R E S E A R C H
Open Access
The fourth power mean of the generalized
two-term exponential sums and its upper and
lower bound estimates
Xiaoxue Li
*and Zhefeng Xu
*Correspondence:
[email protected] Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China
Abstract
In this paper, we use the analytic method and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the generalized two-term exponential sums, and give an exact computational formula for it.
MSC: Primary 11L40; 11F20
Keywords: generalized two-term exponential sums; fourth power mean; computational formula
1 Introduction
Letq≥ be a positive integer. For any integersmandn, the generalized two-term expo-nential sumC(m,n,k,χ;q) is defined by
C(m,n,k,χ;q) =
q
a=
χ(a)e
mak+na q
,
whereχdenotes any Dirichlet charactermodq, ande(y) =eπiy.
Regarding the upper bound estimate ofC(m,n,k,χ;q), many authors have studied it and obtained a series of important results; related contents can be found in [–] and []. For example, from Weil’s classical work [] one can deduce the estimate
C(m, , ,χ;p)≤·p
for (m,p) = .
Recently, Wang [] studied the computational problem of the fourth power mean of
C(m,n,k,χ;p), and proved the following conclusion:
Letpbe an odd prime withp= a+ . Then, for any integermwith (m,p) = , we have the identity
p
n=
C(m,n, ,χ;p)
=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
p(p– p– ) ifχis the principal charactermodp;
p(p– ) ifχis the Legendre symbolmodp;
p(p– ) otherwise.
Wang [] studied the hybrid power mean of the generalized Kloosterman sumspa–=λ(a)×
e(map+a) and ap=–χ(a+a), whereλ denotes a Dirichlet charactermodp, and gave an interesting asymptotic formula for it. That is, she proved the following result:
Letpbe an odd prime. Then, for any non-principal even characterχmodpand any characterλmodpwithλ= (∗p), we have the asymptotic formula
p–
m=
p–
a=
λ(a)e
ma+a p
·
p–
b=
χ(mb+b)
= p+Op.
In this paper, as a note of [] and [], we found that there exists a close relationship between the fourth power mean ofC(m,n, ,χ;p) and|pb=–χ(nb+b)|. The main purpose of this paper is to show this point. That is, we shall prove the following theorem.
Theorem Let p be an odd prime.Then,for any characterχmodp,we have the identity
p
m=
p–
a=
χ(a)e
ma+a
p
=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
p– p+ (–p)p–p– (–p)p ifχ=χ;
p– p ifχ
(–) = –;
p– (–p)·p– p–p· |ap=–χ(a+a)| ifχ=χandχ(–) = ,
whereχdenotes the principal charactermodp,a·a≡modp,and(∗p)is the Legendre
symbol.
From this theorem we may immediately deduce the following corollary.
Corollary Let p be an odd prime.Then,for any non-principal characterχmodp,we have
the inequalities
p– p≤ p
m=
p–
a=
χ(a)e
ma+a p
≤p+p.
2 Several lemmas
In this section, we shall give several lemmas, which are necessary in the proof of our the-orem. Hereinafter, we shall use many properties of character sums and Gauss sums, all of these can be found in reference [], so they will not be repeated here. First we have the following.
Lemma Let p be an odd prime.Then,for any integers m and n with(mn,p) = ,we have the identity
p–
b=
e
mb+nb
p
=
m p
e
–mn
p
p–
a=
e
a
p
,
Proof See Lemma in [].
Lemma Let p be an odd prime,χbe any fixed charactermodp.Then,for any non-real
characterχmodp,we have the identity
p–
m=
χ(m)
p–
a=
χ(a)e
ma+a
p
=p·
p–
a=
χ(a+ )χ( + a)
.
Proof From the properties of Gauss sums, we have
p–
m=
χ(m)
p–
a=
χ(a)e
ma+a p
=
p–
a=
p–
b=
χ(ab)
p–
m=
χ(m)e
m(a–b) + (a–b)
p
=
p–
a=
p–
b=
χ(a)
p–
m=
χ(m)e
mb(a– ) +b(a– )
p
=τ(χ)
p–
a=
χ(a)
p–
b=
χba– e
b(a– )
p
=τ(χ)τχ p–
a=
χ(a)χ
a– χ(a– )
=τ(χ)τχ p–
a=
χ(a)χ(a+ )χ(a– )
=τ(χ)τχ p–
a=
χ(a+ )χ( + a). ()
Note thatχ(p– + ) = ,χis a non-real charactermodp, soχis also a non-principal
character modp. Therefore,|τ(χ)|=|τ(χ)|=√p, so from () we may immediately
de-duce Lemma .
Lemma Let p be an odd prime,χbe any non-principal charactermodp withχ(–) = .
Then we have
p–
a=
χ(a+a)
=
–
p
·
p–
a=
χ(a)
a–
p
and
p–
a=
χ(a+a)
≤·
√
Proof From the properties of quadratic residuemodp, we have
p–
a=
χ(a+a) =
p–
b=
χ(b)
p–
a=
a+a≡bmodp
=
p–
b=
χ(b)
p–
a=
a–ba+≡modp
=
p–
b=
χ(b)
p–
a= (a–b)≡b–modp
=χ()
p–
b=
χ(b)
p–
a=
a≡b–modp
=χ()·
p–
b=
χ(b)
+
b–
p
=χ()·
p–
b=
χ(b)
b–
p
. ()
Note that
p–
b=
χ(b)
b–
p
=
p–
b=
χ(b)
b–
p
=
–
p
p–
b=
χ(b)
b–
p
,
so from () we may immediately deduce the identity
p–
a=
χ(a+a)
=
–
p
p–
a=
χ(a)
a–
p
.
The estimate
p–
a=
χ(a+a)
≤·
√
p
follows from Lemma of []. This proves Lemma .
3 Proof of Theorem
In this section, we shall give two different proofs of our theorem. First, ifχ is a
non-principal charactermodp, then from Lemma we have
p–
a=
χ(a)e
ma+a p
=
p–
a=
p–
b=
χ(ab)e
m(a–b) +a–b p
=
p–
a=
p–
b=
χ(a)e
mb(a– ) +b(a– )
p
=p– +χ(–)
p–
b=
e
–b p
+
p–
a=
χ(a)
p–
b=
e
mb(a– ) +b(a– )
p
=p– –χ(–) +
p–
a=
χ(a)
p–
b=
e
mb(a– ) +b(a– )
p
–
p–
a=
=p+
p–
a=
χ(a)
p–
b=
e
m(a+ )a– ·b+b
p
–
p–
a=
χ(a)
=p+G(p)·
p–
a=
χ(a)
m(a+ )a–
p
e
m·a+ (a– )
p
=p+G(p)·
p–
a=
χ(a)
m(a– )
p
e
m·a+ (a– )
p
, ()
whereG(p) =pa–=e(ap) andG(p) = (–
p)·p(see Theorem .. of []).
From () and the definition of Gauss sums, we may immediately deduce
p–
m=
p–
a=
χ(a)e
ma+a
p
=p(p– ) + pG(p)
p–
m=
p–
a=
χ(a)
m(a– )
p
e
m·a+ (a– )
p
+G(p)
p–
a=
p–
b=
χ(ab)
(a– )(b– )
p
×
p–
m=
e
m·(a+ (a– ) +b+ (b– ))
p
=p(p– ) + pG(p)
p–
a=
χ(a)
(a– )(a– )a+
p
+pG(p)
p–
a=
χ()
(a– )(a– )
p
–G(p)
p–
a=
χ(a)
a–
p
=p(p– ) + pG(p)
p–
a=
χ(a) +pG(p)
–
p
(p– )
–G(p)
p–
a=
χ(a)
a–
p
. ()
Ifχis a non-principal charactermodpwithχ(–) = –, then note that
p–
a=
χ(a)
a–
p
=
p–
a=
χ(a)
a–
p
= ,
G(p) =
p–
a=
e
a p
=
–
p
·p
and
p–
a=
χ(a)e
·a+a
p
=
p–
a=
χ(a)e
a p
From () we have p– m= p– a=
χ(a)e
ma+a
p
= p– p. ()
Ifχis a non-principal charactermodpwithχ(–) = , then from () and Lemma we
have p– m= p– a=
χ(a)e
ma+a
p
= p–
–
p
·p– p–
–
p
·p·
p–
a=
χ(a)
a–
p
= p–
–
p
·p– p–p·
p– a=
χ(a+a)
. ()
Ifχ=χis the principal charactermodp, then from the method of proving () and ()
we have p– m= p– a= e
ma+a
p
=p– p+
–
p
p–p–
–
p
p. ()
Combining (), () and (), we may immediately deduce our theorem.
The second proof of Theorem. First, from the orthogonality of characters modp, we have
χmodp
p– m=
χ(m)
p– a=
χ(a)e
ma+a
p
= (p– )·
p– m= p– a=
χ(a)e
ma+a
p . ()
On the other hand, from Lemma we have
χmodp
p– m=
χ(m)
p– a=
χ(a)e
ma+a p
=p·
χmodp
p– a=
χ(a+ )χ( + a) + p– m= p– a=
χ(a)e
ma+a
p + p– m= m p p– a=
χ(a)e
ma+a
p
–p·
p– a=
χ(a+ )χ( + a)
–p·
p– a=
χ(a+ )
+ a p
Applying the orthogonality of charactersmodp, we can easily deduce that
A=
χmodp
p–
a=
χ(a+ )χ( + a)
= (p– )(p– ), ()
B=
p–
m=
p–
a=
χ(a)e
ma+a
p
=
⎧ ⎨ ⎩
(p– p– ) ifχ=χ;
p(p– –χ
(–)) ifχ=χ.
()
From the definition and properties of Gauss sums, we have
C=
p–
m=
m p
p–
a=
χ(a)e
ma+a p
=p·
p–
a=
χ(a)
a–
p
, ()
D=
p–
a=
χ(a+ )χ( + a)
=p– , ()
E=
p–
a=
χ(a+ )
+ a p
=
p–
a=
χ(a)
a–
p
. ()
Note that ifχ(–) = –, then
p–
a=
χ(a)
a–
p
= .
Combining ()-() and Lemma , we may immediately deduce the identity
p–
m=
p–
a=
χ(a)e
ma+a p
=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
p– p+ (–
p)p–p– (
–
p)p– ifχ=χ;
p– p ifχ(–) = –;
p– (–
p)·p– p–p· |
p–
a=χ(a+a)| ifχ=χandχ(–) = .
This completes another proof of our theorem. The corollary follows from Theorem and Lemma .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LX carried out the part of Introduction, XZ carried out the proof of some lemmas, LX with XZ carried out the theorem’s proof. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referee for his/her very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N.S.F. (11071194) of P.R. China.
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