# New Mixed Exponential Sums and Their Application

Authors: Yu Zhan; Xiaoxue Li
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The main purpose of this paper is to introduce a new mixed exponential sums and then use the analytic methods and the properties of Gauss sums to study the computational problems of the mean value involving these sums and give an interesting computational formula and a sharp upper bound estimate for these mixed exponential sums. As an application, we give a new asymptotic formula for the fourth power mean of DirichletL-functions with the weight of these mixed exponential sums.


---

## Body

## 1. Introduction

Letq
≥
3 be an integer, and let χ be a Dirichlet character mod
q. Then, for any integer n, the famous Gauss sums G
(
χ
,
n
) are defined as follows:
(1)
G
(
χ
,
n
)
=
∑
a
=
1
q
χ
(
a
)
·
e
(
n
a
q
)
,
where e
(
y
)
=
e
2
π
i
y.This sum and the other exponential sums (such as Kloosterman sums) play very important role in the study of analytic number theory, and many famous number theoretic problems are closely related to it. For example, the distribution of primes, Goldbach problem, the estimate of character sums, and the properties of DirichletL-functions are some good examples.From the properties of characters modq, we known that if χ is a primitive character mod q, then G
(
χ
,
n
)
=
χ
¯
(
n
)
·
G
(
χ
,
1
)
≡
χ
¯
(
n
)
·
τ
(
χ
), and |
τ
(
χ
)
|
=
q. Many other properties of G
(
χ
,
n
) and τ
(
χ
) can also be found in [1–4].In this paper, we introduce new mixed exponential sums as follows:(2)
G
(
χ
,
c
,
m
,
n
;
q
)
=
∑
a
=
0
q
-
1
∑
b
=
0
q
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
q
)
,
where c, m, and n are any integers.We will study the arithmetical properties ofG
(
χ
,
c
,
m
,
n
;
q
). About this problem, it seems that none has studied it yet; at least we have not seen any related results before. The problem is interesting, because this sum has a close relationship with the general Kloosterman sums, and it is also analogous to famous Gauss sums, so it must have many properties similar to these sums. It can also help us to further understand and study Kloosterman sums and Gauss sums.The main purpose of this paper is using the analytic method and the properties of Gauss sums to study the fourth power mean ofG
(
χ
,
c
,
m
,
n
;
p
) and its upper bound estimate and prove the following three conclusions.Theorem 1.
Letp be an odd prime; let χ be any nonprincipal character mod
p. Then, for any integers c, m, and n, one has the estimate
(3)
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
≤
2
p
.Theorem 2.
Letp be an odd prime; let χ be any nonprincipal character mod
p. Then, for any integers m and n, one has the identity
(4)
∑
c
=
0
p
-
1
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
4
=
{
3
p
5
-
8
p
4
,
i
f
χ
i
s
t
h
e
L
e
g
e
n
d
r
e
s
y
m
b
o
l
mod
p
;
p
4
(
2
p
-
7
)
,
i
f
χ
i
s
a
c
o
m
p
l
e
x
c
h
a
r
a
c
t
e
r
mod
p
.Theorem 3.
Letp be an odd prime. Then, for any integers c, m, and n with (
c
,
m
2
+
n
2
-
m
n
,
p
)
≠
p, one has the asymptotic formula
(5)
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
|
∑
a
=
0
p
-
1
‍
∑
b
=
0
p
-
1
‍
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
2
·
|
L
(
1
,
χ
)
|
4
=
5
72
·
π
4
·
p
3
+
O
(
p
5
/
2
·
exp
⁡
(
4
ln
⁡
⁡
ln
⁡
⁡
p
ln
⁡
⁡
p
)
)
,
where χ
0 is the principal character mod
p, (
l
,
m
,
n
) denotes the greatest common divisor of l, m, and n, and exp
⁡
(
y
)
=
e
y.In Theorem1, we only discussed the case, in which there exist two variables. For general case (with k(≥
3) variable), whether there exists a sharp estimate for the sums is an interesting problem.Letk
≥
3; whether there exists an exact computational formula for the 2
kth power mean,
(6)
∑
c
=
0
p
-
1
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
2
k
,
is also an open problem.


## 2. Several Lemmas

In this section, we will give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we will use many properties of character sums, Kloosterman sums, and Gauss sums; all of these can be found in [1, 5–7], so they will not be repeated here. First, we have the following.Lemma 4.
Letp be an odd prime; then, for any integers c, m, and n, one has the identity
(7)
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
=
p
·
|
∑
r
=
1
p
-
1
χ
¯
(
r
)
·
e
(
c
r
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
|
,
where r
¯ denotes the solution of the congruence equation r
·
x
≡
1
mod
⁡
p.Proof.
If(
n
,
p
)
=
1, then, from the properties of Gauss sums and quadratic residue mod
p, we have
(8)
∑
a
=
0
p
-
1
e
(
n
a
2
p
)
=
1
+
∑
a
=
1
p
-
1
e
(
n
a
2
p
)
=
1
+
∑
a
=
1
p
-
1
(
1
+
(
a
p
)
)
·
e
(
n
a
p
)
=
∑
a
=
0
p
-
1
e
(
n
a
p
)
+
∑
a
=
1
p
-
1
(
a
p
)
·
e
(
n
a
p
)
=
(
n
p
)
∑
a
=
1
p
-
1
(
a
p
)
·
e
(
a
p
)
=
(
n
p
)
·
τ
(
χ
2
)
,
where χ
2
=
(
*
/
p
) denotes the Legendre symbol.
Sinceχ is a nonprincipal Dirichlet character mod p, from (8), the properties of Gauss sums, and complete residue system mod p, we have
(9)
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
=
1
τ
(
χ
¯
)
·
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
∑
r
=
1
p
-
1
χ
¯
(
r
)
e
(
r
(
a
2
+
a
b
+
b
2
+
c
)
p
)
·
e
(
m
a
+
n
b
p
)
=
1
τ
(
χ
¯
)
·
∑
r
=
1
p
-
1
χ
¯
(
r
)
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
e
(
r
a
2
+
(
b
r
+
m
)
a
+
r
b
2
+
n
b
+
c
r
p
)
=
1
τ
(
χ
¯
)
∑
r
=
1
p
-
1
‍
χ
¯
(
r
)
×
∑
b
=
0
p
-
1
∑
a
=
0
p
-
1
e
(
4
¯
r
(
2
a
+
b
+
r
¯
m
)
2
+
r
b
2
+
r
c
+
n
b
-
4
¯
r
(
b
+
r
¯
m
)
2
p
)
=
1
τ
(
χ
¯
)
·
∑
r
=
1
p
-
1
χ
¯
(
r
)
×
∑
b
=
0
p
-
1
∑
a
=
0
p
-
1
e
(
r
a
2
+
3
r
(
2
¯
b
)
2
+
(
2
n
-
m
)
2
¯
b
+
r
c
-
4
¯
r
¯
m
2
p
)
=
τ
(
χ
2
)
τ
(
χ
¯
)
·
∑
r
=
1
p
-
1
‍
χ
¯
(
r
)
χ
2
(
r
)
×
∑
b
=
0
p
-
1
e
(
3
r
4
¯
(
b
+
3
¯
r
¯
(
2
n
-
m
)
)
2
+
r
c
-
4
¯
r
¯
m
2
-
12
¯
r
¯
(
2
n
-
m
)
2
p
)
=
τ
(
χ
2
)
τ
(
χ
¯
)
·
∑
r
=
1
p
-
1
χ
¯
(
r
)
χ
2
(
r
)
×
∑
b
=
0
p
-
1
e
(
3
r
b
2
+
r
c
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
=
τ
2
(
χ
2
)
τ
(
χ
¯
)
·
χ
2
(
3
)
·
∑
r
=
1
p
-
1
χ
¯
(
r
)
e
(
c
r
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
.
For any nonprincipal character χ
mod
p, we have |
τ
(
χ
)
|
=
p. So, from (9) and noting that |
χ
¯
2
(
3
)
|
=
1, we have
(10)
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
=
p
·
|
∑
r
=
1
p
-
1
χ
¯
(
r
)
·
e
(
c
r
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
|
.
This proves Lemma 4.Lemma 5.
Letp be an odd prime; let χ be any nonprincipal character mod
p. Then, for any integers m and n, one has the estimate
(11)
∑
a
=
1
p
-
1
χ
(
a
)
·
e
(
m
a
+
n
a
¯
p
)
≤
2
p
.Proof.
Sinceχ is a nonprincipal character mod
p, if p
∣
m and p
∣
n, then
(12)
∑
a
=
1
p
-
1
χ
(
a
)
·
e
(
m
a
+
n
a
¯
p
)
=
∑
a
=
1
p
-
1
χ
(
a
)
=
0
.
If p
∣
m and (
p
,
n
)
=
1 or p
∣
n and (
p
,
m
)
=
1 or (
m
n
,
p
)
=
1, then, from the results of Weil [8], Malyšev [7], and Estermann [6], with some minor modifications, we can deduce the estimate
(13)
|
∑
a
=
1
p
-
1
χ
(
a
)
·
e
(
m
a
+
n
a
¯
p
)
|
≤
2
p
·
(
m
,
n
,
p
)
1
/
2
=
2
p
,
where (
m
,
n
,
p
) denotes the greatest common divisor of m, n, and p.
Now Lemma5 follows from (12) and (13).Lemma 6.
Letp be an odd prime; then, for any integer n with (
n
,
p
)
=
1, one has the calculating formula
(14)
∑
m
=
1
p
-
1
‍
|
∑
a
=
1
p
-
1
‍
χ
(
a
)
·
e
(
m
a
+
n
a
¯
p
)
|
4
=
{
2
p
3
-
3
p
2
-
3
p
-
1
,
i
f
χ
i
s
t
h
e
p
r
i
n
c
i
p
a
l
c
h
a
r
a
c
t
e
r
mod
p
;
3
p
3
-
8
p
2
,
i
f
χ
i
s
t
h
e
Legendre’s
s
y
m
b
o
l
mod
p
;
p
2
(
2
p
-
7
)
,
i
f
χ
i
s
a
n
o
n
-
r
e
a
l
c
h
a
r
a
c
t
e
r
mod
p
.Proof.
See [9] or Corollary  2 of [10].Lemma 7.
Letp be an odd prime and let χ be the Dirichlet character mod
p. Then one has the estimate
(15)
∑
a
=
1
p
-
1
|
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
χ
(
a
)
|
L
(
1
,
χ
)
|
4
|
=
O
(
p
·
exp
⁡
(
4
ln
⁡
⁡
ln
⁡
⁡
p
ln
⁡
⁡
p
)
)
.Proof.
See Lemma  5 of [11].


## 3. Proof of the Theorems

In this section, we will complete the proof of our theorems. First we prove Theorem1. In fact, from Lemmas 4 and 5, we may immediately deduce the estimate
(16)
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
=
p
·
|
∑
r
=
1
p
-
1
χ
¯
(
r
)
·
e
(
c
r
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
|
≤
2
p
.
This proves Theorem 1.Theorem2 follows from Lemmas 4 and 6. In fact, from these two lemmas, we have
(17)
∑
c
=
1
p
-
1
‍
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
4
=
p
2
·
∑
c
=
1
p
-
1
‍
|
∑
r
=
1
p
-
1
‍
χ
¯
(
r
)
·
e
(
c
r
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
|
4
=
{
3
p
5
-
8
p
4
,
if
χ
is
the
Legendre
symbol
mod
⁡
p
;
p
4
(
2
p
-
7
)
,
if
χ
is
a
complex
character
mod
⁡
p
.
This proves Theorem 2.Now, we prove Theorem3. Note that the asymptotic formula is
(18)
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
|
L
(
1
,
χ
)
|
4
=
5
72
·
π
4
·
p
+
O
(
exp
⁡
(
4
ln
⁡
⁡
ln
⁡
⁡
p
ln
⁡
⁡
p
)
)
,
and the estimate for Kloosterman sums (see [6]) is as follows:
(19)
∑
a
=
1
p
-
1
e
(
m
a
+
n
a
¯
p
)
≪
p
·
(
m
,
n
,
p
)
1
/
2
;
from Lemmas 4 and 7 and the method of proving the theorem in [11], we have
(20)
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
|
∑
a
=
0
p
-
1
∑
b
=
0
p
-
1
χ
(
a
2
+
a
b
+
b
2
+
c
)
·
e
(
m
a
+
n
b
p
)
|
2
·
|
L
(
1
,
χ
)
|
4
=
p
·
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
|
∑
r
=
1
p
-
1
‍
χ
¯
(
r
)
·
e
(
c
r
-
4
¯
r
¯
(
m
2
+
n
2
-
m
n
)
p
)
|
2
·
|
L
(
1
,
χ
)
|
4
=
p
·
∑
r
=
1
p
-
1
∑
s
=
1
p
-
1
e
(
c
(
r
-
s
)
-
4
¯
(
r
¯
-
s
¯
)
(
m
2
+
n
2
-
m
n
)
p
)
·
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
χ
¯
(
r
s
¯
)
|
L
(
1
,
χ
)
|
4
=
p
·
∑
r
=
1
p
-
1
∑
s
=
1
p
-
1
e
(
c
s
(
r
-
1
)
-
4
¯
s
¯
(
r
¯
-
1
)
(
m
2
+
n
2
-
m
n
)
p
)
·
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
χ
¯
(
r
)
|
L
(
1
,
χ
)
|
4
=
p
(
p
-
1
)
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
|
L
(
1
,
χ
)
|
4
+
O
(
p
3
/
2
·
∑
r
=
2
p
-
1
|
∑
χ
mod
⁡
p
‍
χ
≠
χ
0
χ
¯
(
r
)
|
L
(
1
,
χ
)
|
4
|
)
=
5
72
·
π
4
·
p
3
+
O
(
p
5
/
2
·
exp
⁡
(
4
ln
⁡
⁡
ln
⁡
⁡
p
ln
⁡
⁡
p
)
)
.
This completes the proof of Theorem 3.



---

*Source: 510534-2014-06-19.xml*