Li and Xu Journal of Inequalities and Applications 2013, 2013:504
http://www.journalofinequalitiesandapplications.com/content/2013/1/504
RESEARCH
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
Let q ≥  be a positive integer. For any integers m and n, the generalized two-term exponential sum C(m, n, k, χ ; q) is defined by
C(m, n, k, χ ; q) =
q
a=
mak + na
,
χ (a)e
q
where χ denotes any Dirichlet character mod q, and e(y) = eπ iy .
Regarding the upper bound estimate of C(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:
Let p be an odd prime with p = a + . Then, for any integer m with (m, p) = , we have
the identity
p
C(m, n, , χ; p)
n=
⎧

⎪
⎪
⎨p(p – p – )

= p (p – )
⎪
⎪
⎩ 
p (p – )
if χ is the principal character mod p;
if χ is the Legendre symbol mod p;
otherwise.
©2013 Li and Xu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
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Page 2 of 8
p–
Wang [] studied the hybrid power mean of the generalized Kloosterman sums a= λ(a)×
p–
) and a= χ (a + a), where λ denotes a Dirichlet character mod p, and gave an
e( ma+a
p
interesting asymptotic formula for it. That is, she proved the following result:
Let p be an odd prime. Then, for any non-principal even character χ mod p and any
character λ mod p with λ = ( p∗ ), we have the asymptotic formula

 p–
p– p–
ma + a λ(a)e
χ (mb + b) = p + O p .
·
p
m= a=
b=
In this paper, as a note of [] and [], we found that there exists a close relationship
p–
between the fourth power mean of C(m, n, , χ; p) and | b= 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 character χ mod p, we have the identity


p p–
ma + a χ (a)e
p
m= a=
⎧
– 
–


⎪
⎪
⎨p – p + ( p )p – p – ( p )p
= p – p
⎪
⎪
p–
⎩ 
p – ( –
) · p – p – p · | a= χ (a + a)|
p
if χ = χ ;
if χ (–) = –;
if χ = χ and χ (–) = ,
where χ denotes the principal character mod p, a · a ≡  mod p, 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 character χ mod p, we have
the inequalities


p p–
ma + a χ (a)e
p – p ≤
≤ p + p .
p


m= a=
2 Several lemmas
In this section, we shall give several lemmas, which are necessary in the proof of our theorem. 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– p– m
–mn a
mb + nb
=
e
,
e
e
p
p
p
p
a=
b=
where ( px ) denotes the Legendre symbol, and mm ≡  mod p.
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
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Proof See Lemma  in [].
Lemma  Let p be an odd prime, χ be any fixed character mod p. Then, for any non-real
character χ mod p, we have the identity
p–
p–
 

ma + a χ (m)
χ (a)e
p
m=
a=
p–

= p · χ (a + )χ ( + a) .
a=
Proof From the properties of Gauss sums, we have
p–
m=
p–


ma + a χ (m)
χ (a)e
p
=
a=
p– p–
a= b=
m(a – b ) + (a – b)
χ (ab)
χ (m)e
p
m=
p–
p– p–
=
χ (a)
a= b=
= τ (χ )
p–
 
mb (a – ) + b(a – )
χ (m)e
p
m=
p–
χ (a)
a=
p–
b(a – )
χ b  a –  e
p
b=
= τ (χ )τ χ 
χ (a)χ a –  χ  (a – )
p–
a=
χ (a)χ (a + )χ (a – )
= τ (χ )τ χ 
p–
a=
χ (a + )χ ( + a).
= τ (χ )τ χ 
p–
()
a=
Note that χ (p –  + ) = , χ is a non-real character mod p, so χ  is also a non-principal
√
character mod p. Therefore, |τ (χ )| = |τ (χ  )| = p, so from () we may immediately deduce Lemma .
Lemma  Let p be an odd prime, χ be any non-principal character mod p with χ (–) = .
Then we have
 p–
p–


–
a –
·
χ (a + a) =
χ (a)
p
p
a=
and
p–
√
χ (a + a) ≤  · p.
a=
a=
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
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Proof From the properties of quadratic residue mod p, we have
p–
χ (a + a) =
a=
p–
=
a=
a+a≡b mod p
b=
=
p–
χ (b)
p–
p–
 = χ()
p–
a=
(a–b) ≡b – mod p
b=
= χ () ·
p–
b=

a=
a –ba+≡ mod p
b=
p–
χ (b)
p–
χ (b)
b=
p–
χ (b)

a=
a ≡b – mod p


p–
b –
b –
= χ () ·
.
χ (b)  +
χ (b)
p
p
()
b=
Note that
p–
b=
b – 
χ (b)
p
=
p–
b=

b –
χ (b)
p
=
p–

– b –
,
χ (b)
p
p
b=
so from () we may immediately deduce the identity
p–
 p–


– a –
χ (a + a) =
χ (a)
.
p
p
a=
a=
The estimate
p–
√
χ (a + a) ≤  · p
a=
follows from Lemma  of []. This proves Lemma .
3 Proof of Theorem
In this section, we shall give two different proofs of our theorem. First, if χ is a nonprincipal character mod p, then from Lemma  we have
p–


ma + a χ (a)e
p
a=
=
p– p–
a= b=
=
p– p–
a= b=
m(a – b ) + a – b
χ (ab)e
p
χ (a)e
mb (a – ) + b(a – )
p
p– p–
p– –b
mb (a – ) + b(a – )
+
e
χ (a)
e
= p –  + χ (–)
p
p
a=
b=
b=
p–
mb (a – ) + b(a – ) –
χ (a)
e
χ (a)
= p –  – χ (–) +
p
a=
a=
p–
p–
b=
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
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p– p–
m(a + )a –  · b + b
–
=p+
χ (a)
e
χ (a)
p
a=
a=
p–
b=
p–
= p + G(p) ·
χ (a)
a=
m(a + )a – 
m · a + (a – )
e
p
p
m(a – )
m · a + (a – )
e
,
χ (a)
= p + G(p) ·
p
p
a=
p–
()
p–

) · p (see Theorem .. of []).
where G(p) = a= e( ap ) and G (p) = ( –
p
From () and the definition of Gauss sums, we may immediately deduce


p– p–
ma + a χ (a)e
p
m= a=
= p (p – ) + pG(p)
p– p–
χ (a)
m= a=
p– p–
+ G (p)
χ (ab)
a= b=
×
m(a – )
m · a + (a – )
e
p
p
(a – )(b – )
p
p– m · (a + (a – ) + b + (b – ))
e
p
m=
p–
(a – )(a – )a + 
χ (a)
= p (p – ) + pG (p)
p
a=


p–


(a – )(a – )
a –

– G (p)
+ pG (p)
χ ()
χ (a)
p
p
a=
a=

p–
–
(p – )
= p (p – ) + pG (p)
χ (a) + pG (p)
p
a=


p–
p–

a – 
– G (p)
χ (a)
p
a=


.
()
If χ is a non-principal character mod p with χ (–) = –, then note that
p–
χ (a)
a=
a – 
p
=
p–
χ (a)
a=
a – 
p
= ,
p–  a
–
·p
G (p) =
e
=
p
p
a=

and
p–
 p–

 · a + a a χ (a)e
χ (a)e
=
= p .
p
p a=
a=
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
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From () we have


p– p–
ma + a χ (a)e
= p – p .
p
()
m= a=
If χ is a non-principal character mod p with χ (–) = , then from () and Lemma  we
have


p– p–
ma + a χ (a)e
p
m= a=
p–


–
–
a –


· p – p –
·p·
= p – 
χ (a)
p
p
p
a=
p–

–
· p – p – p · = p – 
χ (a + a) .
p

()
a=
If χ = χ is the principal character mod p, then from the method of proving () and ()
we have

p– p– 
– 
–
ma + a p –p–
p.
e
= p – p + 
p
p
p
()
m= a=
Combining (), () and (), we may immediately deduce our theorem.
The second proof of Theorem. First, from the orthogonality of characters mod p, we
have
p–
p–
 

ma + a χ (m)
χ (a)e
p
χ mod p m=
a=


p– p–
ma + a = (p – ) ·
χ (a)e
.
p
()
m= a=
On the other hand, from Lemma  we have
p–
p–
 

ma + a χ (m)
χ (a)e
p
χ mod p m=
a=
 p– p–
p–
 

ma
+
a
= p ·
χ (a + )χ ( + a) +
χ (a)e
p
χ mod p a=
m= a=
p– p–
p–

 

m ma + a  +
χ (a)e
χ (a + )χ ( + a)
–p ·
p a=
p
m=
a=
p–

 + a  –p ·
χ (a + )
p
a=
≡ p A + B + C – p D – p E.
()
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
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Applying the orthogonality of characters mod p, we can easily deduce that

p–
χ (a + )χ ( + a) = (p – )(p – ),
A=
()
χ mod p a=
p– p–
 

ma + a B=
χ (a)e
p
m= a=
⎧
⎨(p – p – )
if χ = χ ;
=
⎩p (p –  – χ (–)) if χ = χ .
()
From the definition and properties of Gauss sums, we have
p–
p– p–
 



m ma + a a –  χ (a)e
χ (a)
C=
=p·
,
p a=
p
p
m=
a=

p–
χ (a + )χ ( + a) = p – ,
D=
()
()
a=
p–
 p–


 + a a –  χ (a + )
χ (a)
E=
=
.
p
p
a=
()
a=
Note that if χ (–) = –, then
p–
χ (a)
a=
a – 
p
= .
Combining ()-() and Lemma , we may immediately deduce the identity


p– p–
ma + a χ (a)e
p
m= a=
⎧
– 
–


⎪
⎪
⎨p – p + ( p )p – p – ( p )p – 
= p – p
⎪
⎪
p–
⎩ 
p – ( –
) · p – p – p · | a= χ (a + a)|
p
if χ = χ ;
if χ (–) = –;
if χ = χ and χ (–) = .
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.
Received: 13 June 2013 Accepted: 2 October 2013 Published: 08 Nov 2013
Li and Xu Journal of Inequalities and Applications 2013, 2013:504
http://www.journalofinequalitiesandapplications.com/content/2013/1/504
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10.1186/1029-242X-2013-504
Cite this article as: Li and Xu: The fourth power mean of the generalized two-term exponential sums and its upper
and lower bound estimates. Journal of Inequalities and Applications 2013, 2013:504
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