text{Kl}_{\unicode[STIX]{x1D708}}(\cdot ,q)$
, taking shapes of Lemmas 2.3 and 2.4 for instance.
For
$\unicode[STIX]{x1D708}=2,$
one may appeal to the classical Kuznetsov–Selberg identity, which reads
$$\begin{eqnarray}\displaystyle S_{2}(m,n;q)=\mathop{\sum }_{d|(m,n,q)}\sqrt{d}S_{2}(mnd^{-2},1;qd^{-1}). & & \displaystyle\end{eqnarray}$$
Such a formula was originally stated by Selberg and the first rigorous proof was given by Kuznetsov [Reference Kuznetsov15] appealing to his trace formula. The study of
$\unicode[STIX]{x1D70F}_{3}$
requires a similar transformation formula for
$\unicode[STIX]{x1D708}=3$
. Smith [Reference Smith21] tried to make an extension of (2.4); as pointed out by Heath-Brown [Reference Heath-Brown11], however, the extension is not successful, and there is an alternative approach by virtue of smooth (or friable) numbers. In Appendix A, we will give an elementary proof of (2.4), and the arguments can be easily extended to yield the following formula for
$S_{3}(\mathbf{n};q)$
.
Lemma 2.1. Let
$q$
be a squarefree number. For all
$m,n,l\in \mathbf{Z},$
we have
$$\begin{eqnarray}\displaystyle S_{3}(m,n,l;q) & = & \displaystyle \mathop{\sum }_{d|(m,n,l,q)}\;\mathop{\sum }_{r|(n/d,l/d,q/d)}\;\mathop{\sum }_{s|(nl/(dr)^{2},m/d,q/dr)}\nonumber\\ \displaystyle & & \displaystyle \times \,d\unicode[STIX]{x1D707}(rs)S_{3}\biggl(\frac{\overline{rs}mnl}{d^{3}r^{2}s^{2}},1,1;\frac{q}{drs}\biggr),\end{eqnarray}$$
where
$\overline{rs}$
is defined to be the multiplicative inverse of
$rs\,(\text{mod}\,q/(drs)).$
The above formula seems complicated at first sight, but it is much simpler in applications since
$d,r,s$
contribute as divisor sums. By virtue of Lemma 2.1, one may transform averages of
$S_{3}(m,n,l;q)$
over
$m,n,l$
to bilinear or trilinear forms of
$S_{3}(\unicode[STIX]{x1D6FE}mnl,1,1;q^{\prime })$
with some
$\unicode[STIX]{x1D6FE}\in \mathbf{Z}$
and
$q^{\prime }|q$
. Note that
$S_{3}(x,1,1;p)=\text{Kl}_{3}(x,p)$
for all
$x\in \mathbf{Z}$
, and thus, Lemmas 2.2–2.4 as given below are applicable if the trace function is taken as
$x\mapsto \text{Kl}_{3}(x,q).$
2.2 Arithmetic exponent pairs for trace functions
Let
$q$
be a squarefree number and
$K=K_{q}$
defined by (2.1). For a given interval
$I$
, we consider the following average of
$K$
over
$I$
:
$$\begin{eqnarray}\displaystyle \mathfrak{S}(K,W)=\mathop{\sum }_{n\in I}K_{q}(n)W_{\unicode[STIX]{x1D6FF}}(n), & & \displaystyle \nonumber\end{eqnarray}$$
where
$W_{\unicode[STIX]{x1D6FF}}$
is an arbitrary function defined over
$\mathbf{Z}/\unicode[STIX]{x1D6FF}\mathbf{Z}$
satisfying
$\Vert W_{\unicode[STIX]{x1D6FF}}\Vert _{\infty }\leqslant 1$
. Roughly speaking, the Pólya–Vinogradov bound is non-trivial for
$|I|>q^{1/2+\unicode[STIX]{x1D700}}$
(at least for
$\unicode[STIX]{x1D6FF}=1$
). The situation is much easier if conditions are imposed on the prime factorization of
$q$
. In [Reference Wu and Xi22], we developed the method of arithmetic exponent pairs for averages of composite trace functions that may go far beyond the Pólya–Vinogradov bound, as long as the factorization of
$q$
is good enough. Such an observation can be at least dated back to Heath-Brown and his proof of Weyl-type subconvex bounds for Dirichlet
$L$
-functions to well-factorable moduli.
To summarize the arithmetic exponent pairs developed in [Reference Wu and Xi22], we define the normalized Fourier transform
$$\begin{eqnarray}\widehat{W}_{\unicode[STIX]{x1D6FF}}(y)=\frac{1}{\sqrt{\unicode[STIX]{x1D6FF}}}\mathop{\sum }_{x\,(\text{mod}\,\unicode[STIX]{x1D6FF})}W_{\unicode[STIX]{x1D6FF}}(x)\text{e}\biggl(\frac{-yx}{\unicode[STIX]{x1D6FF}}\biggr).\end{eqnarray}$$
Recall that a middle-extension sheaf
${\mathcal{F}}_{p}$
on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
is said to be
$d$
-amiable if it is pointwise pure of weight
$0$
and no geometrically irreducible component of
${\mathcal{F}}_{p}$
is geometrically isomorphic to an Artin–Schreier sheaf of the form
${\mathcal{L}}_{\unicode[STIX]{x1D713}(P)},$
where
$P\in \mathbf{F}_{\!p}[X]$
is a polynomial of degree
${\leqslant}d.$
In such a case, we also say the associated trace function
$K_{p}$
is
$d$
-amiable. The composite trace function
$K_{q}$
is called
$d$
-amiable if
$K_{p}$
is
$d$
-amiable for each
$p\mid q.$
In addition, a sheaf (or its associated trace function) is said to be
$\infty$
-amiable if it is amiable for any fixed
$d\geqslant 1.$
We are ready to recall the main results in [Reference Wu and Xi22, §§3–6]. Let
$\unicode[STIX]{x1D702}>0$
be a sufficiently small number. Suppose
$q$
is a squarefree number with no prime factors exceeding
$q^{\unicode[STIX]{x1D702}}$
, and
$K=K_{q}(\cdot )$
is an
$\infty$
-amiable trace function
$\,(\text{mod}\,q)$
. For
$|I|<q\unicode[STIX]{x1D6FF},$
one expects the estimate
$$\begin{eqnarray}\displaystyle \mathfrak{S}(K,W)\ll q^{\unicode[STIX]{x1D700}}\biggl(\frac{q}{|I|}\biggr)^{\unicode[STIX]{x1D705}}|I|^{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D708}}\Vert \widehat{W}_{\unicode[STIX]{x1D6FF}}\Vert _{\infty }^{\unicode[STIX]{x1D707}} & & \displaystyle\end{eqnarray}$$
holds for some tuple
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})$
, where
$\unicode[STIX]{x1D700}>0$
and the implied constant depends only on
$\unicode[STIX]{x1D700}$
and the conductor
$\mathfrak{c}(K)$
of
$K$
. If this is the case, we call
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})$
an exponent pair for
$K$
. In particular, one may take
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})=(0,1,0,0)$
trivially.
As shown in [Reference Wu and Xi22], other exponent pairs can be produced starting from the trivial one
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})=(0,1,0,0)$
by virtue of
$A$
- and
$B$
-processes in the
$q$
-analogue of the van der Corput method. Since we do not input extra conditions on
$W_{\unicode[STIX]{x1D6FF}}$
, it is not a bad choice to utilize the trivial bound
$\Vert \widehat{W}_{\unicode[STIX]{x1D6FF}}\Vert _{\infty }\leqslant \sqrt{\unicode[STIX]{x1D6FF}}.$
The values of
$\unicode[STIX]{x1D708},\unicode[STIX]{x1D707}$
are usually not too large and they do not play essential roles in many applications. For instance, we will require the constraints
$0\leqslant \unicode[STIX]{x1D708}+\unicode[STIX]{x1D707}/2-\unicode[STIX]{x1D705}\leqslant 1$
in the proof of Lemma 2.3 below, which hold obviously upon many choices of exponent pairs.
Lemma 2.2. Keep the notation and convention as above. The estimate (2.6) holds for every
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})$
produced by
$A$
- and
$B$
-processes applied to
$(0,1,0,0)$
with
$$\begin{eqnarray}\displaystyle 0\leqslant \unicode[STIX]{x1D705}\leqslant {\textstyle \frac{1}{2}}\leqslant \unicode[STIX]{x1D706}\leqslant 1. & & \displaystyle\end{eqnarray}$$
In particular, one may take
$$\begin{eqnarray}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})=({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}},1),\qquad ({\textstyle \frac{1}{6}},{\textstyle \frac{2}{3}},{\textstyle \frac{1}{2}},0)\quad \text{and}\quad ({\textstyle \frac{2}{7}},{\textstyle \frac{4}{7}},{\textstyle \frac{2}{7}},1).\end{eqnarray}$$
2.3 General bilinear forms of trace functions
For any coefficients
$\boldsymbol{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}_{m}),\boldsymbol{\unicode[STIX]{x1D6FD}}=(\unicode[STIX]{x1D6FD}_{n})$
supported in some intervals of length
$M,N$
, respectively, we introduce the bilinear form
$$\begin{eqnarray}\displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)=\mathop{\sum }_{m}\mathop{\sum }_{n}\unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D6FD}_{n}K_{q}(mn). & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 2.3. With the same notation and assumptions as in Lemma 2.2, we have
$$\begin{eqnarray}\displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K) & \ll & \displaystyle (q\mathit{MN})^{\unicode[STIX]{x1D700}}(\mathit{MN})^{1/2}\Vert \boldsymbol{\unicode[STIX]{x1D6FC}}\Vert \Vert \boldsymbol{\unicode[STIX]{x1D6FD}}\Vert \nonumber\\ \displaystyle & & \displaystyle \times \,(\unicode[STIX]{x1D700}_{1}q^{\unicode[STIX]{x1D705}/2}M^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}-1)/2}+q^{-1/4}+N^{-1/2})\nonumber\end{eqnarray}$$
for
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706},\unicode[STIX]{x1D708},\unicode[STIX]{x1D707})$
with
$0\leqslant \unicode[STIX]{x1D708}+\unicode[STIX]{x1D707}/2-\unicode[STIX]{x1D705}\leqslant 1$
, where
$\unicode[STIX]{x1D700}_{1}=1$
if
$M<q$
and vanishes otherwise.
Remark 1. As one will see later, the proof of Theorem 1.1 only utilizes the choice
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706})=(\frac{1}{2},\frac{1}{2})$
, in which case
$q$
is not required to have only small prime factors. Lemma 2.3 presents an estimate in which the exponent pair is displayed explicitly, and this might be useful on some other occasions.
Proof. By Cauchy’s inequality, we have
$$\begin{eqnarray}\displaystyle |\mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)|^{2}\leqslant \Vert \boldsymbol{\unicode[STIX]{x1D6FC}}\Vert ^{2}\mathop{\sum }_{m}\biggl|\mathop{\sum }_{n}\unicode[STIX]{x1D6FD}_{n}K_{q}(mn)\biggr|^{2}. & & \displaystyle \nonumber\end{eqnarray}$$
Squaring out and switching summations, we get
$$\begin{eqnarray}\displaystyle |\mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)|^{2}\leqslant \Vert \boldsymbol{\unicode[STIX]{x1D6FC}}\Vert ^{2}\mathop{\sum }_{n_{1}}\mathop{\sum }_{n_{2}}|\unicode[STIX]{x1D6FD}_{n_{1}}||\unicode[STIX]{x1D6FD}_{n_{2}}||\unicode[STIX]{x1D6F4}(n_{1},n_{2};q)|, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}(n_{1},n_{2};q)=\mathop{\sum }_{m}K_{q}(mn_{1})\overline{K_{q}(mn_{2})}. & & \displaystyle \nonumber\end{eqnarray}$$
For
$n_{1}=n_{2}$
, we utilize the trivial bound
$\unicode[STIX]{x1D6F4}(n_{1},n_{2};q)\ll Mq^{\unicode[STIX]{x1D700}}$
, giving a contribution to the right-hand side of (2.8) at most
$O(q^{\unicode[STIX]{x1D700}}M\Vert \boldsymbol{\unicode[STIX]{x1D6FC}}\Vert ^{2}\Vert \boldsymbol{\unicode[STIX]{x1D6FD}}\Vert ^{2})$
. We now assume
$n_{1}\neq n_{2}$
and put
$d=(n_{1}-n_{2},q),$
which is also a squarefree number and
$(d,q/d)=1.$
From the Chinese remainder theorem, it follows that
$$\begin{eqnarray}K_{q}(mn_{1})\overline{K_{q}(mn_{2})}=|K_{d}(mn_{1})|^{2}K_{q/d}(mn_{1})\overline{K_{q/d}(mn_{2})}.\end{eqnarray}$$
One may check that
$m\mapsto K_{q/d}(mn_{1})\overline{K_{q/d}(mn_{2})}$
is an
$\infty$
-amiable trace function
$\,(\text{mod}\,q/d)$
as long as
$q/d$
is large enough. From Lemma 2.2 it follows that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}(n_{1},n_{2};q)\ll (qM)^{\unicode[STIX]{x1D700}}\biggl(\frac{q}{dM}\biggr)^{\unicode[STIX]{x1D705}}M^{\unicode[STIX]{x1D706}}d^{\unicode[STIX]{x1D708}+\unicode[STIX]{x1D707}/2} & & \displaystyle \nonumber\end{eqnarray}$$
if
$q>M;$
otherwise we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}(n_{1},n_{2};q)\ll (qM)^{\unicode[STIX]{x1D700}}\biggl[\frac{M}{q}\biggr](q/d)^{1/2}d\ll (qM)^{\unicode[STIX]{x1D700}}M(d/q)^{1/2}. & & \displaystyle \nonumber\end{eqnarray}$$
Hence, we conclude that
$$\begin{eqnarray}\displaystyle |\mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)|^{2} & \ll q^{\unicode[STIX]{x1D700}}M\Vert \boldsymbol{\unicode[STIX]{x1D6FC}}\Vert ^{2}\Vert \boldsymbol{\unicode[STIX]{x1D6FD}}\Vert ^{2}+\unicode[STIX]{x1D700}_{1}\mathscr{B}_{1}+(1-\unicode[STIX]{x1D700}_{1})\mathscr{B}_{2}, & \displaystyle \nonumber\end{eqnarray}$$
where
Note that
$0\leqslant \unicode[STIX]{x1D708}+\unicode[STIX]{x1D707}/2-\unicode[STIX]{x1D705}\leqslant 1$
. We then have
On the other hand, we have
This establishes Lemma 2.3 by collecting the above estimates of
$\mathscr{B}_{1}$
and
$\mathscr{B}_{2}$
.◻
2.4 Double
$q$
-van der Corput method
The classical van der Corput method can also be applied to the two-dimensional exponential sums of the shape
where
$\unicode[STIX]{x1D6FA}$
is a rectangle, or some set bounded by simple algebraic curves. As an analogue of
$q$
-van der Corput, one can also develop a double
$q$
-van der Corput method to study the double sums of trace functions
where
$\mathscr{K}$
is the trace function of an
$\ell$
-adic middle-extension sheaf on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}\times \mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
. In applications to Theorem 1.1, we need the simpler case
$\mathscr{K}(m,n)=K(mn)$
with
$K$
being the amiable trace function given as above. This leads to the study of the bilinear form
$\mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)$
, and we will prove the following alternative estimate for
$\mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)$
with smooth coefficients
$\boldsymbol{\unicode[STIX]{x1D6FC}}$
and
$\boldsymbol{\unicode[STIX]{x1D6FD}}$
, provided that
$K$
is the trace function of certain Kloosterman sheaves.
Lemma 2.4. Suppose
$U,V\in \mathscr{C}^{\infty }(\mathbf{R}^{+})$
are two bounded smooth functions with compact support and let
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{m}=U\biggl(\frac{m}{M}\biggr),\qquad \unicode[STIX]{x1D6FD}_{n}=V\biggl(\frac{n}{N}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Let
$a$
be a fixed integer and
$q\geqslant 1$
a squarefree number which has a factor
$q_{1}\in (q^{1/3-\unicode[STIX]{x1D700}},q^{1/3+\unicode[STIX]{x1D700}})$
. For
$K=\text{Kl}_{3}(a\cdot ,q),$
we have
$$\begin{eqnarray}\displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K) & \ll (q\mathit{MN})^{\unicode[STIX]{x1D700}}\mathit{MN}\{q^{-1/3}+M^{-1/2}+N^{-1/2}+q^{1/3}(\mathit{MN})^{-1/2}\}. & \displaystyle \nonumber\end{eqnarray}$$
Remark 2. If
$\max \{M,N\}\ll q^{2/3}$
, the upper bound in Lemma 2.4 can be reduced to
$$\begin{eqnarray}\displaystyle (q\mathit{MN})^{\unicode[STIX]{x1D700}}\mathit{MN}\cdot (\mathit{MN})^{-1/2}q^{1/3}=(q\mathit{MN})^{\unicode[STIX]{x1D700}}\biggl(\frac{q}{M}\biggr)^{\unicode[STIX]{x1D705}}\biggl(\frac{q}{N}\biggr)^{\unicode[STIX]{x1D705}}M^{\unicode[STIX]{x1D706}}N^{\unicode[STIX]{x1D706}} & & \displaystyle \nonumber\end{eqnarray}$$
with
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706})=(\frac{1}{6},\frac{2}{3})$
. This coincides with the two-dimensional van der Corput method in the classical case; see Conjecture Q in [Reference Graham and Kolesnik10, p. 73] for details.
One can see that Lemma 2.4 is non-trivial for
$M>q^{\unicode[STIX]{x1D700}},N>q^{\unicode[STIX]{x1D700}}$
and
$\mathit{MN}>q^{2/3+\unicode[STIX]{x1D700}}$
. This can be compared with the estimate of Fouvry, Kowalski and Michel [Reference Fouvry, Kowalski and Michel5, Theorem 1.16] in the case of prime moduli
$p$
, where they have to assume
$\mathit{MN}>p^{3/4+\unicode[STIX]{x1D700}}$
.
We believe Lemma 2.4 holds for general trace functions
$K$
. The situation is much simplier if
$K$
is a non-trivial Dirichlet character mod
$q$
, in which case the bilinear form becomes the product of two linear forms, so that arithmetic exponent pairs developed in [Reference Wu and Xi22] apply.
Proof. The lemma becomes trivial if
$\max \{M,N\}<2q^{1/3}$
. We first assume
$M<2q^{1/3}\leqslant N<q,$
in which case one may apply the arithmetic exponent pairs to the sum over
$n$
with
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706})=(\frac{1}{6},\frac{2}{3})$
, getting
$$\begin{eqnarray}\displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)\ll (q\mathit{MN})^{\unicode[STIX]{x1D700}}M\biggl(\frac{q}{N}\biggr)^{1/6}N^{2/3}\ll (q\mathit{MN})^{\unicode[STIX]{x1D700}}q^{1/6}\mathit{MN}^{1/2}. & & \displaystyle \nonumber\end{eqnarray}$$
This is at most
$(q\mathit{MN})^{\unicode[STIX]{x1D700}}q^{1/3}(\mathit{MN})^{1/2}$
since
$M<2q^{1/3}.$
It remains to consider the case
$\min \{M,N\}\geqslant 2q^{1/3},$
which we henceforth assume.
Suppose
$q=rs$
. For any
$l_{1},l_{2}\in \mathbf{Z}$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)\nonumber\\ \displaystyle & & \displaystyle =\mathop{\sum }_{m}\mathop{\sum }_{n}U\biggl(\frac{m+l_{1}r}{M}\biggr)V\biggl(\frac{n+l_{2}r}{N}\biggr)K_{rs}((m+l_{1}r)(n+l_{2}r))\nonumber\\ \displaystyle & & \displaystyle =\mathop{\sum }_{m}\mathop{\sum }_{n}U\biggl(\frac{m+l_{1}r}{M}\biggr)V\biggl(\frac{n+l_{2}r}{N}\biggr)K_{r}(mn)K_{s}((m+l_{1}r)(n+l_{2}r)),\nonumber\end{eqnarray}$$
where
$K_{s},K_{r}$
are the corresponding trace functions
$\,(\text{mod}\,s)$
and
$\,(\text{mod}\,r)$
, respectively, both of which are also amiable as long as
$r,s$
are both large enough.
Summing over
$l_{1}\leqslant L_{1}$
and
$l_{2}\leqslant L_{2}$
with
$L_{1}\leqslant 1+[M/r],L_{2}\leqslant 1+[N/r]$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)\ll \frac{r^{\unicode[STIX]{x1D700}}}{L_{1}L_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \mathop{\sum }_{m\ll M}\,\mathop{\sum }_{n\ll N}\biggl|\mathop{\sum }_{l_{1}\leqslant L_{1}}\,\mathop{\sum }_{l_{2}\leqslant L_{2}}U\biggl(\frac{m+l_{1}r}{M}\biggr)V\biggl(\frac{n+l_{2}r}{N}\biggr)K_{s}((m+l_{1}r)(n+l_{2}r))\biggr|.\nonumber\end{eqnarray}$$
By Fourier inversion, we have
$$\begin{eqnarray}\displaystyle U\biggl(\frac{m+l_{1}r}{M}\biggr) & = & \displaystyle \int _{\mathbf{R}}\widehat{U}(y)\text{e}\biggl(\frac{(m+l_{1}r)y}{M}\biggr)\,dy,\nonumber\\ \displaystyle V\biggl(\frac{n+l_{2}r}{N}\biggr) & = & \displaystyle \int _{\mathbf{R}}\widehat{V}(y)\text{e}\biggl(\frac{(n+l_{2}r)y}{N}\biggr)\,dy,\nonumber\end{eqnarray}$$
from which we may derive that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)\ll \frac{r^{\unicode[STIX]{x1D700}}}{L_{1}L_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\mathop{\sum }_{m\ll M}\,\mathop{\sum }_{n\ll N}\biggl|\mathop{\sum }_{l_{1}\leqslant L_{1}}\,\mathop{\sum }_{l_{2}\leqslant L_{2}}\text{e}(l_{1}y_{1}+l_{2}y_{2})K_{s}((m+l_{1}r)(n+l_{2}r))\biggr|\nonumber\end{eqnarray}$$
for some
$y_{1},y_{2}\in \mathbf{R}.$
By Cauchy’s inequality, we find
with
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}(l_{1},l_{2})=\mathop{\sum }_{m\in \mathscr{M}(l_{1})}\,\mathop{\sum }_{n\in \mathscr{N}(l_{2})}K_{s}(mn)\overline{K_{s}((m+l_{1}r)(n+l_{2}r))}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\mathscr{M}(l_{1}),\mathscr{N}(l_{2})$
are some intervals with
$|\mathscr{M}(l_{1})|\ll M$
and
$|\mathscr{N}(l_{2})|\ll N.$
Here we have used the estimates
$$\begin{eqnarray}\displaystyle |\widehat{U}(y)|,\qquad |\widehat{V}(y)|\ll (1+|y|)^{-A},\quad A\geqslant 0 & & \displaystyle \nonumber\end{eqnarray}$$
from integration by parts.
We claim, for all
$l_{1},l_{2}\in \mathbf{Z},$
that
$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6F1}(l_{1},l_{2})|\ll (\mathit{MN}s)^{\unicode[STIX]{x1D700}}\{s^{-1}\mathit{MN}(l_{1},l_{2},s)+M+N+s\}. & & \displaystyle\end{eqnarray}$$
Hence,
$$\begin{eqnarray}\displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)^{2} & \ll & \displaystyle \frac{\mathit{MN}q^{\unicode[STIX]{x1D700}}}{L_{1}L_{2}}\mathop{\sum }_{0\leqslant |l_{1}|\leqslant L_{1}}\,\mathop{\sum }_{0\leqslant |l_{2}|\leqslant L_{2}}\{s^{-1}\mathit{MN}(l_{1},l_{2},s)+M+N+s\}\nonumber\\ \displaystyle & \ll & \displaystyle \frac{\mathit{MN}q^{\unicode[STIX]{x1D700}}}{L_{1}L_{2}}\{\mathit{MN}+L_{1}L_{2}(s^{-1}\mathit{MN}+M+N+s)\}.\nonumber\end{eqnarray}$$
By taking
$L_{1}=1+[M/r],L_{2}=1+[N/r]$
, we get
$$\begin{eqnarray}\displaystyle \mathscr{B}(\boldsymbol{\unicode[STIX]{x1D6FC}},\boldsymbol{\unicode[STIX]{x1D6FD}};K)\ll (\mathit{MN})^{1/2+\unicode[STIX]{x1D700}}\{r+(s^{-1}\mathit{MN}+M+N+s)^{1/2}\}, & & \displaystyle \nonumber\end{eqnarray}$$
provided that
$\min \{M,N\}\gg r.$
The lemma follows by choosing
$r,s$
such that
$q^{1/3-\unicode[STIX]{x1D700}}\ll r\ll q^{1/3+\unicode[STIX]{x1D700}}$
.
It remains to prove (2.9). By the completion method, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle |\unicode[STIX]{x1D6F1}(l_{1},l_{2})|\nonumber\\ \displaystyle & & \displaystyle \quad \ll \frac{1}{s^{2}}\mathop{\sum }_{|h_{1}|<s/2}\,\mathop{\sum }_{|h_{2}|<s/2}\!\min \biggl\{M,\frac{s}{|h_{1}|}\biggr\}\!\min \biggl\{N,\frac{s}{|h_{2}|}\biggr\}|\unicode[STIX]{x1D6F4}(h_{1},h_{2};l_{1},l_{2};s)|,\nonumber\end{eqnarray}$$
where
Then (2.9) follows from an upper bound for
$\unicode[STIX]{x1D6F4}(h_{1},h_{2};l_{1},l_{2};s).$
From the Chinese remainder theorem, it suffices to consider the case of prime moduli, which has been studied in Theorem B.1 in Appendix B. More precisely, put
$s_{1}=(s,a)$
and
$s_{2}=s/s_{1}$
, we then have
$(as_{1},s_{2})=1$
since
$s$
is squarefree. From the Chinese remainder theorem, we may decompose
Note that
$|K_{s_{1}}(0)|^{2}=1/s_{1}^{2}$
. Thus,
$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6F4}(h_{1},h_{2};l_{1},l_{2};s)|\leqslant |\unicode[STIX]{x1D6F4}(\overline{s_{1}}h_{1},\overline{s_{1}}h_{2};l_{1},l_{2};s_{2})|. & & \displaystyle \nonumber\end{eqnarray}$$
By Theorem B.1 we then have
$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6F4}(h_{1},h_{2};l_{1},l_{2};s)|\ll s_{2}^{1+\unicode[STIX]{x1D700}}(h_{1},h_{2},l_{1},l_{2},s_{2})\leqslant s^{1+\unicode[STIX]{x1D700}}(h_{1},h_{2},l_{1},l_{2},s), & & \displaystyle \nonumber\end{eqnarray}$$
from which we derive that
$$\begin{eqnarray}\displaystyle & & \displaystyle |\unicode[STIX]{x1D6F1}(l_{1},l_{2})|\nonumber\\ \displaystyle & & \displaystyle \quad \ll \frac{1}{s^{2}}\mathop{\sum }_{|h_{1}|\leqslant s/2}\,\mathop{\sum }_{|h_{2}|\leqslant s/2}\!\min \biggl\{M,\frac{s}{|h_{1}|}\biggr\}\!\min \biggl\{N,\frac{s}{|h_{2}|}\biggr\}s^{1+\unicode[STIX]{x1D700}}(h_{1},h_{2},l_{1},l_{2},s)\nonumber\\ \displaystyle & & \displaystyle \quad \ll (\mathit{MN}s)^{\unicode[STIX]{x1D700}}\{s^{-1}\mathit{MN}(l_{1},l_{2},s)+M+N+s\}\nonumber\end{eqnarray}$$
as claimed in (2.9), and thus completing the proof of Lemma 2.4. ◻
3 Reduction of the problem
In what follows, we assume
$X\geqslant 3$
is large. Recall that
$$\begin{eqnarray}\displaystyle \mathscr{D}_{3}(X;q,a)=\mathop{\sum }_{\substack{ n_{1}n_{2}n_{3}\leqslant X \\ n_{1}n_{2}n_{3}\equiv a\,(\text{mod}\,q)}}1. & & \displaystyle\end{eqnarray}$$
A fundamental tool in the transformation of
$\mathscr{D}_{3}(X;q,a)$
is the Poisson summation formula. Hence, we would like to remove the condition
$n_{1}n_{2}n_{3}\leqslant X,$
so that the three variables
$n_{1},n_{2},n_{3}$
are independent and smooth. To this end, we follow the arguments in [Reference Fouvry, Kowalski and Michel6] and appeal to the following lemma as a smooth partition of unity, which works well in our situation (see [Reference Fouvry3, Lemme 2] for instance).
Lemma 3.1. For every
$\unicode[STIX]{x1D6E5}>1,$
there exists a sequence
$\{b_{l,\unicode[STIX]{x1D6E5}}\}_{l\geqslant 0}$
of smooth functions with support included in
$[\unicode[STIX]{x1D6E5}^{l-1},\unicode[STIX]{x1D6E5}^{l+1}]$
, such that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{l\geqslant 0}b_{l,\unicode[STIX]{x1D6E5}}(x)=1\quad \text{for all }x\geqslant 1, & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle x^{j}b_{l,\unicode[STIX]{x1D6E5}}^{(j)}(x)\ll (1-\unicode[STIX]{x1D6E5}^{-1})^{-j}\quad \text{for all }x\geqslant 1\text{ and }j,l\geqslant 0. & & \displaystyle\end{eqnarray}$$
In our applications, we take
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}=1+(\log X)^{-B} & & \displaystyle \nonumber\end{eqnarray}$$
for some parameter
$B\geqslant 1.$
From now on, we denote by
$N_{j},1\leqslant j\leqslant 3$
, some parameters of the form
$$\begin{eqnarray}\displaystyle N_{j}=\unicode[STIX]{x1D6E5}^{l}=(1+(\log X)^{-B})^{l} & & \displaystyle\end{eqnarray}$$
for some
$l\geqslant 1.$
For such a variable
$N_{j}=\unicode[STIX]{x1D6E5}^{l}$
, we define
$$\begin{eqnarray}\displaystyle V_{j}=b_{l,\unicode[STIX]{x1D6E5}} & & \displaystyle\end{eqnarray}$$
as given in Lemma 3.1. Thus, the derivatives of
$V_{j}$
satisfy
$$\begin{eqnarray}\displaystyle x^{l}V_{j}^{(l)}(x)\ll (\log X)^{Bl} & & \displaystyle\end{eqnarray}$$
for all
$l\geqslant 0$
, where the implied constant may depend on
$l$
. Moreover, integration by parts gives
$$\begin{eqnarray}\displaystyle \widehat{V}_{j}(\unicode[STIX]{x1D709})\ll N_{j}(\log X)^{Bl}(1+|\unicode[STIX]{x1D709}|N_{j})^{-l} & & \displaystyle\end{eqnarray}$$
for any fixed
$l\geqslant 0$
, and the implied constant also depends on
$l$
.
For
$\boldsymbol{N}=(N_{1},N_{2},N_{3})$
, we now consider the smoothed sum
Define
We appeal to the following lemma for the separation of variables; the proof can be found in [Reference Fouvry, Kowalski and Michel6, Lemma 4.2].
Lemma 3.2. For any
$A>0,$
one can choose
$B\geqslant 1$
such that
$$\begin{eqnarray}\displaystyle \mathscr{D}_{3}(X;q,a)-\mathscr{M}_{3}(X;q,a) & = & \displaystyle \mathop{\sum }_{\boldsymbol{N}}\biggl\{\mathscr{D}_{3}(\boldsymbol{N};q,a)-\frac{1}{\unicode[STIX]{x1D711}(q)}\mathscr{D}_{3}(\boldsymbol{N};q)\biggr\}\nonumber\\ \displaystyle & & \displaystyle +\,O\biggl(\frac{X}{q}(\log X)^{-A}\biggr),\nonumber\end{eqnarray}$$
where
$\boldsymbol{N}=(N_{1},N_{2},N_{3})$
runs over triples of
$\boldsymbol{N}$
as above such that
$$\begin{eqnarray}\displaystyle X(\log X)^{-B}\leqslant N_{1}N_{2}N_{3}\leqslant X. & & \displaystyle\end{eqnarray}$$
Due to the symmetry of the problem, we assume
$$\begin{eqnarray}\displaystyle N_{1}\leqslant N_{2}\leqslant N_{3}. & & \displaystyle\end{eqnarray}$$
Since the number of triples
$\boldsymbol{N}$
satisfying (3.8) with
$N_{i}$
of the form (3.3) is
$O((\log X)^{3B+3})$
, Lemma 3.2 shows that Theorem 1.1 will follow if we can show that for any
$\unicode[STIX]{x1D703}<\frac{9}{17}$
and
$A>0$
, we have
$$\begin{eqnarray}\displaystyle \mathscr{D}_{3}(\boldsymbol{N};q,a)-\frac{1}{\unicode[STIX]{x1D711}(q)}\mathscr{D}_{3}(\boldsymbol{N};q)\ll \frac{X}{q}(\log X)^{-A} & & \displaystyle\end{eqnarray}$$
uniformly for all triples
$\boldsymbol{N}=(N_{1},N_{2},N_{3})$
satisfying (3.8), (3.9), and (3.3) and for all integers
$a$
coprime to
$q$
satisfying
$q\asymp X^{\unicode[STIX]{x1D703}}$
.
To evaluate
$\mathscr{D}_{3}(\boldsymbol{N};q,a),$
we would like to invoke the following Poisson summation formula.
Lemma 3.3. Suppose that
$g$
and
$\widehat{g}$
are both in
$L^{1}(\mathbf{R})$
. For any fixed integers
$d$
and
$a$
, we have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ n\in \mathbf{Z} \\ n\equiv a\,(\text{mod}\,d)}}g(n)=\frac{1}{d}\mathop{\sum }_{h\in \mathbf{Z}}\widehat{g}\biggl(\frac{h}{d}\biggr)\text{e}\biggl(\frac{ha}{d}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Note that
The following lemma can be obtained by applying Lemma 3.3 in different situations. The first identity follows from the applications of Poisson summation to each variable, and this is essentially [Reference Fouvry, Kowalski and Michel6, Corollary 2.3]. The second one is concluded by applying Poisson twice, and the third is produced by only one Poisson to
$n_{3}$
; both of them are complementary to (3.11).
Lemma 3.4. With the notation as above, we have
We would like to evaluate
$\mathscr{D}_{3}(\boldsymbol{N};q,a)$
by virtue of (3.11), (3.12), and (3.13). The main term is expected to come from the terms with zero frequency. To do so, we define
and
$$\begin{eqnarray}\displaystyle \mathscr{E}_{j}(\boldsymbol{N};q,a)=\mathscr{D}_{3}(\boldsymbol{N};q,a)-\mathscr{M}_{j}(\boldsymbol{N}),\quad j=1,2,3. & & \displaystyle\end{eqnarray}$$
For each
$j\in \{1,2,3\}$
, we may evaluate
$\mathscr{M}_{j}$
asymptotically and each of them gives the expected main term
$\mathscr{D}_{3}(\boldsymbol{N};q)/\unicode[STIX]{x1D711}(q).$
The evaluations can be stated as follows.
Proposition 3.1. For each
$j\in \{1,2,3\}$
, we have
$$\begin{eqnarray}\displaystyle \mathscr{M}_{j}(\boldsymbol{N}) & =\frac{1}{\unicode[STIX]{x1D711}(q)}\mathscr{D}_{3}(\boldsymbol{N};q)+O\biggl(\frac{X}{qN_{j}}\unicode[STIX]{x1D70F}(q)(\log X)^{3B}\biggr). & \displaystyle \nonumber\end{eqnarray}$$
Proof. From Poisson summation (Lemma 3.3), it follows that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ n_{3}\in \mathbf{Z} \\ (n_{3},q)=1}}V_{3}(n_{3}) & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{m_{3}\in \mathbf{Z}}\widehat{V}_{3}\biggl(\frac{m_{3}}{q}\biggr)S_{2}(0,m_{3};q)\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x1D711}(q)}{q}\widehat{V}_{3}(0)+\frac{1}{\sqrt{q}}\mathop{\sum }_{|m_{3}|\geqslant 1}\widehat{V}_{3}\biggl(\frac{m_{3}}{q}\biggr)S_{2}(0,m_{3};q),\nonumber\end{eqnarray}$$
where
$S_{2}(0,m;q)$
satisfies the bound
$|S_{2}(0,m;q)|\leqslant (m,q)/\sqrt{q}$
. From (3.6), the second summand is bounded by
$$\begin{eqnarray}\displaystyle \ll \frac{N_{3}(\log X)^{2B}}{q}\biggl(\mathop{\sum }_{m_{3}\leqslant q/N_{3}}(m_{3},q)+\frac{q^{2}}{N_{3}^{2}}\mathop{\sum }_{m_{3}>q/N_{3}}\frac{(m_{3},q)}{m_{3}^{2}}\biggr)\ll \unicode[STIX]{x1D70F}(q)(\log X)^{2B}. & & \displaystyle \nonumber\end{eqnarray}$$
Hence, we conclude that
$$\begin{eqnarray}\displaystyle \widehat{V}_{3}(0)=\frac{q}{\unicode[STIX]{x1D711}(q)}\mathop{\sum }_{\substack{ n_{3}\in \mathbf{Z} \\ (n_{3},q)=1}}V_{3}(n_{3})+O(\unicode[STIX]{x1D70F}(q)(\log X)^{3B}), & & \displaystyle \nonumber\end{eqnarray}$$
where we have used the elementary fact
$\unicode[STIX]{x1D711}(q)\gg q/\!\log q.$
It now follows that
$$\begin{eqnarray}\displaystyle \mathscr{M}_{3}(\boldsymbol{N})=\frac{1}{\unicode[STIX]{x1D711}(q)}\mathscr{D}_{3}(\boldsymbol{N};q)+O\biggl(\frac{N_{1}N_{2}}{q}\unicode[STIX]{x1D70F}(q)(\log X)^{3B}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
On the other hand, from the inclusion–exclusion principle, we may write
By Poisson summation, we derive that
In a similar way, we can also conclude
$$\begin{eqnarray}\displaystyle \mathscr{M}_{1}(\boldsymbol{N})=\frac{1}{\unicode[STIX]{x1D711}(q)}\mathscr{D}_{3}(\boldsymbol{N};q)+O\biggl(\frac{N_{2}N_{3}}{q}\unicode[STIX]{x1D70F}(q)(\log X)^{3B}\biggr).\Box & & \displaystyle \nonumber\end{eqnarray}$$
The next section will be devoted to the estimates for
$\mathscr{E}_{j}(\boldsymbol{N};q,a),j=1,2,3$
.
4 Estimates for
$\mathscr{E}_{j}(\boldsymbol{N};q,a)$
The estimates for
$\mathscr{E}_{j}(\boldsymbol{N};q,a)$
form the essential part of this paper. Roughly speaking, we will apply Lemma 2.3 to
$\mathscr{E}_{1}(\boldsymbol{N};q,a)$
by re-grouping variables and this argument works well if the three variables are of comparable sizes. When there are reasonably long summations, we would apply Lemma 2.4 to
$\mathscr{E}_{1}(\boldsymbol{N};q,a)$
. On the other hand, the treatments to
$\mathscr{E}_{2}(\boldsymbol{N};q,a)$
and
$\mathscr{E}_{3}(\boldsymbol{N};q,a)$
are both trivial, and these are equivalent to the known levels of distributions for
$\unicode[STIX]{x1D70F}_{2}$
and
$\unicode[STIX]{x1D70F}_{1}.$
Put
$$\begin{eqnarray}N_{j}=X^{\unicode[STIX]{x1D707}_{j}},\quad j=1,2,3,\end{eqnarray}$$
so that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{1}\leqslant \unicode[STIX]{x1D707}_{2}\leqslant \unicode[STIX]{x1D707}_{3}. & & \displaystyle\end{eqnarray}$$
The following estimates for
$\mathscr{E}_{j}(\boldsymbol{N};q,a)$
will work complementarily to conclude the final level in Theorem 1.1.
Proposition 4.1. With the notation as above, we have
$$\begin{eqnarray}\displaystyle \mathscr{E}_{1}(\boldsymbol{N};q,a)\ll \frac{X}{q(\log X)^{A}} & & \displaystyle \nonumber\end{eqnarray}$$
if one of the following two conditions holds:
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\leqslant \unicode[STIX]{x1D703}<\frac{4}{7},\qquad \frac{(2\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}+2)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}-1}{\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}+1}<\unicode[STIX]{x1D707}_{3}<2-3\unicode[STIX]{x1D703}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\leqslant \unicode[STIX]{x1D703}<\frac{3}{5},\qquad \unicode[STIX]{x1D707}_{2}<2-3\unicode[STIX]{x1D703},\qquad \unicode[STIX]{x1D707}_{3}>\frac{8}{3}\unicode[STIX]{x1D703}-1. & \displaystyle\end{eqnarray}$$
Proposition 4.2. With the notation as above, we have
$$\begin{eqnarray}\displaystyle \mathscr{E}_{2}(\boldsymbol{N};q,a)\ll \frac{X}{q(\log X)^{A}} & & \displaystyle \nonumber\end{eqnarray}$$
if
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{2}+\unicode[STIX]{x1D707}_{3}>{\textstyle \frac{3}{2}}\unicode[STIX]{x1D703}.\end{eqnarray}$$
Proposition 4.3. With the notation as above, we have
$$\begin{eqnarray}\displaystyle \mathscr{E}_{3}(\boldsymbol{N};q,a)\ll \frac{X}{q(\log X)^{A}} & & \displaystyle \nonumber\end{eqnarray}$$
if
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{3}>\unicode[STIX]{x1D703}.\end{eqnarray}$$
We now start the proof of the above three propositions. Recall that
According to (3.6), the Fourier transform
$\widehat{V_{i}}(\unicode[STIX]{x1D709})$
is essentially supported in the interval
$|\unicode[STIX]{x1D709}|<X^{\unicode[STIX]{x1D700}}qN_{i}^{-1}$
. By dyadic decomposition, it suffices to consider the sum
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{1}:=\frac{N_{1}N_{2}N_{3}}{q^{2}}\mathop{\sum }_{m_{1}\sim M_{1}}\,\mathop{\sum }_{m_{2}\sim M_{2}}\,\mathop{\sum }_{m_{3}\sim M_{3}}\unicode[STIX]{x1D6FC}_{1}(m_{1})\unicode[STIX]{x1D6FC}_{2}(m_{2})\unicode[STIX]{x1D6FC}_{3}(m_{3})S_{3}(am_{1},m_{2},m_{3};q), & & \displaystyle \nonumber\end{eqnarray}$$
where
$0<M_{j}<X^{\unicode[STIX]{x1D700}}qN_{j}^{-1}$
for
$1\leqslant j\leqslant 3$
, and we write
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{j}(m_{j})=\frac{1}{N_{j}}\widehat{V}_{j}\biggl(\frac{m_{j}}{q}\biggr),\quad j=1,2,3, & & \displaystyle\end{eqnarray}$$
so that
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{j}(m_{j})\ll (\log X)^{B},\quad j=1,2,3.\end{eqnarray}$$
We may regard
$\unicode[STIX]{x1D6E5}_{1}$
as a trilinear form of Kloosterman sums, and by re-grouping variables, we can apply the estimates for bilinear forms in Lemma 2.3 to produce non-trivial bounds. To do so, one has to transform
$S_{3}(am_{1},m_{2},m_{3};q)$
to the shape
$S_{3}(am_{1}m_{2}m_{3},1,1;q)$
, for which we appeal to the Kuznetsov–Selberg-type identity as given in Lemma 2.1, getting
Define
Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{1} & = & \displaystyle \frac{N_{1}N_{2}N_{3}}{q^{2}}\mathop{\sum }_{drs|q}d\unicode[STIX]{x1D707}(rs)\mathop{\sum }_{M_{1}M_{2}/(dr)^{2}s<l\leqslant 4M_{1}M_{2}/(dr)^{2}s}\;\mathop{\sum }_{m_{3}\sim M_{3}/ds}\nonumber\\ \displaystyle & & \displaystyle \times \,\unicode[STIX]{x1D6FE}(ls,dr)\unicode[STIX]{x1D6FC}_{3}(m_{3}ds)\text{Kl}_{3}\biggl(\overline{rs}lm_{3},\frac{q}{drs}\biggr).\nonumber\end{eqnarray}$$
For fixed
$d,r,s$
with
$drs|q$
, we are now in a position to apply Lemma 2.3 to the inner sums over
$l,m_{3}$
with
$$\begin{eqnarray}\displaystyle q\rightarrow \frac{q}{drs},\qquad M\asymp \frac{M_{1}M_{2}}{(dr)^{2}s},\qquad N\asymp \frac{M_{3}}{ds}, & & \displaystyle \nonumber\end{eqnarray}$$
getting
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{1} & \ll & \displaystyle \frac{N_{1}N_{2}N_{3}}{q^{2}}\mathop{\sum }_{drs|q}d\unicode[STIX]{x1D707}^{2}(rs)\biggl|\mathop{\sum }_{l}\mathop{\sum }_{m_{3}}\unicode[STIX]{x1D6FE}(ls,dr)\unicode[STIX]{x1D6FC}_{3}(m_{3}ds)\text{Kl}_{3}\biggl(\overline{rs}lm_{3},\frac{q}{drs}\biggr)\biggr|\nonumber\\ \displaystyle & \ll & \displaystyle X^{\unicode[STIX]{x1D700}}\frac{N_{1}N_{2}N_{3}}{q^{2}}\mathop{\sum }_{drs|q}d\unicode[STIX]{x1D707}^{2}(rs)\frac{M_{1}M_{2}M_{3}}{d^{3}(rs)^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl\{\unicode[STIX]{x1D700}_{1}\biggl(\frac{q}{drs}\biggr)^{\unicode[STIX]{x1D705}/2}\biggl(\frac{M_{1}M_{2}}{(dr)^{2}s}\biggr)^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}-1)/2}+\biggl(\frac{q}{drs}\biggr)^{-1/4}+\biggl(\frac{M_{3}}{ds}\biggr)^{-1/2}\biggr\},\nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D700}_{1}=1$
if
$M_{1}M_{2}/(dr)^{2}s<q/drs$
and vanishes otherwise. Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{1} & \ll & \displaystyle X^{\unicode[STIX]{x1D700}}N_{1}N_{2}N_{3}(M_{1}M_{2})^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}+1)/2}M_{3}q^{\unicode[STIX]{x1D705}/2-2}\mathop{\sum }_{drs|q}(dr)^{\unicode[STIX]{x1D705}/2-\unicode[STIX]{x1D706}-1}s^{-(\unicode[STIX]{x1D706}+3)/2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{X^{\unicode[STIX]{x1D700}}N_{1}N_{2}N_{3}M_{1}M_{2}M_{3}}{q^{2}}(q^{-1/4}+M_{3}^{-1/2})\nonumber\\ \displaystyle & \ll & \displaystyle X^{\unicode[STIX]{x1D700}}N_{1}N_{2}N_{3}(M_{1}M_{2})^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}+1)/2}M_{3}q^{\unicode[STIX]{x1D705}/2-2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{X^{\unicode[STIX]{x1D700}}N_{1}N_{2}N_{3}M_{1}M_{2}M_{3}}{q^{2}}(q^{-1/4}+M_{3}^{-1/2}).\nonumber\end{eqnarray}$$
Recall that
$0<M_{j}<X^{\unicode[STIX]{x1D700}}qN_{j}^{-1}$
for
$1\leqslant j\leqslant 3$
. Hence, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{1}\ll X^{\unicode[STIX]{x1D700}}\{q^{\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}/2}X^{(\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}+1)/2}N_{3}^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}-1)/2}+q^{3/4}+(qN_{3})^{1/2}\}. & & \displaystyle \nonumber\end{eqnarray}$$
This would yield
$$\begin{eqnarray}\displaystyle \mathscr{E}_{1}(\boldsymbol{N};q,a)\ll \frac{X}{q}(\log X)^{-A} & & \displaystyle \nonumber\end{eqnarray}$$
as long as
$$\begin{eqnarray}\displaystyle q^{\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}/2}X^{(\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}+1)/2}N_{3}^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}-1)/2}+q^{3/4}+(qN_{3})^{1/2}\ll \frac{X^{1-\unicode[STIX]{x1D700}}}{q}. & & \displaystyle \nonumber\end{eqnarray}$$
This is implied by (4.2a) while noting the definitions of
$\unicode[STIX]{x1D707}_{3}$
and
$\unicode[STIX]{x1D703}$
.
Applying Lemma 2.4 to the
$m_{1},m_{2}$
-sums in (4.6), we get
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{1}\ll X^{\unicode[STIX]{x1D700}}\{q^{2/3}+(qN_{2})^{1/2}+q^{1/3}(N_{1}N_{2})^{1/2}\}. & & \displaystyle \nonumber\end{eqnarray}$$
This is acceptable if (4.2b) holds. This completes the proof of Proposition 4.1.
We now start from (3.12), which is obtained by applying Poisson summation twice. According to the decays of Fourier transforms in (3.6), it suffices to consider the sum
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{2}:=\frac{N_{2}N_{3}}{q^{3/2}}\mathop{\sum }_{\substack{ n_{1}\in \mathbf{Z} \\ (n_{1},q)=1}}\mathop{\sum }_{m_{2}\sim M_{2}}\,\mathop{\sum }_{m_{3}\sim M_{3}}V_{1}(n_{1})\unicode[STIX]{x1D6FC}_{2}(m_{2})\unicode[STIX]{x1D6FC}_{3}(m_{3})S_{2}(an_{1}m_{2},m_{3};q), & & \displaystyle \nonumber\end{eqnarray}$$
where
$0<M_{j}<X^{\unicode[STIX]{x1D700}}qN_{j}^{-1}$
for
$2\leqslant j\leqslant 3$
, and
$\unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FC}_{3}$
are defined by (4.5). Proposition 4.2 follows from Weil’s bound for
$S_{2}$
. In fact, from (2.2) it follows that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{2}\ll \frac{X^{\unicode[STIX]{x1D700}}N_{1}N_{2}N_{3}\unicode[STIX]{x1D70F}(q)}{q^{3/2}}\mathop{\sum }_{m_{2}\sim M_{2}}\,\mathop{\sum }_{m_{3}\sim M_{3}}(m_{2},m_{3},q)^{1/2}\ll X^{1+\unicode[STIX]{x1D700}}q^{1/2}(N_{2}N_{3})^{-1}. & & \displaystyle \nonumber\end{eqnarray}$$
This yields
$$\begin{eqnarray}\displaystyle \mathscr{E}_{2}(\boldsymbol{N};q,a) & \ll \frac{X}{q(\log X)^{A}} & \displaystyle \nonumber\end{eqnarray}$$
under the condition of (4.3).
Proposition 4.3 follows from trivial treatments to each summand.
5 Proof of Theorem 1.1
As argued in §3, it suffices to prove (3.10) uniformly for all triples
$\boldsymbol{N}=(N_{1},N_{2},N_{3})$
, satisfying (3.8), (3.9), and (3.3) and for all integers
$a$
coprime to
$q$
satisfying
$q\asymp X^{\unicode[STIX]{x1D703}}$
.
First, we show all possible cases for
$(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2},\unicode[STIX]{x1D707}_{3})$
(and thus for
$\boldsymbol{N}=(N_{1},N_{2},N_{3})$
) can be covered by Propositions 4.1 and 4.2. We now assume
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{2}+\unicode[STIX]{x1D707}_{3}\leqslant {\textstyle \frac{3}{2}}\unicode[STIX]{x1D703},\quad \unicode[STIX]{x1D707}_{3}\leqslant \unicode[STIX]{x1D703};\end{eqnarray}$$
otherwise, one can employ Propositions 4.2 or 4.3.
Furthermore, we assume (4.2b) is not valid, and it suffices to show that the resultant
$\unicode[STIX]{x1D707}_{2},\unicode[STIX]{x1D707}_{3}$
fall into the area given by (4.2a).
Case 1:
$\unicode[STIX]{x1D707}_{2}\geqslant 2-3\unicode[STIX]{x1D703}$
. From (5.1) it follows that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{3}\leqslant {\textstyle \frac{3}{2}}\unicode[STIX]{x1D703}-(2-3\unicode[STIX]{x1D703})<2-3\unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x1D707}_{2} & & \displaystyle \nonumber\end{eqnarray}$$
if
$\frac{1}{2}<\unicode[STIX]{x1D703}<\frac{8}{15},$
in which case we have a contradiction to (4.1). We thus find that
$\unicode[STIX]{x1D707}_{2}\geqslant 2-3\unicode[STIX]{x1D703}$
does not hold if
$\frac{1}{2}<\unicode[STIX]{x1D703}<\frac{8}{15}$
.
Case 2:
$\unicode[STIX]{x1D707}_{2}<2-3\unicode[STIX]{x1D703}$
and
$\unicode[STIX]{x1D707}_{3}\leqslant \frac{8}{3}\unicode[STIX]{x1D703}-1$
. Note that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{3}\leqslant {\textstyle \frac{8}{3}}\unicode[STIX]{x1D703}-1<2-3\unicode[STIX]{x1D703} & & \displaystyle \nonumber\end{eqnarray}$$
if
$\frac{1}{2}<\unicode[STIX]{x1D703}<\frac{9}{17}$
.
It remains to consider the restriction
$$\begin{eqnarray}\frac{(2\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}+2)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}-1}{\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}+1}<\unicode[STIX]{x1D707}_{3},\end{eqnarray}$$
and we would like to show this is redundant by choosing suitable
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706}).$
Note that
$\unicode[STIX]{x1D707}_{3}\geqslant \frac{1}{3}$
. For
$(\unicode[STIX]{x1D705},\unicode[STIX]{x1D706})=(\frac{1}{2},\frac{1}{2})$
, we have
$$\begin{eqnarray}\frac{(2\unicode[STIX]{x1D706}-\unicode[STIX]{x1D705}+2)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}-1}{\unicode[STIX]{x1D705}-\unicode[STIX]{x1D706}+1}=\frac{5}{2}\unicode[STIX]{x1D703}-1,\end{eqnarray}$$
which is clearly smaller than
$\frac{1}{3}$
if
$\frac{1}{2}<\unicode[STIX]{x1D703}<\frac{8}{15}$
.
Therefore, we may conclude
$$\begin{eqnarray}\mathscr{E}_{j}(\boldsymbol{N};q,a)\ll \frac{X}{q(\log X)^{A}}\end{eqnarray}$$
for
$j=1,2,3$
as long as
$\frac{1}{2}<\unicode[STIX]{x1D703}<\frac{9}{17}$
. To conclude Theorem 1.1, it remains to show, in view of Proposition 3.1 and the above estimate, that
$$\begin{eqnarray}\displaystyle \frac{X}{qN_{j}}\unicode[STIX]{x1D70F}(q)(\log X)^{3B}\ll \frac{X}{q(\log X)^{A}},\quad j=1,2,3 & & \displaystyle\end{eqnarray}$$
holds while applying Propositions 4.1, 4.2, or 4.3. In fact, we will show that
$N_{j}$
is at least a certain positive power of
$X$
. Thanks to Heath-Brown [Reference Heath-Brown11], we may assume
$\frac{1}{2}+\frac{1}{82}\leqslant \unicode[STIX]{x1D703}<\frac{9}{17}$
.
The case
$j=3$
is obvious since
$N_{3}\geqslant X^{1/3-\unicode[STIX]{x1D700}}$
due to (3.8) and (3.9).
While applying Proposition 4.2, one requires that
$N_{3}\ll X^{\unicode[STIX]{x1D703}}$
and
$N_{2}N_{3}\gg X^{3/2\unicode[STIX]{x1D703}}$
, which in fact implies
$N_{2}\gg X^{1/2\unicode[STIX]{x1D703}}\gg X^{1/4}$
, and thus gives (5.2) for
$j=2$
.
It remains to consider the case
$j=1.$
Under the condition (4.2a) and assuming that
$\unicode[STIX]{x1D707}_{2}+\unicode[STIX]{x1D707}_{3}\leqslant \frac{3}{2}\unicode[STIX]{x1D703}$
, one has
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{1}\geqslant 1-(\unicode[STIX]{x1D707}_{2}+\unicode[STIX]{x1D707}_{3})-B\frac{\log \log X}{\log X}\geqslant 1-\frac{3}{2}\unicode[STIX]{x1D703}-B\frac{\log \log X}{\log X}>1-\frac{3}{2}\cdot \frac{11}{17}=\frac{1}{34},\end{eqnarray}$$
provided that
$X$
is large enough. On the other hand, under the condition (4.2b) and assuming that
$\unicode[STIX]{x1D707}_{3}\leqslant \unicode[STIX]{x1D703}$
, one has similarly
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{1}\geqslant 1-(2-3\unicode[STIX]{x1D703})-\unicode[STIX]{x1D703}-B\frac{\log \log X}{\log X}=2\unicode[STIX]{x1D703}-1-B\frac{\log \log X}{\log X}\geqslant \frac{1}{42}.\end{eqnarray}$$
Each condition implies the lower bound
$N_{1}\gg X^{1/42}$
, which yields (5.2) for
$j=1$
, completing the proof of Theorem 1.1.
Acknowledgements
This work is supported in part by NSFC (No. 11601413). I am grateful to the referee for valuable comments and suggestions, which led to a more polished version of this paper.
A. Appendix. Kuznetsov–Selberg-type identity for Kloosterman sums
Let us recall the classical Kuznetsov–Selberg identity
$$\begin{eqnarray}\displaystyle S_{2}(m,n;q)=\mathop{\sum }_{d|(m,n,q)}\sqrt{d}S_{2}(mnd^{-2},1;qd^{-1}). & & \displaystyle \nonumber\end{eqnarray}$$
Since Kuznetsov’s proof is based on the trace formula, it is interesting to ask whether automorphic forms are really necessary. In fact, there are at least two existing proofs, both of which are supposed to be elementary. The first one was given by Matthes [Reference Matthes17] using the Chinese remainder theorem and Möbius inversion; and the second is due to Andersson [Reference Andersson1], however the theory of Hecke operators was applied. Here we give a completely elementary and direct proof, which only requires the orthogonality of additive characters.
A.1 An elementary proof of (2.4)
Our method works for the following slightly more general sum
In particular, we have
$\unicode[STIX]{x1D6EF}_{1}(m,n;q)=S_{2}(m,n;q).$
We shall prove
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{k}(m,n;q)=\mathop{\sum }_{d|(m,n,q)}\sqrt{d}S_{2}(mnd^{-2},k;qd^{-1}) & & \displaystyle\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{k}(m,n;q) & =\mathop{\sum }_{d|(m,k,q)}\sqrt{d}S_{2}(mkd^{-2},n;qd^{-1}). & \displaystyle\end{eqnarray}$$
Thus, (2.4) follows by taking
$k=1$
in (A.1).
Remark A.1. One may see from (A.1) and (A.2) that
$\unicode[STIX]{x1D6EF}_{k}(m,n;q)$
is stable under permutations among
$m,n,k.$
This is at least observed by Andersson [Reference Andersson1].
Using the orthogonality of additive characters to detect the congruence condition, we have
We evaluate the
$x$
-sum by orthogonality, getting
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{k}(m,n;q)=\frac{1}{\sqrt{q}}\mathop{\sum }_{d|q}\,\mathop{\sum }_{\substack{ a\,(\text{mod}\,q/d) \\ (a,q/d)=1}}\,\mathop{\sum }_{\substack{ y\,(\text{mod}\,q) \\ ady\equiv -m\,(\text{mod}\,q)}}\text{e}\biggl(\frac{ny-kad}{q}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
For each fixed
$d|q$
, from the condition
$ady\equiv -m\,(\text{mod}\,q)$
it follows that
$d|m$
. In a similar way, we also conclude that
$d|n$
. We thus have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{k}(m,n;q) & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(m,n,q)}\,\mathop{\sum }_{\substack{ a\,(\text{mod}\,q/d) \\ (a,q/d)=1}}\,\mathop{\sum }_{\substack{ y\,(\text{mod}\,q) \\ y\equiv -\overline{a}m/d\,(\text{mod}\,q/d)}}\text{e}\biggl(\frac{ny-kad}{q}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(m,n,q)}d\mathop{\sum }_{\substack{ a\,(\text{mod}\,q/d) \\ (a,q/d)=1}}\text{e}\biggl(\frac{\overline{a}mn/d+kad}{q}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{d|(m,n,q)}\sqrt{d}S_{2}(mnd^{-2},k;qd^{-1}).\nonumber\end{eqnarray}$$
This establishes (A.1).
On the other hand, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{k}(m,n;q) & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(k,q)}\,\mathop{\sum }_{\substack{ y\,(\text{mod}\,q) \\ (y,q)=d}}\text{e}\biggl(\frac{ny}{q}\biggr)\mathop{\sum }_{\substack{ x\,(\text{mod}\,q) \\ xy\equiv k\,(\text{mod}\,q)}}\text{e}\biggl(\frac{mx}{q}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(k,q)}\,\mathop{\sum }_{\substack{ y\,(\text{mod}\,q/d) \\ (y,q/d)=1}}\text{e}\biggl(\frac{ny}{q/d}\biggr)\mathop{\sum }_{\substack{ x\,(\text{mod}\,q) \\ xyd\equiv k\,(\text{mod}\,q)}}\text{e}\biggl(\frac{mx}{q}\biggr).\nonumber\end{eqnarray}$$
For each
$y$
with
$(y,q/d)=1$
and
$d|(k,q)$
, the innermost sum can be rewritten as
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ x\,(\text{mod}\,q) \\ xy\equiv k/d\,(\text{mod}\,q/d)}}\text{e}\biggl(\frac{mx}{q}\biggr)=\mathop{\sum }_{\substack{ x\,(\text{mod}\,q) \\ x\equiv \overline{y}k/d\,(\text{mod}\,q/d)}}\text{e}\biggl(\frac{mx}{q}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Write
$x=\overline{y}k/d+l\cdot q/d$
with
$l\,(\text{mod}\,d)$
, the above exponential sum is
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{l\,(\text{mod}\,d)}\text{e}\biggl(\frac{m(\overline{y}k/d+l\cdot q/d)}{q}\biggr)=\text{e}\biggl(\frac{m\overline{y}k/d}{q}\biggr)\mathop{\sum }_{l\,(\text{mod}\,d)}\text{e}\biggl(\frac{ml}{d}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
By orthogonality of additive characters, this must vanish unless
$d|m$
, in which case it is equal to
$$\begin{eqnarray}\displaystyle d\cdot \text{e}\biggl(\frac{\overline{y}mkd^{-2}}{q/d}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Hence, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{k}(m,n;q) & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(m,k,q)}d\mathop{\sum }_{\substack{ y\,(\text{mod}\,q/d) \\ (y,q/d)=1}}\text{e}\biggl(\frac{ny+\overline{y}mkd^{-2}}{q/d}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{d|(m,k,q)}\sqrt{d}S_{2}(mkd^{-2},n;qd^{-1})\nonumber\end{eqnarray}$$
as claimed in (A.2).
A.2 Extension to rank three Kloosterman sums
We now prove (2.5) following the above strategy. Using the orthogonality of additive characters to detect the congruence condition, we have
We evaluate the
$x$
-sum by orthogonality, getting
For each fixed
$d|q$
, from
$adyz\equiv -m\,(\text{mod}\,q)$
it follows that
$d|m$
. In a similar way, we also conclude that
$d|(n,l)$
. We thus have
Applying (A.1) twice, we may derive that
$$\begin{eqnarray}\displaystyle S_{3}(m,n,l;q) & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(m,n,l,q)}\;\mathop{\sum }_{r|(n/d,l/d,q/d)}d^{3/2}r^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{\substack{ a\,(\text{mod}\,q/d) \\ (a,q/d)=1}}\text{e}\biggl(\frac{a}{q/d}\biggr)S_{2}(\overline{a}nl/(dr)^{2},m/d;q/(dr))\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\sqrt{q}}\mathop{\sum }_{d|(m,n,l,q)}\;\mathop{\sum }_{r|(n/d,l/d,q/d)}\;\mathop{\sum }_{s|(nl/(dr)^{2},m/d,q/dr)}d^{3/2}(rs)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{\substack{ a\,(\text{mod}\,q/d) \\ (a,q/d)=1}}\text{e}\biggl(\frac{a}{q/d}\biggr)S_{2}\biggl(\frac{\overline{a}mnl}{d^{3}r^{2}s^{2}},1;\frac{q}{drs}\biggr).\nonumber\end{eqnarray}$$
Note that
$q$
is squarefree, and thus
$(rs,q/(drs))=1.$
From the Chinese remainder theorem, the last sum over
$a$
can be expressed as
$$\begin{eqnarray}\displaystyle \biggl(\frac{q}{drs}\biggr)^{1/2}\unicode[STIX]{x1D707}(rs)S_{3}\biggl(\frac{\overline{rs}mnl}{d^{3}r^{2}s^{2}},1,1;\frac{q}{drs}\biggr), & & \displaystyle \nonumber\end{eqnarray}$$
from which we conclude that
$$\begin{eqnarray}\displaystyle S_{3}(m,n,l;q) & = & \displaystyle \mathop{\sum }_{d|(m,n,l,q)}\;\mathop{\sum }_{r|(n/d,l/d,q/d)}\;\mathop{\sum }_{s|(nl/(dr)^{2},m/d,q/dr)}\nonumber\\ \displaystyle & & \displaystyle \times \,d\unicode[STIX]{x1D707}(rs)S_{3}\biggl(\frac{\overline{rs}mnl}{d^{3}r^{2}s^{2}},1,1;\frac{q}{drs}\biggr).\nonumber\end{eqnarray}$$
This establishes (2.5).
B. Appendix. Estimate for two-dimensional exponential sums
Let
$p$
be an odd prime and
${\mathcal{F}}$
a middle-extension sheaf on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
, pointwise pure of weight 0 and geometrically irreducible. Suppose
$T$
is the trace function of
${\mathcal{F}}$
in the sense of Katz [Reference Katz14, §7.3.7]. For
$\mathbf{l}=(l_{1},l_{2})$
and
$\mathbf{h}=(h_{1},h_{2}),$
we consider the double sum
The following estimate presents a square-root cancellation when
${\mathcal{F}}$
is a Kloosterman sheaf of rank three.
Theorem B.1. For
$T(x)=\text{Kl}_{3}(x,p),$
we have
$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6F1}(\mathbf{l},\mathbf{h};p)|\ll p\cdot (h_{1},h_{2},l_{1},l_{2},p). & & \displaystyle \nonumber\end{eqnarray}$$
Proof. The bound is trivial if
$p\mid (h_{1},h_{2},l_{1},l_{2}).$
We now assume
$p\nmid (h_{1},h_{2},l_{1},l_{2}).$
For fixed
$u,v\,(\text{mod}\,p),$
define
so that
By the Parseval (discrete Plancherel) identity, we have
where
$$\begin{eqnarray}\displaystyle \widehat{T}(s)=\frac{1}{\sqrt{p}}\mathop{\sum }_{u\,(\text{mod}\,p)}T(u)\text{e}\biggl(\frac{-\mathit{su}}{p}\biggr) & & \displaystyle \nonumber\end{eqnarray}$$
and
After a change of variable, one may write
$$\begin{eqnarray}\displaystyle H(u,v)=\sum ^{\ast }_{\substack{ a\,(\text{mod}\,p) \\ l_{2}a+l_{1}u\overline{a}\equiv v-u-l_{1}l_{2}\,(\text{mod}\,p)}}\text{e}\biggl(\frac{h_{1}a+h_{2}u\overline{a}}{p}\biggr), & & \displaystyle \nonumber\end{eqnarray}$$
from which and the orthogonality of additive characters, it follows that
$$\begin{eqnarray}\displaystyle & & \displaystyle \widehat{H}(s,t)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{p}\mathop{\sum }_{u\,(\text{mod}\,p)}~\sum ^{\ast }_{a\,(\text{mod}\,p)}\text{e}\biggl(\frac{h_{1}a+h_{2}u\overline{a}}{p}\biggr)\text{e}\biggl(\frac{-\mathit{su}-t(l_{2}a+l_{1}u\overline{a}+u+l_{1}l_{2})}{p}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =\sum ^{\ast }_{\substack{ a\,(\text{mod}\,p) \\ (h_{2}-tl_{1})\overline{a}\equiv s+t\,(\text{mod}\,p)}}\text{e}\biggl(\frac{(h_{1}-tl_{2})a-tl_{1}l_{2}}{p}\biggr).\nonumber\end{eqnarray}$$
This yields
$$\begin{eqnarray}\displaystyle \widehat{H}(s,t)=\text{e}\biggl(\frac{(h_{1}-tl_{2})(h_{2}-tl_{1})\overline{(s+t)}-tl_{1}l_{2}}{p}\biggr) & & \displaystyle \nonumber\end{eqnarray}$$
if
$s+t\not \equiv 0\,(\text{mod}\,p);$
and otherwise,
$\widehat{H}(s,t)$
vanishes unless
$h_{2}\equiv tl_{1}\,(\text{mod}\,p)$
, in which case the variable
$a$
runs freely over
$(\mathbf{Z}/p\mathbf{Z})^{\times }.$
Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}(\mathbf{l},\mathbf{h};p)=\unicode[STIX]{x1D6F1}_{1}(\mathbf{l},\mathbf{h};p)+\unicode[STIX]{x1D6F1}_{2}(\mathbf{l},\mathbf{h};p), & & \displaystyle \nonumber\end{eqnarray}$$
where
and
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{2}(\mathbf{l},\mathbf{h};p)=\mathop{\sum }_{\substack{ t\,(\text{mod}\,p) \\ h_{2}\equiv tl_{2}\,(\text{mod}\,p)}}|\widehat{T}(t)|^{2}\sum ^{\ast }_{a\,(\text{mod}\,p)}\text{e}\biggl(\frac{(h_{1}-tl_{2})a-tl_{1}l_{2}}{p}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Note that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{2}(\mathbf{l},\mathbf{h};p)=(p-1)\mathop{\sum }_{\substack{ t\,(\text{mod}\,p) \\ h_{2}\equiv tl_{1}\,(\text{mod}\,p) \\ h_{1}\equiv tl_{2}\,(\text{mod}\,p)}}|\widehat{T}(t)|^{2}\text{e}\biggl(\frac{-tl_{1}l_{2}}{p}\biggr)+O(p)\ll p & & \displaystyle \nonumber\end{eqnarray}$$
since at least one of
$h_{1},h_{2},l_{1},l_{2}$
is coprime to
$p$
.
It remains to prove
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{1}(\mathbf{l},\mathbf{h};p)\ll p. & & \displaystyle \nonumber\end{eqnarray}$$
From now on, we assume
$T=\text{Kl}_{3}$
so that
$$\begin{eqnarray}\displaystyle \widehat{T}(x)=\left\{\begin{array}{@{}ll@{}}\text{Kl}_{2}(\overline{x},p),~~\quad & p\nmid x,\\ 0,\quad & p\mid x.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
Hence,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{1}(\mathbf{l},\mathbf{h};p) & =\sqrt{p}\sum ^{\ast }_{s\,(\text{mod}\,p)}\text{Kl}_{2}(\overline{s},p)W(s,\mathbf{l},\mathbf{h};p), & \displaystyle \nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & & \displaystyle W(s,\mathbf{l},\mathbf{h};p)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{\sqrt{p}}\mathop{\sum }_{\substack{ t\,(\text{mod}\,p) \\ t\not \equiv 0,s\,(\text{mod}\,p)}}\text{Kl}_{2}(\overline{t},p)\text{e}\biggl(\frac{(h_{1}-tl_{2})(h_{2}-tl_{1})\overline{(t-s)}-tl_{1}l_{2}}{p}\biggr).\nonumber\end{eqnarray}$$
Given a tuple
$(\mathbf{l},\mathbf{h})=(l_{1},l_{2},h_{1},h_{2})\in \mathbf{Z}^{4},$
we would like to show
$s\mapsto W(s,\mathbf{l},\mathbf{h};p)$
is a good trace function up to some controllable tails. To do so, we would like to appeal to the following quantitative characterization due to Fouvry, Kowalski and Michel [Reference Fouvry, Kowalski and Michel4, Corollary 5.3].
Proposition B.1. Let
$\mathbf{F}_{\!p}$
be a finite field and
$\ell \not =p$
a prime. Let
$f\in \mathbf{F}_{\!p}(X,Y)$
be a rational function given by
$f=f_{1}/f_{2}$
with
$f_{i}\in \mathbf{F}_{\!p}[X,Y]$
coprime polynomials, and assume that
$\mathfrak{c}(f):=1+\deg (f_{1})+\deg (f_{2})<p$
. Let
${\mathcal{F}}$
be a middle-extension sheaf on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
which is pointwise pure of weight
$0$
and
$f$
-disjoint in the sense of [Reference Fouvry, Kowalski and Michel4, Definition 5.2].
Then there exists an absolute constant
$A\geqslant 1$
, independent of
$f$
and
${\mathcal{F}}$
, such that for all
$x\in \mathbf{F}_{\!p}$
, we have
$$\begin{eqnarray}\frac{1}{\sqrt{p}}\mathop{\sum }_{\substack{ y\in \mathbf{F}_{\!p} \\ f_{2}(x,y)\not =0}}t_{{\mathcal{F}}}(y)\unicode[STIX]{x1D713}(f(x,y))=-t_{0}(x)+t_{1}(x)+t_{2}(x),\end{eqnarray}$$
where
$t_{0}$
is the trace function of a middle-extension sheaf
${\mathcal{G}}_{0}$
of weight
$0$
on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
with
$$\begin{eqnarray}\mathfrak{c}({\mathcal{G}}_{0})\leqslant (2\mathfrak{c}(f)\mathfrak{c}({\mathcal{F}}))^{A}\end{eqnarray}$$
the function
$t_{1}$
is zero except for a set of values of
$x\in \mathbf{F}_{\!p}$
of size at most
$(2\mathfrak{c}(f)\mathfrak{c}({\mathcal{F}}))^{A}$
, and it satisfies
$$\begin{eqnarray}\Vert t_{1}\Vert _{\infty }\leqslant (2\mathfrak{c}(f)\mathfrak{c}({\mathcal{F}}))^{A},\end{eqnarray}$$
while the function
$t_{2}$
satisfies
$$\begin{eqnarray}\Vert t_{2}\Vert _{\infty }\leqslant (2\mathfrak{c}(f)\mathfrak{c}({\mathcal{F}}))^{A}p^{-1/2}.\end{eqnarray}$$
Remark B.1. It is useful to make a comment on the
$f$
-disjointness here. According to [Reference Fouvry, Kowalski and Michel4, Definition 5.2], a middle-extension sheaf
${\mathcal{F}}$
is called
$f$
-disjoint, if
$H_{c}^{2}(\mathbf{A}^{1}\times \overline{\mathbf{F}}_{p},{\mathcal{F}}\otimes {\mathcal{L}}_{x})$
vanishes for all
$x\in \overline{\mathbf{F}}_{p}$
, where
${\mathcal{L}}_{x}=j_{x}^{\ast }{\mathcal{L}}_{\unicode[STIX]{x1D713}(f(X,Y))}$
with
$\unicode[STIX]{x1D713}$
an additive character and
$j_{x}:\{x\}\times \mathbf{A}^{1}{\hookrightarrow}\mathbf{A}^{2}$
being the closed immersion. Therefore,
${\mathcal{F}}$
is always
$f$
-disjoint if it is irreducible of rank at least two (e.g. Kloosterman sheaves in one or more variables).
Taking
${\mathcal{F}}=[-1]^{\ast }{\mathcal{K}}\ell _{2}$
and
$f(x,y)=(h_{1}-yl_{2})(h_{2}-yl_{1})/(y-x)-yl_{1}l_{2}$
with
$$\begin{eqnarray}f_{1}(x,y)=(h_{1}-yl_{2})(h_{2}-yl_{1})-(y-x)yl_{1}l_{2},\qquad f_{2}(x,y)=y-x\end{eqnarray}$$
in Proposition B.1, we then find
$f_{1}$
and
$f_{2}$
are coprime mod
$p$
since
$p\nmid (l_{1},l_{2},h_{1},h_{2})$
. From Proposition B.1 we conclude that
$$\begin{eqnarray}\displaystyle W(s,\mathbf{l},\mathbf{h};p)=-W_{0}(s,\mathbf{l},\mathbf{h};p)+W_{1}(s,\mathbf{l},\mathbf{h};p)+W_{2}(s,\mathbf{l},\mathbf{h};p), & & \displaystyle \nonumber\end{eqnarray}$$
where, for some absolute
$C>0$
,
$s\mapsto W_{0}(s,\mathbf{l},\mathbf{h};p)$
is the trace function of a middle-extension sheaf of weight
$0$
on
$\mathbf{A}_{\mathbf{F}_{\!p}}^{1}$
with the conductor at most
$C$
, the function
$s\mapsto W_{1}(s,\mathbf{l},\mathbf{h};p)$
is zero except for a set
${\mathcal{D}}$
of values of
$s\in \mathbf{F}_{\!p}$
of size
$|{\mathcal{D}}|\leqslant C$
,
$$\begin{eqnarray}\Vert W_{1}\Vert _{\infty }\leqslant C,\qquad \Vert W_{2}\Vert _{\infty }\leqslant Cp^{-1/2}.\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{1}(\mathbf{l},\mathbf{h};p) & = & \displaystyle -\sqrt{p}\sum ^{\ast }_{s\,(\text{mod}\,p)}\text{Kl}_{2}(\overline{s},p)W_{0}(s,\mathbf{l},\mathbf{h};p)\nonumber\\ \displaystyle & & \displaystyle +\,\sqrt{p}\mathop{\sum }_{s\in {\mathcal{D}}}\text{Kl}_{2}(\overline{s},p)W_{1}(s,\mathbf{l},\mathbf{h};p)+O(p)\nonumber\\ \displaystyle & = & \displaystyle -\sqrt{p}\sum ^{\ast }_{s\,(\text{mod}\,p)}\text{Kl}_{2}(\overline{s},p)W_{0}(s,\mathbf{l},\mathbf{h};p)+O(p).\nonumber\end{eqnarray}$$
Since
$\text{Kl}_{2}(\overline{x},p)$
has a singularity at
$x=0$
, however
$$\begin{eqnarray}\displaystyle W(0,\mathbf{l},\mathbf{h};p) & = & \displaystyle \frac{1}{\sqrt{p}}\text{e}\biggl(\frac{-(h_{1}l_{1}+h_{2}l_{2})}{p}\biggr)\sum ^{\ast }_{t\,(\text{mod}\,p)}\text{Kl}_{2}(t,p)\text{e}\biggl(\frac{h_{1}h_{2}t}{p}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \text{e}\biggl(\frac{-(h_{1}l_{1}+h_{2}l_{2})}{p}\biggr)\biggl\{\mathop{\sum }_{h_{1}h_{2}x+1\equiv 0\,(\text{mod}\,p)}\text{e}\biggl(\frac{x}{p}\biggr)+\frac{1}{p}\biggr\},\nonumber\end{eqnarray}$$
which yields
$W_{0}(x,\mathbf{l},\mathbf{h};p)$
has no singularity at
$x=0$
. We then find the associated sheaves with
$\text{Kl}_{2}(\overline{x},p)$
and
$W_{0}(x,\mathbf{l},\mathbf{h};p)$
have no geometrically irreducible components that are geometrically isomorphic. The Riemann hypothesis of Deligne (see [Reference Deligne2] or [Reference Fouvry, Kowalski and Michel8, Theorem 4.1]) then yields
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{1}(\mathbf{l},\mathbf{h};p)\ll p & & \displaystyle \nonumber\end{eqnarray}$$
with an absolute implied constant. This completes the proof of Theorem B.1 by combining the above arguments.◻
