1 Introduction
1.1 Motivation
For a prime p and an integer n, we define the s-dimensional Kloosterman sum
$$ \begin{align*} {\mathcal K}_{s, p}(n) = p^{-(s-1)/2} \sum_{\substack{x_1, \ldots, x_s=1\\ x_1\cdots x_s \equiv n \pmod p}}^{p-1} {\mathbf{\,e}}_p(x_1 +\cdots + x_s), \end{align*} $$
where
${\mathbf {\,e}}_p(x) = \exp (2\pi ix/p)$
. The celebrated result of Deligne [Reference Deligne7] gives the bound
$$ \begin{align} |{\mathcal K}_{s, p}(n)| \leqslant s. \end{align} $$
(See also [Reference Iwaniec and Kowalski16, Section 11.11].) In the classical case of
$s=2$
, we denote
$$ \begin{align*} {\mathcal K}_{p}(n) = {\mathcal K}_{2, p}(n). \end{align*} $$
Recently, there has been active interest in estimating sums of Kloosterman sums either over sequences of parameters n of arithmetic interest or twisted by arithmetic functions, such as
$$ \begin{align} P_{s, p}(L) = \sum_{\substack{\ell \leqslant L\\ \ell~\text{prime}}} {\mathcal K}_{s, p}(\ell) \quad\mbox{and}\quad M_{s, p}(f;N) = \sum_{n \leqslant N} f(n){\mathcal K}_{s, p}(n), \end{align} $$
with some multiplicative function
$f(n)$
such as the Möbius function
$\mu (n)$
or the divisor function
$\tau (n)$
. (See, for example, [Reference Blomer, Fouvry, Kowalski, Michel and Milićević5, Reference Fouvry, Kowalski and Michel8, Reference Korolev and Shparlinski18, Reference Liu, Shparlinski and Zhang21].)
These results rely on recent progress on bounds of bilinear Type-I and Type-II sums with Kloosterman sums (see [Reference Bag and Shparlinski2, Reference Blomer, Fouvry, Kowalski, Michel and Milićević4, Reference Blomer, Fouvry, Kowalski, Michel and Milićević5, Reference Fouvry, Kowalski and Michel8, Reference Kerr, Shparlinski, Wu and Xi17, Reference Kowalski, Michel and Sawin19, Reference Kowalski, Michel and Sawin20, Reference Shparlinski22, Reference Shparlinski and Zhang23]). In particular, [Reference Fouvry, Kowalski and Michel8, Theorem 1.5] implies power-saving bounds on the sums
$P_{s, p}(L)$
given by (1.2) with
$L \geqslant p^{3/4+\varepsilon }$
for an arbitrary fixed
$\varepsilon> 0$
. These bounds have been subsequently improved in [Reference Blomer, Fouvry, Kowalski, Michel and Milićević5, Theorem 1.8] in the case of
$s=2$
and for the same range of L.
For the sums
$M_{s, p}(f; N)$
given by (1.2), [Reference Fouvry, Kowalski and Michel8, Theorem 1.7] gives power-saving bounds on
$M_{s, p}(\mu; N)$
, provided that
$N\geqslant p^{3/4+\varepsilon }$
for an arbitrary fixed
$\varepsilon> 0$
. Similar bounds are given by [Reference Kowalski, Michel and Sawin20, Corollary 1.4] on
$M_{s, p}(\tau; N)$
, provided that
$N\geqslant p^{2/3+\varepsilon }$
. These thresholds have both been reduced to
$N\geqslant p^{1/2+\varepsilon }$
in [Reference Korolev and Shparlinski18], however with a logarithmic saving instead of a power saving in the bound.
1.2 Outline of our results
1.2.1 Sums of Kloosterman sums over square-free numbers
We first consider sums of Kloosterman sums over square-free numbers (that is, over numbers which are not divisible by the square of a prime):
$$ \begin{align} Q_{s, p}(N) = \sum_{n \leqslant N} |\mu(n)| {\mathcal K}_{s, p}(n) = \sum_{\substack{n \leqslant N\\ n~\text{square-free}}} {\mathcal K}_{s, p}(n). \end{align} $$
The trivial bound
$$ \begin{align} Q_{s, p}(N) \ll N \end{align} $$
is implied by (1.1). Our goal is twofold: to obtain a nontrivial bound (possibly with a power saving) for N as small as possible and to have a bound as good as possible for any given N. Our main result in this direction is given by Theorem 2.1, which is proved in Section 4.
A rather straightforward approach (see the remarks following the statement of Theorem 2.1), implies that if
$p^{1/2+\varepsilon } \leqslant N \leqslant p$
for an arbitrary fixed
$\varepsilon>0$
, then
$$ \begin{align} Q_{s, p}(N) \ll N^{1/2} p^{1/4+o(1)}, \end{align} $$
which already improves the trivial bound (1.4) in this range. Our Theorem 2.1 provides a power-saving improvement upon (1.5) for any given N in the range
$p^{1/2+\varepsilon } \leqslant N \leqslant p$
. For example, if
$N=p$
, then Theorem 2.1 implies
$$ \begin{align} Q_{s,p}(p) \ll p^{3/4-1/232+o(1)}, \end{align} $$
while (1.5) only gives
$ Q_{s,p}(p) \ll p^{3/4+o(1)}$
.
1.2.2 Sums of Kloosterman sums over smooth numbers
Next, we recall that an integer
$n \geqslant 1$
is called y-smooth if
$P(n) \leqslant y$
, where
$P(n)$
denotes the largest prime divisor of n. See [Reference Granville, Buhler and Stevenhagen13, Reference Hildebrand and Tenenbaum15] for background and classical estimates on smooth numbers. For
$N \geqslant y \geqslant 2$
, we denote by
${\mathcal S}(N,y)$
the set of y-smooth positive integers
$n \leqslant N$
and, as usual, we denote
$\Psi (N,y) = \# {\mathcal S}(N,y)$
.
We consider the sum
$$ \begin{align*} R_{s.p}(N,y) = \sum_{n \in {\mathcal S}(N,y)} {\mathcal K}_{s, p}(n) , \end{align*} $$
for which we have, in analogy to (1.4), the trivial bound
$$ \begin{align*} R_{s, p}(N,y) \ll \Psi(N,y) \leqslant N. \end{align*} $$
As in the case of squarefree numbers, our goal is to obtain nontrivial bounds for N as small as possible and, at the same time, to have a bound as good as possible for any given N and y. Additionally, we are also interested in obtaining nontrivial bounds for y as small as possible.
In fact, it turns out that the estimates we have for
$R_{s, p}(N,y) $
hold in a much broader context of trace functions (see Section 1.3). Thus, given a trace function
$\mathsf {K}(n)$
, we consider the generalisation of
$R_{s, p}(N,y)$
,
$$ \begin{align} R_{\mathsf{K}}(N,y) = \sum_{n \in \mathcal{S}(N,y)} \mathsf{K}(n), \end{align} $$
for which, if the values of
$\mathsf {K}(n)$
are uniformly bounded, we have, in analogy to (1.4), the trivial bound
$$ \begin{align*} R_{\mathsf{K}}(N,y) \ll \Psi(N,y) \leqslant N. \end{align*} $$
Our main result in this direction is given by Theorem 2.2, which is proved in Section 4. As a special case of Theorem 2.2, if
$\log y/\!\log \log N\rightarrow \infty $
, then
$$ \begin{align} R_{\mathsf{K}}(N,y) \ll \Psi(N,y) y^{1/2} p^{1/8} N^{-1/4+o(1)}, \end{align} $$
which is nontrivial when
$N> y^2 p^{1/2+\varepsilon }$
. (See the remarks following the statement of Theorem 2.2.)
1.3 Trace functions
Many of the results that are mentioned in Section 1.1 also apply to more general sums involving trace functions (apart from several exceptions described in [Reference Blomer, Fouvry, Kowalski, Michel and Milićević5, Reference Kerr, Shparlinski, Wu and Xi17, Reference Kowalski, Michel and Sawin20–Reference Shparlinski and Zhang23]). The exact definition of these trace functions requires some notions from algebraic geometry, which go beyond the more analytic frameworks of this work. Instead, we refer to [Reference Fouvry, Kowalski and Michel8–Reference Fouvry, Kowalski and Michel10] and especially [Reference Fouvry, Kowalski, Michel, Bucur and Zureick-Brown11] for a general background, and precise definitions and properties of trace functions.
It turns out that Kloosterman sums are representatives of a much richer class of isotypic trace functions
$\mathsf {K}(n)$
, which are associated with isotypic trace sheaves
$\mathfrak {F}$
modulo p of bounded conductor (we refer to [Reference Fouvry, Kowalski and Michel8, Reference Fouvry, Kowalski, Michel and Zannier9] for precise definitions and properties of trace functions). In addition to Kloosterman sums, other concrete examples of isotypic trace functions include:
-
• traces of Frobenius of elliptic curves modulo p;
-
• exponential functions of the form
${\mathbf {\,e}}(\psi (n)/p)$
with rational functions
$\psi (Z) \in {\mathbb Q}(Z)$
and similar values of multiplicative characters as well as their products.
Some of our principal tools, such as Lemmas 3.1 and 3.2 below, are presented for trace functions. Because of these observations, in Theorem 2.2, we present a bound on
$R_{\mathsf {K}}(N,y)$
for a large class of trace functions, including Kloosterman sums, rather than a bound on just
$R_{s,p}(N,y)$
.
However, the main ingredient in the proof of Theorem 2.1, Lemma 3.3, is based on [Reference Kowalski, Michel and Sawin20, Theorem 4.3], which, at the moment, is only known for Kloosterman sums. Thus, it is not clear how to extend Theorem 2.1 to arbitrary trace functions (though an analogue of the trivial bound (1.5) still holds).
1.4 Structure of the paper
The rest of the paper is organised as follows. In Section 2, we formulate our main results, and also show how the bounds (1.6) and (1.8) follow from them. Then, in Section 3.2, we state some preliminary results including estimates for sums of trace functions and a variant of the Type-I estimate for Kloosterman sums, which are crucial in our treatment of smooth numbers and square-free numbers. Then, in Section 4, we prove Theorems 2.1 and 2.2.
2 Main results
2.1 Sums of Kloosterman sums over square-free numbers
Our main result in this direction is the following bound. Recall the definition of
$Q_{s,p}(N)$
from (1.3).
Theorem 2.1. For any positive integer
$s \geqslant 2$
and any even positive integer
$\ell $
, if
$p^{1/2+2/\ell } \leqslant N \leqslant p$
, then
$$ \begin{align*} |Q_{s, p}(N)| \leqslant N^{1/2} p^{1/4} \bigg(\frac{p^{1/2+2/\ell}}{N}\bigg)^{{1}/{2(4\ell-3)}} p^{o(1)}. \end{align*} $$
For a specific value of N, one can optimise the bound in Theorem 2.1 by making the choice of
$\ell $
that minimises the term
$$ \begin{align*} \bigg(\frac{p^{1/2+2/\ell}}{N}\bigg)^{{1}/{2(4\ell-3)}}. \end{align*} $$
For example, if
$N=p$
, then by choosing
$\ell = 8$
, we derive (1.6). We also note that the bound (1.5) can be quickly obtained by combining (4.1) and (4.2) in our proof of Theorem 2.1.
Our approach works for
$N> p$
as well. However, the optimisation of our argument becomes more cluttered in such generality. Since we are mostly interested in short sums and to exhibit our ideas in the simplest possible form, we focus on the case
$N \leqslant p$
.
2.2 Sums of trace function over smooth numbers
Before formulating our main result in this direction, we recall some well-known estimates in the theory of smooth numbers. Let
$\alpha (N,y)$
be the saddle point corresponding to the y-smooth numbers up to N as discussed in [Reference Harper14, Reference Hildebrand and Tenenbaum15]. In particular,
$\alpha (N,y)$
satisfies
$$ \begin{align*} \alpha(N,y) = (1+o(1))\frac{\log(1+y/\!\log N)}{\log y}, \end{align*} $$
provided that
$y \leq N$
and
$y \to \infty $
(see [Reference Hildebrand and Tenenbaum15, Theorem 2]). In particular, if
$\log y/\!\log \log N \to \infty $
, then
$\alpha (N,y) \to 1$
and if
$y = (\log N)^K$
for some
$K \geq 1$
, then
$\alpha (N,y) = 1-1/K + o(1)$
. We have
$$ \begin{align} \Psi(N,y) = N^{\alpha(N,y) + o(1)} \end{align} $$
(see, for example, [Reference Harper14, Section 2] or [Reference Hildebrand and Tenenbaum15, Theorem 1]).
Our main result in this direction is the following bound. Recall the definition of
$R_{\mathsf {K}}(N,y)$
from (1.7).
Theorem 2.2. Let
$\mathsf {K}$
be a nonexceptional isotypic trace function associated to some sheaf
$\mathfrak {F}$
modulo a prime p of bounded conductor. For
$N \geq p^{1/2}$
and
$y \geq \log N$
,
$$ \begin{align*} |R_{\mathsf{K}}(N,y) | \leqslant \Psi(N,y) y^{1/2} p^{\beta} N^{-\gamma + o(1)} , \end{align*} $$
where
$$ \begin{align*} \beta = \frac{1}{4(1+\alpha(N,y))} \quad\mbox{and}\quad \gamma= \frac{\alpha(N,y)^2}{2(1+\alpha(N,y))}. \end{align*} $$
Some remarks on the bound are in order. If
$y = N^{o(1)}$
, then
$y^{1/2}$
can be dropped from the bound giving
$$ \begin{align*} R_{\mathsf{K}}(N,y) \ll \Psi(N,y) p^{\beta} N^{-\gamma+o(1)}, \end{align*} $$
which is nontrivial when
$N \geqslant p^{\beta /\gamma + \varepsilon } = p^{1/2\alpha (N,y)^2 + \varepsilon }$
for arbitrary fixed
$\varepsilon> 0$
. In the special case
$N=p$
, we have a nontrivial bound for
$R_{\mathsf {K}}(N,y)$
, provided that
$\alpha (N,y)> 1/\sqrt {2}$
or
$y \geq (\log N)^{2+\sqrt {2}+\varepsilon }$
.
However, if
$\log y/\!\log \log N\rightarrow \infty $
, then
$\alpha (N,y) = 1 + o(1)$
. Hence,
$\beta = 1/8+o(1)$
and
$\gamma = 1/4+o(1)$
. Thus, Theorem 2.2 yields (1.8).
3 Preparations
3.1 Notation
We use the standard notation
$U\ll V$
and
$V \gg U$
as equivalent to the statement
$|U|\leq c V$
, for some constant
$c> 0$
, which, throughout this paper, may depend only on the integer parameters
$\ell $
and s.
For a finite set
${\mathcal S}$
, we use
$\# {\mathcal S}$
to denote its cardinality.
The variables of summation d, k, m and n are always positive integers.
We also follow the convention that fractions of the shape
$1/ab$
mean
$1/(ab)$
(rather than
$b/a$
as their formal interpretation might suggest).
The letter p always denotes a prime number and we use
${\mathbb F}_p$
to denote the finite field of p elements.
We denote by
$p(\ell )$
and
$P(\ell )$
the smallest and the largest prime factors of an integer
$\ell \ne 0$
, respectively. We adopt the convention that
$p(1) = P(1) = +\infty $
.
Finally, we write
$\sum _{k\leqslant K}$
to denote the summation over positive integers
$k \leqslant K$
.
3.2 Preliminary bounds on sums of trace functions and Kloosterman sums
We recall the following bound, which is a combination of [Reference Fouvry, Kowalski and Michel8, Proposition 6.2] and [Reference Fouvry, Kowalski and Michel8, Theorem 6.3] (see [Reference Fouvry, Kowalski and Michel10–Reference Fouvry, Kowalski, Michel, Raju, Rivat and Soundararajan12] for several other variations of this result).
Lemma 3.1. Let
$\mathsf {K}$
be a nonexceptional isotypic trace function associated to some sheaf
$\mathfrak {F}$
modulo a prime p of bounded conductor. There exists a set
${\mathcal E}_{\mathfrak {F}} \subseteq {\mathbb F}_p$
of cardinality
$\# {\mathcal E}_{\mathfrak {F}} \ll 1$
, such that uniformly over
$d \in {\mathbb F}_p \setminus {\mathcal E}_{\mathfrak {F}}$
and
$h \in {\mathbb Z}$
,
$$ \begin{align*} \sum_{x \in{\mathbb F}_p} \mathsf{K}(x) \mathsf{K}(d x) {\mathbf{\,e}}_p(hx) \ll p^{1/2}. \end{align*} $$
The standard completion technique (see [Reference Iwaniec and Kowalski16, Section 12.2]) shows that Lemma 3.1 implies the following bound on incomplete sums.
Lemma 3.2. In the notation of Lemma 3.1, for any
$K \leqslant p$
, uniformly over
$d, e \in {\mathbb F}_p$
satisfying
$d \ne 0$
and
$e/d \notin {\mathcal E}_{\mathfrak {F}}$
,
$$ \begin{align*} \sum_{n \leqslant K} \mathsf{K}(dn) \mathsf{K}(en) \ll p^{1/2} \log p. \end{align*} $$
Furthermore, only in the case of Kloosterman sums, we need the following estimate of the Pólya–Vinogradov type: for any
$K \leqslant p$
, uniformly over
$d \in {\mathbb F}_p^*$
,
$$ \begin{align} \sum_{n \leqslant K} {\mathcal K}_{s,p}(dn) \ll p^{1/2} \log p. \end{align} $$
(See [Reference Fouvry, Kowalski, Michel, Bucur and Zureick-Brown11, Theorem 6.2]. The result also follows from [Reference Fouvry, Kowalski and Michel10, Corollary 1.6] and the completion techniques of [Reference Iwaniec and Kowalski16, Section 12.2].)
We now record a bound on a variant of Type-I sums of Kloosterman sums. Instead of Type-I sums with
${\mathcal K}_{s,p}(mn)$
, we consider sums with
${\mathcal K}_{s,p}(m^{r} n)$
, where r is an arbitrary integer (if r is negative, we consider the argument of the corresponding Kloosterman sum modulo p). This is obtained by a slight extension of the argument of [Reference Kowalski, Michel and Sawin20]. As we have mentioned, this result does not immediately extend to general trace functions. (See [Reference Banks and Shparlinski3, Reference Blomer, Fouvry, Kowalski, Michel and Milićević4, Reference Fouvry, Kowalski and Michel8, Reference Kerr, Shparlinski, Wu and Xi17, Reference Kowalski, Michel and Sawin19, Reference Kowalski, Michel and Sawin20] for several other bounds on related sums.)
Lemma 3.3. Fix an integer
$r \neq 0$
and an even integer
$\ell \geqslant 2$
. Let
$D, N \leqslant p$
be positive integers with
$N> 2p^{1/\ell }$
. For each
$d \leqslant D$
, let
${\mathcal N}_d \subseteq [1,N]$
be an interval. Then, for any complex weights
$\boldsymbol {\alpha }=\{\alpha _d\}_{d \leqslant D}$
with
$\alpha _d \ll 1$
,
$$ \begin{align*} \sum_{d\leqslant D} \sum_{n\in {\mathcal N}_d} \alpha_d {\mathcal K}_{s,p}(d^r n) \ll DN\bigg(N^{-1} + \frac{p^{1+1/\ell}}{DN^2}\bigg)^{1/(2\ell)} p^{o(1)}. \end{align*} $$
Proof. We follow the proof of [Reference Kowalski, Michel and Sawin20, Theorem 4.3], which is conveniently summarised in [Reference Banks and Shparlinski3, Section 4.3] and also extended to sums when the nonsmooth variable runs through an arbitrary set. Let
$$ \begin{align*} S = \sum_{d\leqslant D} \sum_{n\in {\mathcal N}_d} \alpha_d {\mathcal K}_{s,p}(d^r n). \end{align*} $$
Let A and B be integer parameters to be chosen later for which
$$ \begin{align} 2AB\leqslant N. \end{align} $$
By introducing averages over
$a \sim A$
and
$b \sim B$
(where
$a \sim A$
denotes the dyadic range
$A \leq a < 2A$
and similarly for
$b \sim B$
), and replacing n by
$n+ab$
,
$$ \begin{align*} \begin{split} S &= \frac{1}{AB} \sum_{a \sim A} \sum_{b \sim B} \sum_{d \leq D} \alpha_d \sum_{\substack{n\in {\mathbb Z}\\ n+ab \in {\mathcal N}_d}} {\mathcal K}_{s,p}(d^r(n+ab)) \\ &= \frac{1}{AB} \sum_{a \sim A} \sum_{d \leq D} \alpha_d \sum_{n\in {\mathbb Z}} \sum_{\substack{b \sim B \\ n+ab \in {\mathcal N}_d}} {\mathcal K}_{s,p}(d^r(n+ab)). \end{split} \end{align*} $$
Since the range for the inner sum over b is an interval, by the completion technique (see [Reference Iwaniec and Kowalski16, Section 12.2]),
$$ \begin{align*} S \ll \frac{\log p}{AB} \sum_{a \sim A} \sum_{d \leq D} \sum_{n=-N}^N \bigg|\kern-2pt\sum_{b \sim B}{\mathcal K}_{s,p}(d^r(n+ab)) {\mathbf{\,e}}(bt) \bigg| \end{align*} $$
for some
$t \in {\mathbb R}$
, where
$ {\mathbf {\,e}}(z) = \exp (2\pi i z)$
. By making a change of variables
$u = d^ra$
and
$v = \overline {a}n$
,
$$ \begin{align*} S \ll \frac{\log p}{AB} \sum_{u, v \in {\mathbb F}_p} \nu(u,v) \bigg|\kern-2pt\sum_{b \sim B} {\mathcal K}_{s,p}(u(v+b)) {\mathbf{\,e}}(bt) \bigg|, \end{align*} $$
where
$\nu (u,v)$
is the number of triples
$(a,m,n)$
with
$a \sim A$
,
$m \in \{d^r:~1\leq d \leq D\}$
and
$n \in [-N, N]$
such that
$$ \begin{align*} u \equiv \overline{a}n \pmod{p} \quad\mbox{and}\quad v \equiv ma\pmod{p}. \end{align*} $$
Following the steps in [Reference Banks and Shparlinski3, Section 4.3] leading to [Reference Banks and Shparlinski3, (4.9), (4.10) and (4.11)] and defining
${\mathcal M} = \{d^r:~1\leq d \leq D\}$
,
$$ \begin{align} S^{2\ell} \ll A^{-2}B^{-2\ell} D^{2\ell-2} N^{2\ell-1} (B^\ell p^2 + B^{2\ell} p) J(2A,{\mathcal M}) p^{o(1)}, \end{align} $$
where
$J(H,{\mathcal M})$
denotes the number of solutions to the congruence
$$ \begin{align} x k\equiv ym\bmod p \end{align} $$
for which
$x,y\in [1,H]$
and
$k,m \in {\mathcal M}$
. Taking
$B = \lfloor p^{1/\ell }\rfloor $
, we see that (3.3) simplifies as
$$ \begin{align} S^{2\ell} \ll A^{-2} D^{2\ell-2} N^{2\ell-1} J(2A,{\mathcal M}) p^{1+o(1)}. \end{align} $$
In the setting of the proof of [Reference Kowalski, Michel and Sawin20, Theorem 4.3], the condition
$$ \begin{align} 2A\max_{m\in {\mathcal M}} m < p \end{align} $$
is satisfied, which allows us to replace (3.4) with an equation
$x k = ym$
in integers. Then, using the classical bound on the divisor function (see [Reference Iwaniec and Kowalski16, (1.81)]), it is shown in [Reference Kowalski, Michel and Sawin20] that, under the condition (3.6), we have
$J(2A,{\mathcal M}) \leqslant (A\# {\mathcal M})^{1+ o(1)}$
. However, (3.6) is too restrictive for our purpose, so we instead use a result of Ayyad, Cochrane and Zheng [Reference Ayyad, Cochrane and Zheng1, Theorem 2] which, similarly to the proof of [Reference Cilleruelo, Shparlinski and Zumalacárregui6, Theorem 4.1], leads to the bound
$$ \begin{align} J(2A,{\mathcal M}) \ll A^2D^2/p + (AD)^{1+ o(1)}. \end{align} $$
$$ \begin{align*} S^{2\ell} \ll (D^{2\ell} N^{2\ell-1} + A^{-1} D^{2\ell-1} N^{2\ell-1}p) p^{o(1)}. \end{align*} $$
Since
$N> 2p^{1/\ell } \geqslant 2B$
, we may choose
$$ \begin{align*} A=\bigg\lfloor \frac{N}{2B}\bigg\rfloor \gg N p^{-1/\ell}, \end{align*} $$
which guarantees that the condition (3.2) is met. We obtain the stated bound after simple calculations.
4 Proofs of the main results
Proof of Theorem 2.1.
Using inclusion–exclusion, we can write
$$ \begin{align*} Q_{s, p}(N) = \sum_{d \leqslant N^{1/2} }\mu(d)\sum_{n\leqslant N/d^2} {\mathcal K}_{s, p} (d^2n). \end{align*} $$
Next, we split the sum
$Q_{s, p}(N)$
into dyadic intervals with respect to some parameter
$D \geqslant 1$
to get
$O(\log N)$
sums of the type
$$ \begin{align*} S(D,N)=\sum_{d\sim D}\mu(d)\sum_{n\leqslant N/d^2} {\mathcal K}_{s, p} (d^2n) \end{align*} $$
for some
$D \leqslant N^{1/2}$
. By (3.1),
$$ \begin{align} S(D,N)\ll Dp^{1/2}\log p. \end{align} $$
By the Deligne bound (1.1),
$$ \begin{align} S(D,N) \ll DN/D^2 = N/D. \end{align} $$
However, for any fixed even integer
$\ell>0$
, by Lemma 3.3,
$$ \begin{align*} S(D,N) \ll DK\bigg(K^{-1} + \frac{p^{1+1/\ell}}{DK^2}\bigg)^{1/(2\ell)} p^{o(1)}, \end{align*} $$
where
$K=N/D^2$
, provided that
$K> 2p^{1/\ell }$
. Note that
$ DK = N/D \leqslant N \leqslant p \leqslant p^{1+1/\ell }$
and thus,
$$ \begin{align*} K^{-1} \leqslant \frac{p^{1+1/\ell}}{DK^2}. \end{align*} $$
It follows that if
$K> 2p^{1/\ell }$
, then
$$ \begin{align} S(D,N) \ll DK\bigg(\frac{p^{1+1/\ell}}{DK^2}\bigg)^{1/(2\ell)}p^{o(1)} = \frac{N}{D}\bigg( \frac{D^3p^{1+1/\ell}}{N^2} \bigg)^{1/(2\ell)}p^{o(1)}. \end{align} $$
Using the bounds (4.1) or (4.3) for
$K> 2p^{1/\ell }$
and the bound (4.2) for
$K \leq 2p^{1/\ell }$
, we arrive at
$$ \begin{align} S(D,N) \ll \min\{f_1(D), f_2(D)\} p^{o(1)} + N^{1/2} p^{1/(2\ell)}, \end{align} $$
where
$$ \begin{align*} f_1(D) = Dp^{1/2} \quad\mbox{and}\quad f_2(D) = \frac{N}{D}\bigg( \frac{D^3p^{1+1/\ell}}{N^2} \bigg)^{1/(2\ell)}. \end{align*} $$
Choose the parameter
$$ \begin{align*} D_0 = \bigg(\frac{N^{2\ell-2}}{p^{\ell-1-1/\ell}}\bigg)^{{1}/{(4\ell-3)}} = N^{1/2}p^{-1/4} \bigg(\frac{p^{1/2+2/\ell}}{N}\bigg)^{{1}/{2(4\ell-3)}} \end{align*} $$
such that
$f_1(D_0) = f_2(D_0)$
. Clearly,
$1 \leqslant D_0 \leqslant N^{1/2}$
since
$N \geq p^{1/2 + 2/\ell }$
by assumption. Hence,
$$ \begin{align*} \min\{f_1(D), f_2(D)\} \leqslant f_1(D_0) \leqslant N^{1/2}p^{1/4} \bigg(\frac{p^{1/2+2/\ell}}{N}\bigg)^{{1}/{2(4\ell-3)}}. \end{align*} $$
It can be easily verified that
$$ \begin{align*} p^{{1}/{2\ell}} \leqslant p^{1/4} \bigg(\frac{p^{1/2+2/\ell}}{p}\bigg)^{{1}/{2(4\ell-3)}} \leqslant p^{1/4} \bigg(\frac{p^{1/2+2/\ell}}{N}\bigg)^{{1}/{2(4\ell-3)}}. \end{align*} $$
Hence, the second term on the right-hand side of (4.4) is dominated by the first term, and
$$ \begin{align*} S(D,N) \ll N^{1/2}p^{1/4} \bigg(\frac{p^{1/2+2/\ell}}{N}\bigg)^{{1}/{2(4\ell-3)}} p^{o(1)}. \end{align*} $$
This concludes the proof.
Proof of Theorem 2.2.
Let
$L_0 \in [1, N]$
be a parameter to be chosen later. Observe that any y-smooth integer in
$(L_0, N]$
can be uniquely factored as
$n = \ell m$
such that
$$ \begin{align*} \ell \in (L_0, yL_0], \quad \frac{\ell}{P(\ell)} \leq L_0,\quad p(m) \geq P(\ell), \end{align*} $$
where
$P(\ell )$
denotes the largest prime factor of
$\ell $
and
$p(m)$
denotes the smallest prime factor of m. Indeed, this factorisation can be obtained by writing
$n = p_1p_2\cdots p_k$
with primes
$p_1 \leqslant \cdots \leqslant p_k$
and by setting
$\ell = p_1p_2\cdots p_r$
, where r is the smallest positive integer such that
$p_1p_2\cdots p_r> L_0$
.
Thus,
$$ \begin{align*}R_{\mathsf{K}}(N,y) = \sum_{\substack{L_0 < \ell \leq P(\ell) L_0 \\ \ell \in S(y)}} \, \sum_{\substack{m \in {\mathcal S}(N/\ell, y) \\ p(m) \geq P(\ell)}} \mathsf{K}(\ell m) + O(L_0). \end{align*} $$
After dyadic partition of the range for
$\ell $
, we see that there is
$L \in (L_0, yL_0]$
such that
$$ \begin{align} R_{\mathsf{K}}(N,y) \ll U \log N + L_0, \end{align} $$
where
$$ \begin{align*} U= \sum_{\substack{L < \ell \leq \min\{P(\ell)L_0, 2L\} \\ \ell \in S(y)}} \, \sum_{\substack{m \in {\mathcal S}(N/\ell, y) \\ p(m) \geq P(\ell)}} \mathsf{K}(\ell m). \end{align*} $$
We now employ the completion technique as in [Reference Iwaniec and Kowalski16, Section 12.2] again. That is, first, we write
$$ \begin{align*} U = \sum_{\substack{L < \ell \leq \min\{P(\ell)L_0, 2L\} \\ \ell \in S(y)}} \, \sum_{\substack{m \in {\mathcal S}(N/L, y) \\ p(m) \geq P(\ell)}} \mathsf{K}(\ell m) \frac{1}{N} \sum_{ 1 \leqslant k \leqslant N/\ell} \sum_{a=1}^N {\mathbf{\,e}}(a (m-k)/N), \end{align*} $$
where, as before,
$ {\mathbf {\,e}}(z) = \exp (2\pi i z)$
. After changing the order of summation and using [Reference Iwaniec and Kowalski16, (8.6)] (similarly to the argument in Section 4), we derive
$$ \begin{align*} U \ll \sum_{\substack{\ell \sim L \\ \ell \in S(y)}} \bigg| \sum_{\substack{m \in {\mathcal S}(N/L, y) \\ p(m) \geq P(\ell)}} \mathsf{K}(\ell m) {\mathbf{\,e}}(\eta m) \bigg| \log N \end{align*} $$
for some real
$\eta \in {\mathbb R}$
.
By the Cauchy–Schwarz inequality,
$$ \begin{align*} U^2 \ll (\log N)^2 \Psi(L, y) \sum_{\substack{\ell \sim L \\ \ell \in S(y)}} \bigg| \sum_{\substack{m \in {\mathcal S}(N/L, y) \\ p(m) \geq P(\ell)}} \mathsf{K}(\ell m) {\mathbf{\,e}}(\eta m) \bigg|^2. \end{align*} $$
Writing
$q = P(\ell )$
and replacing
$\ell $
by
$q\ell $
,
$$ \begin{align*}U^2 \ll (\log N)^2 \Psi(L, y) \sum_{q \leq y} \sum_{\ell \sim L/q} \bigg| \sum_{\substack{m \in {\mathcal S}(N/L, y) \\ p(m) \geq q}} \mathsf{K}(q \ell m) {\mathbf{\,e}}(\eta m) \bigg|^2, \end{align*} $$
where we have dropped the primality condition on q. Expanding the square,
$$ \begin{align} U^2 \ll (\log N)^2 \Psi(L,y) \sum_{q \leq y} \sum_{m_1,m_2 \in {\mathcal S}(N/L,y)} \bigg| \sum_{\ell \sim L/q} \mathsf{K}(q\ell m_1) \overline{\mathsf{K}(q\ell m_2)} \bigg|. \end{align} $$
The contribution
$Y_1$
to the right-hand side of (4.6) from terms with
$m_1/m_2\in {\mathcal E}_{\mathfrak {F}}$
is
$$ \begin{align*} Y_1 \ll (\log N)^2 \Psi(L,y) \sum_{q \leq y} \Psi(N/L, y) \frac{L}{q} \ll L N^{-\alpha} \Psi(N,y)^2 (\log N)^3, \end{align*} $$
where
$\alpha = \alpha (N,y)$
and we have also used the standard bounds (see [Reference Harper14, Section 2])
$$ \begin{align*} \Psi(L, y) \ll \bigg(\frac{L}{N}\bigg)^{\alpha}\Psi(N,y),\quad\Psi(N/L,y) \ll \frac{1}{L^{\alpha}} \Psi(N,y). \end{align*} $$
To estimate the contribution
$Y_2$
to the right-hand side of (4.6) from terms with
$m_1/m_2\not \in {\mathcal E}_{\mathfrak {F}}$
, we observe that Lemma 3.2 implies that
$$ \begin{align*} \bigg| \sum_{\ell \sim L/q} \mathsf{K}(q\ell m_1) \overline{\mathsf{K}(q\ell m_2)} \bigg| \ll p^{1/2} \log p \end{align*} $$
when
$m_1/m_2\not \in {\mathcal E}_{\mathfrak {F}}$
. Hence,
$$ \begin{align*} Y_2 \ll (\log N)^2 \Psi(L,y)\cdot y \Psi(N/L, y)^2 p^{1/2}\log p \ll y p^{1/2} L^{-\alpha} \Psi(N,y)^2 N^{o(1)}. \end{align*} $$
Overall, with
$\alpha = \alpha (N,y)$
, then,
$$ \begin{align*} U^2 & \ll Y_1 + Y_2 \leqslant (LN^{-\alpha} + yp^{1/2}L^{-\alpha}) \Psi(N,y)^2 N^{o(1)} \\ & \leqslant y (L_0N^{-\alpha} + p^{1/2}L_0^{-\alpha}) \Psi(N,y)^2 N^{o(1)}. \end{align*} $$
Choosing
$L_0 = p^{1/2(1+\alpha )} N^{\alpha /(1+\alpha )}$
to balance the two terms depending on
$L_0$
, we conclude that
$$ \begin{align*}U \leqslant y^{1/2} p^{{1}/{4(1+\alpha)}} N^{-{\alpha^2}/{2(1+\alpha)}} \Psi(N,y) N^{o(1)}. \end{align*} $$
Hence, we see from (4.5) that
$$ \begin{align*} R_{\mathsf{K}}(N,y) & \ll \Psi(N,y) y^{1/2} p^{{1}/{4(1+\alpha)}} N^{-{\alpha^2}{2(1+\alpha)}+o(1)} + p^{1/2(1+\alpha)} N^{\alpha/(1+\alpha)}\\ & = \Psi(N,y) y^{1/2} p^{\beta} N^{-\gamma + o(1)} + p^{2\beta} N^{\alpha -2 \gamma}. \end{align*} $$
We may assume that
$p^{\beta } \leqslant N^{\gamma }$
since otherwise, the claimed bound is trivial. Using (2.1),
$$ \begin{align*} p^{2\beta} N^{\alpha -2 \gamma} \leqslant p^{\beta} N^{\alpha -\gamma} = \Psi(N,y)p^{\beta} N^{-\gamma + o(1)} \end{align*} $$
and the result follows.
Acknowledgement
The authors would like to thank the referee for very encouraging comments.
Open access
