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Comptes Rendus
Mathématique
Mayank Pandey
Moment estimates for the exponential sum with higher divisor functions
Volume 360 (2022), p. 419-424
<https://doi.org/10.5802/crmath.45>
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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Comptes Rendus
Mathématique
2022, Vol. 360, p. 419-424
https://doi.org/10.5802/crmath.45
Number theory / Théorie des nombres
Moment estimates for the exponential sum
with higher divisor functions
Mayank Pandeya
a Department of Mathematics, California Institute of Technology, Pasadena, CA 91125 USA
E-mail: mpandey@caltech.edu
Manuscript received 12 August 2019, revised 31 December 2019 and 26 March 2020, accepted 28 March 2020.
1. Introduction
For a sequence (an)nÊ1 of arithmetic interest, it is often desirable to have estimates for the Lp
norms of the exponential sum M(α) = ∑nÉX ane(nα) as X grows. Such estimates are useful in
applications of the circle method. In addition, sufficiently strong estimates for them can yield
estimates for the distribution function {α ∈ [0,1] : |M(α)| Êλ} for λ in appropriate ranges.
In the case that an is 1 if n is a kth power and 0 otherwise, such estimates have connections
to Waring’s problem, and the consequences of conjectured estimates for
∫ 1
0 |M(α)|s dα for s in
various ranges have been studied by Vaughan and Wooley [6].
This problem was also studied by Keil [4] in the case of the indicator function of k-free
numbers, and the size of
∫ 1
0 |M(α)|s dα was estimated up to a constant factor for all s 6= 1+ 1k ,
and in the case s = 1+ 1k , it was only determined up to a factor of log X .
In general, when higher values of s are considered, as long as the sequence in question has
some structure in arithmetic progressions, the bulk of the contribution ends up coming from
narrow regions near a small number of points (typically rationals with small denominator). For
this reason, one typically expects that
∫ 1
0 |M(α)|s dα is between X −εAs (X ) and X εAs (X ) with
As (X ) equal to either Xα1s or Xα1s + Xα2s−σ1 for some α1 < α2, and some σ1 > 0. The second
case is what happens in the case of k-free numbers, as shown in [4], as well what is conjectured
in the case of kth powers (see [6] for more discussion of this). In the case of the Möbius function,
the first case is conjectured (it is implied by Mertens conjecture that |M(α)|¿ X 1/2+ε).
In this paper, we study the case of divisor functions and high moments. In particular, let k Ê 2
be some integer, and s > 2 be real. Then, let
τk (n) =
∑
d1...dk=n
1
and
M(α) = ∑
nÉX
τk (n)e(nα).
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
420 Mayank Pandey
Our main result is the following (we refer the reader to the end of this section for a specification
of our notational conventions regarding asymptotic notation).
Theorem 1. We have∫ 1
0
|M(α)|s dα= X s−1(log X )s(k−1) ∑
`Ê0
γ`,s,k
(log X )`
+Os,k (X s−1−δs,k+ε)
with
δs,k =
2(s −2)
(s +7)(k +1)+2 .
for some coefficients γs,k,` satisfying the bound |γs,k,`|¿ exp(O(`)), with γs,k,0 > 0.
We prove this with a straightforward application of the circle method. For such high moments,
the contribution near rationals with small denominator (the major arcs) dominates. We therefore
require bounds for the remaining points (the minor arcs).
The minor arc bounds we use follow from a decomposition of τk into type I and type II sums.
Vaughan’s identity in the proof of analogous bounds for the von Mangoldt function provides
this decomposition, though the convolution structure of τk makes the decomposition somewhat
more straightforward. The major arc estimates follow from standard estimates for partial sums of
τk (n)χ(n) coming from Voronoi summation (in particular, Theorem 4.16 in [2]).
In the course of dealing with the main term, we use a result on the order of magnitude of higher
moments of Dirichlet kernels, which we state here. A proof of this will be given in a later section.
Here, we write
v(β) = ∑
nÉX
e(nβ).
Proposition 2. We have that for s > 2∫ 1
0
|v(β)|s dβ= As X s−1 +Os
(
X s−2
)
.
where
As = 2
π
∫ ∞
0
|sin t |s
t s
dt .
Our methods likely generalize straightforwardly to the case of χ1 ∗ ·· · ∗ χk for some fixed
Dirichlet characters χ1, . . . ,χk , and yield a similar result. The case of Fourier coefficients of GL(k)
cusp forms is quite distinct however, since it is expected, and was shown by Jutila [3] for some of
the GL(2) case, that the relevant exponential sum is small everywhere. Consequently, the bulk of
the contribution should not be expected to come from the major arcs, so the method used here
fails.
We have not taken much care to optimize the sizes of the error terms. In particular, the error
terms in Proposition 6 can likely be improved quite cheaply. However, an error term qualitatively
superior to δs,k ¿s 1k is likely quite hard to obtain.
1.1. Notation and conventions
X is some sufficiently large real number that should be thought of as going to ∞, and ε > 0 is
some sufficiently small constant. s > 2 is a fixed real number, and k Ê 2 is some fixed integer. As
usual, we use the notation A ¿ B ⇐⇒ A = O(B) ⇐⇒ B À A to denote that |A| É C B for some
absolute constant C . Any further paramters on which C might depend will be noted in subscript
below the O or ¿. For the rest of this paper, this implied constant may depend on s,k,ε, and any
further parameters on which it may depend will be noted in a subscript. We write a ∼ A to denote
that A < a É 2A, and a ³ A to denote that A ¿ a ¿ A. Summations are by default assumed to be
over the natural numbersN= {1,2, . . .} unless specified otherwise.
Mayank Pandey 421
2. Setup
Take P = X η, with η = 2s−2δs,k . It is easy to see that we have the bound η É 25 . η also has the
property that 2k+1 −
(
9
2 + 1k+1
)
η = 12η(s − 2). The significance of this will become clear later on
when we are collecting various error terms. Also, let M be the union of
M(q, a) = {α ∈ [0,1] : |α−a/q | É P X −1}
for q É P, (a, q) = 1, and m= [0,1] \M. Note that for large X , all the M(q, a) are disjoint. It is easy
to see by Dirichlet’s approximation theorem that for all α ∈m, there exist P < q É X /P, (a, q) = 1
so that |α−a/q | É q−2. Then, the main result follows if we can prove the following estimates for
the contribution of the major and minor arcs.
Proposition 3. We have that∫
M
|M(α)|s dα= X s−1(log X )s(k−1) ∑
`Ê0
γ`,s,k
(log X )`
+O(X s−1−δs,k+ε)
where γs,k,`,δs,k are as in the statement of Theorem 1.
Proposition 4. Supposed that α, a, q are so that (a, q) = 1, |α− aq | É 1/q2. Then, we have that∣∣∣∣∣ ∑nÉX τk (n)e(nα)
∣∣∣∣∣¿
(√
q X + Xp
q
+X 4/5
)
(log X )O(1)
Proposition 5. We have the bound∫
m
|M(α)|s dα¿ X s−1−δs,k (log X )O(1).
Proof of Proposition 5. This follows immediately from Proposition 4, whose proof we defer to
the last section, and Parseval. Indeed, note that since η É 2/5, it follows from Proposition 4 that
supα∈m |M(α)|¿ X 1−η/2(log X )O(1), and therefore∫
m
|M(α)|s dα¿
(
sup
α∈m
|M(α)|
)s−2 ∫ 1
0
|M(α)|2dα
¿ (X 1−η/2)s−2X (log X )O(1) ¿ X s−1− 12η(s−2)(log X )O(1).
The proposition follows upon noting that 12η(s −2) = δs,k . 
In the next section, we shall prove Propositions 3. Proposition 4 is proven in the last section.
The main theorem clearly follows from Propositions 3 and 5.
3. Major arc estimates for higher divisor functions
Our main major arc estimate is the following.
Proposition 6. Suppose that q Ê 1,(a, q) = 1. Then, we have that∑
nÉX
τk (n)e
(
an
q
)
= X Pk,q (log X )+O(q
1
2 + kk+1 X
k−1
k+1 (q X )ε)
where Pk,q (log X ) is a polynomial of degree k −1 in log X with coefficients of size ¿ τ2(q)O(1)/q. In
addition, the coefficient of (log X )k−1 is nonnegative and À 1/q.
Proof. This follows from the method in the proof of Proposition 4.2 in [5], though we may use
Theorem 4.16 in [2] to achieve the above error terms. 
422 Mayank Pandey
From partial summation, we then obtain the following.
Corollary 7. Suppose that q Ê 1,(a, q) = 1, |β| É 1. Then, we have that∑
nÉX
τk (n)e
(
an
q
+nβ
)
=Qk,q (log X )v(β)+O((1+|β|X )q
1
2 + kk+1 X
k−1
k+1 (q X )ε)
where Qk,q (log X ) is a polynomial of degree k −1 in log X with coefficients of size ¿ τ2(q)O(1)/q.
In addition, the leading coefficient is nonnegative and À 1/q.
Before we start dealing with the main term, we shall prove Proposition 2.
Proof of Proposition 2. Our proof here uses the method in a MathStackExchange post of daniel-
fischer [1], though we take some care here to track the error terms.
We shall suppose for simplicity that X is an integer, as it can be easily checked that adjusting
X by O(1) does not alter the main term on the RHS by a quantity that can’t be absorbed into the
error term.
It is well-known then that |v(β)| =
∣∣∣ sin(π(X+1)β)sin(πβ) ∣∣∣, which can be shown by summing the geomet-
ric series that v(β) is. Taylor expanding sinπβ, we have that for β ∈ [0,1/2]
πβ
sinπβ
= 1+O(β2)
and it can be easily checked that |πβ/sin(πβ)−1| É 3/4 for β ∈ [0,1/2] (taking πβ/sin(πβ) = 1 at
β= 0). Therefore, we have that(
πβ
sinπβ
)s
=
(
1+
(
πβ
sinπβ
−1
))s
= 1+O(β2)
so ∫ 1
0
|v(β)|s dβ= 2
∫ 1/2
0
|sin((X +1)πβ)|s (πβ)−s dβ+O
(∫ 1/2
0
|sin((X +1)πβ)|sβ2−s dβ
)
.
By the bound |sin((X +1)πβ)|s ¿ min(1,(βX )s ) we have that the term inside the O( · ) is
¿
∫ 1/X
0
(βX )sβ2−s dβ+
∫ 1/2
1/X
β2−s dβ¿ X s−2.
Now, by a change of variables, the main term equals
2
π(X +1)
∫ (X+1)π/2
0
|sin t |s (t/(X +1))−s dt = 2
π
(X +1)s−1
∫ (X+1)π/2
0
|sin t |s
t s
dt
= 2
π
X s−1
∫ ∞
0
|sin t |s
t s
dt +O(X s−2).
as we have by a trivial bound that∫ (X+1)π/2
0
|sin t |s
t s
dt =
∫ ∞
0
|sin t |s
t s
dt +O(X 1−s ).
The desired result follows. 
We will now prove Proposition 3 using Proposition 2.
Proof of Proposition 3. From the definition of M, we have that∫
M
|M(α)|s dα= ∑
qÉP
∑∗
a(q)
∫ P/X
−P/X
∣∣∣∣M ( aq +β
)∣∣∣∣s dβ
= ∑
qÉP
ϕ(q)Qk,q (log X )
s
∫ P/X
−P/X
|v(β)|s dβ+O(X s−1P 92 + 1k+1 X − 2k+1 +ε).
Mayank Pandey 423
We may extend the range of integration to [−1/2,1/2] at a total loss of ¿ P (X /P )s−1(log X )O(1) ¿
X s−1−(s−2)η(log X )O(1) by the bound v(β) ¿ min(X ,‖β‖−1). Applying Proposition 2 then yields
that the above equals
As X
s−1 ∑
qÉP
ϕ(q)Qk,q (log X )
s +O(X s−1P 92 + 1k+1 X − 2k+1 +ε+X s−1X −(s−2)η+ε).
Now, writing Qk,q (log X ) =α0(q)+·· ·+αk−1(q)(log X )k−1, we obtain that
Qk,q (log X )
s = (log X )s(k−1)α0(q)s
(
1+ α1(q)α0(q)
−1
log X
+·· ·+ αk−1(q)α0(q)
−1
(log X )k−1
)s
=α0(q)s (log X )s(k−1)
∑
`Ê0
(s) . . . (s −`+1)
`!
· β`(q)
(log X )`
for some coefficients β`(q) with β0(q) = 1, and |β`(q)| ¿ τ2(q)O(`) for ` Ê 1. Here, we have use
the fact that α0(q) is nonnegative and À 1/q . Executing the summation over q , we thus obtain
that ∫
M
|M(α)|s dα= X s−1(log X )s(k−1) ∑
`Ê0
γs,k,`
(log X )`
+O(X s−1+
( 9
2 + 1k+1
)
η− 2k+1 +ε+X s−1−(s−2)η+ε)
for some coefficients γs,k,` satisfying the bound |γs,k,`| ¿ exp(O(`)). The desired result follows
from our choice of η, as (s −2)η= 2δs,k , and as we noted previously(
9
2
+ 1
k +1
)
η− 2
k +1 =−
s −2
2
η=−δs,k . 
4. The minor arcs
We shall now prove Proposition 4, from which our minor arc bound follows. Proposition 4 is
essentially the same bound one obtains in the case of the von Mangoldt function. Our proof
proceeds in the same manner as this case, through a decomposition of τk (n) into type I and type
II sums.
It is easy to see by splitting into dyadic intervals that it suffices to show the result with a sum
over n ∼ X , so we shall assume this from now on.
Our decomposition into type I and II sums will follow straightforwardly from the structure of
τk as a Dirichlet convolution. Note that
τk1[1,2X ] =
(
1[1,2X ] ∗·· ·∗1[1,2X ]︸ ︷︷ ︸
k times
)
1[X ,2X ].
Decomposing [1,2X ] into dyadic intervals then yields that this is a linear combination (with
coefficients of size O(1)) of O((log X )O(1)) summands of the form
(1I1 ∗·· ·∗1Ik )1[X ,2X ]
where I j is of the form either [N j ,2N j ] or [N j ,2X ] (with N j Ê X in the second case) for all j ,
for some 1 É N1 É ·· · É Nk satisfying N1 . . . Nk ³ X . It suffices then to show the bound in the
proposition for sums of the form ∑
n∼X
(1I1 ∗·· ·∗1Ik )(n)e(nα).
We have two cases. If all the Nk É X 1/5, then there exists a j so that X 2/5 ¿ N1 . . . N j , N j+1 . . . Nk ¿
X 3/5 so it follows that the sum equals ∑
m³N1...N j
n³N j+1...Nk
a(m)b(n)e(αmn)
424 Mayank Pandey
for some coefficients a(m),b(n) so that |a(m)|¿ τ j (m), |b(n)|¿ τk− j (n). The bound then follows
from a standard bound for type II sums (Lemma 13.8 in [2], for example, suffices).
Otherwise, we have that Nk > X 1/5, so the sum equals∑
n³Nk ,m³N1...Nk−1
a(m)e(αmn)
for some coefficients a(m) bounded by τk−1(n), so the desired result then follows standard
bounds on type I sums, such as Lemma 13.7 in [2].
Acknowledgements
The author would like to thank the anonymous referee for various corrections as well as for point-
ing out the simplified decomposition into type I and II sums used in the proof of Proposition 4.
References
[1] daniel-fischer, “Lp norm of the Dirichlet Kernel”, Math StackExchange. https://math.stackexchange.com/questions/
2031547 (version from 11-27-2016).
[2] H. Iwaniec, E. Kowalski, Analytic number theory, Colloquium Publications, vol. 53, American Mathematical Society,
2004, xi+615 pages.
[3] M. Jutila, “On exponential sums involving the Ramanujan function”, Proc. Indian Acad. Sci., Math. Sci. 97 (1987),
no. 1-3, p. 157-166.
[4] E. Keil, “Moment estimates for exponential sums over k-free numbers”, Int. J. Number Theory 9 (2013), no. 3, p. 607-
619.
[5] K. Matomäki, M. Radziwiłł, T. Tao, “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges”,
https://arxiv.org/abs/1707.01315, 2017.
[6] R. C. Vaughan, T. D. Wooley, “On the Distribution of Generating Functions”, Bull. Lond. Math. Soc. 30 (1998), no. 2,
p. 113-122.


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