Zhu and Han Journal of Inequalities and Applications 2014, 2014:116
http://www.journalofinequalitiesandapplications.com/content/2014/1/116
RESEARCH
Open Access
On the fourth power mean of the two-term
exponential sums
Minhui Zhu1 and Di Han2*
A retraction article was published for this article. It is available from the following link:
http://www.journalofinequalitiesandapplications.com/content/2014/1/344.
RETRACTED 3rd SEPTEMBER 2014 doi:10.1186/1029-242X-2014-344
*
Correspondence:
[email protected]
2
Department of Mathematics,
Northwest University, Xi’an, Shaanxi,
P.R. China
Full list of author information is
available at the end of the article
Abstract
The main purpose of this paper is using the analytic methods and the properties of
Gauss sums to study the computational problem of one kind of fourth power mean
of two-term exponential sums, and to give an interesting identity and asymptotic
formula for it.
MSC: 11L05
Keywords: the two-term exponential sums; fourth power mean; Gauss sums;
identity; asymptotic formula
1 Introduction
Let q ≥  be a positive integer. For any integers m and n, the two-term exponential sum
C(m, n, k; q) is defined as follows:
k
q
ma + na
C(m, n, k; q) =
,
e
q
a=
where e(y) = eπ iy .
About the various properties of C(m, n, k; q), some authors had studied it, and they obtained a series of results, some related works can be found in references [–]. For example,
Gauss’ classical work proved the remarkable formula (see [])
⎧√
⎪
⎪ q,
⎪
⎨ ,
√
q( + i)  + e(–q/) = √
C(, , ; q) =
⎪ i q,

⎪
⎪
⎩
√
( + i) q,
if q ≡  mod ,
if q ≡  mod ,
if q ≡  mod ,
if q ≡  mod ,
where i = –.
√
Generally, for any odd number q and (m, q) = , the exact value of |C(m, n, ; q)| is q
(e.g. see Berndt, Evans, Williams and Apostol’s related works). Cochrane and Zheng []
show for the general sum that
C(m, n, k; q) ≤ k ω(q) q  ,
where ω(q) denotes the number of all distinct prime divisors of q.
© 2014 Zhu and Han; 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.
Zhu and Han Journal of Inequalities and Applications 2014, 2014:116
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In this paper, we study the fourth power mean of the two-term exponential sum
C(m, n, k; q) as follows:
q
C(m, n, k; q) ,
(.)
m=
RETRACTED 3rd SEPTEMBER 2014 doi:10.1186/1029-242X-2014-344
where n is any integer with (n, q) = .
As regards this problem, it seems that none has yet studied it, at least we have not seen
any related result before. The problem is interesting, because it can reflect that the mean
value of C(m, n, k; q) is well behaved. The main purpose of this paper is to show this point.
That is, we shall prove the following conclusion.
Theorem Let p >  be a prime. Then for any integer n with (n, p) = , we have the identity

p p– 
ma + na e
p
m= a=
p – p – p,
if  p – ,
=




p – p – p + τ (χ ) + τ (χ  ), if |p – ,
where χ is any -order character mod p.
Note that for any non-principal character χ mod p, we have |τ (χ)| =
theorem we may immediately deduce the following.
√
p, so from our
Corollary Let p >  be a prime with |p – . Then for any integer n with (n, p) = , we have
the asymptotic formula

p p– 

ma + na e
= p – p + O p  .
p
m= a=
It seems that our method can also be used to deal with (.) for all prime p and integer
k ≥ . But this time, the computing is very complex.
For any integer h ≥ , whether there exists an exact computational formula for
h
p p– 
ma + na e
,
p
m= a=
where p is an odd prime and (n, p) = , is an open problem.
2 Several lemmas
In this section, we will give several lemmas which are necessary in the proof of our theorem. In the proving process of all lemmas, we used many properties of Gauss sums;
all these can be found in [], we will not be repeat them here. First we have the following.
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Lemma  Let p be an odd prime, χ be any non-principal character mod p. Then for any
integer n with (n, p) = , we have the identity
RETRACTED 3rd SEPTEMBER 2014 doi:10.1186/1029-242X-2014-344
p– p–
ma + na  χ(m)
e
p
m=
a=
p–
χ (a + a + )|,
if χ is not a -order character mod p,
p|
= √ a= p–
p| –  + a= χ(a( – a))|, if χ is a -order character mod p.
Proof Note that χ is a non-principal character mod p, so if χ is not a -order character
mod p (that is, χ  = χ , the principal character mod p), then from the properties of Gauss
sums we have
p– p–
ma + na 
χ(m)
e
p
m=
a=
m(a – b ) + n(a – b)
χ(m)e
p
m=
p– p– p–
=
a= b=
=
 
mb (a – ) + nb(a – )
χ(m)e
p
m=
p– p– p–
a= b=
p–
p–
 
nb(a – )
= τ (χ)
χ a –
χ (b)e
p
a=
b=
 
χ a –  χ n(a – )
= τ (χ)τ χ 
p–
a=
= χ  (n)τ (χ)τ χ 
χ (a + ) –  χ a
p–
a=
χ (a + ) – a
= χ  (n)τ (χ)τ χ 
p–
a=

χ a + a +  ,
= χ  (n)τ (χ)τ χ 
p–
(.)
a=
p–
where τ (χ) = a= χ(a)e( ap ) denotes the classical Gauss sum.
If χ is a -order character mod p, then χ  = χ ; note that for any integer a with (a, p) = ,
we have
, if a is a third residue mod p,
χ  (a) + χ(a) +  =
, otherwise.
From the method of proving (.) we have the identity
p–
m=
p– ma + na 
χ(m)
e
p
a=
p–

nb(a – )
χ a – 
χ (b)e
p
a=
p–
= τ (χ)
b=
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p–
p– p–
 nb(a – )
= –τ (χ)
= τ (χ)
χ a –
e
χ a – 
p
a=
a=
b=
p–
= –τ (χ)
χ  (a) + χ(a) +  χ(a – )
a=
= –τ (χ)
p–
p–
p–
χ(a – ) +
χ a( – a) +
χ ( – a)
a=
a=
χ a( – a)
p–
= –τ (χ) – +
a=
.
(.)
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a=
Now note that |τ (χ)| =
Lemma .
√
p, if χ = χ . From (.) and (.) we may immediately deduce
Lemma  Let p be an odd prime, χ be any non-real character mod p. Then we have the
identity
p–
τ  (χ)
.
χ a(a – ) =
τ (χ  )
a=
Therefore,
p–
√
χ a(a – ) = p.
a=
Proof From the definition and properties of the classical Gauss sums we have
τ  (χ) =
p– p–
a= b=
p–
p–
a+b
b(a + )
χ(a)χ(b)e
χ(a)
χ  (b)e
=
p
p
a=
b=
χ(a)χ (a + ) = τ χ 
χ(a – )χ a
= τ χ
p–
p
a=
a=
= τ χ
χ ( – a)a = τ χ 
χ a( – a)
p–
p–
a=
a=
or
p–
τ  (χ)
.
χ a(a – ) =
τ (χ  )
a=
This proves Lemma .
3 Proof of the theorem
In this section, we shall complete the proof of our theorem. First from the orthogonality
of characters mod p we have
p– p–

p– p– ma + na  

ma + na χ(m)
e
e
= (p – )
.
p
p
χ mod p m=
a=
m= a=
(.)
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On the other hand, if  p–, then any non-principal character χ is not a -order character
mod p. Note that
p–
m=
p– ma + na 
χ (m)
e
p
a=
mb (a – ) + nb(a – ) =
e
p
a=
m=
p– p– p–
b=
= (p – ) + p –  = p – p – .
(.)
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From (.) and Lemma  we have
p– p–
ma + na  
χ(m)
e
p
χ mod p m=
a=
p–
p– p– p–
ma + na  
ma + na  
=
χ (m)
e
χ(m)
e
+
p
p
m=
a=

= p –p–

χ mod p m=
χ=χ
+
χ mod p
χ=χ

= p –p–


+p
a=
p–


p ·
χ a + a +  
a=

p–

χ a + a +  χ mod p a=

–p
p–

χ a + a + 

a=

= p –p–


+ p (p – )
p– p–
 – p (p – )
a= b=
a +a≡b +b mod p

= p –p–


+ p (p – )
p– p–
 – p (p – )
a= b=
(a–b)(a+b+)≡ mod p

= p – p –  + p (p – )(p –  + p –  – ) – p (p – )
= (p – ) p – p – p –  .
(.)
If  p – , then combining (.) and (.) we may immediately deduce the identity

p– p– 
ma + na e
= p – p – p – 
p
m= a=
or

p p– 
ma + na e
= p – p – p.
p
m= a=
(.)
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If |p – , let χ = χ be a -order character mod p, then χ  = χ is also a -order character
mod p; this time note that
p–
m=
p– ma + na 
χ (m)
e
= p(p – ) – p –  = p – p – 
p
a=
and
p–

χ (a + ) – a

=
a=
p–

χ (a + ) –  =
p–
a=
=
χ a  – 
a=
p–
RETRACTED 3rd SEPTEMBER 2014 doi:10.1186/1029-242X-2014-344
χ  (a) + χ  (a) +  χ (a – )
a=
= – +
p–
χ a( – a) ,
a=
and from Lemma  and the method of proving (.) we have
p– p–
ma + na  
χ(m)
e
p
χ mod p m=

= p – p – 
a=

+
p–


p χ a + a +  
a=
χ mod p
χ=χ ,χ ,χ

p–
+ p– +
χ a( – a) a=
= p – p – 


p–
+ p (p – )(p – ) – p χ (a + ) – a a=
+ p p +  – 
= p – p – 

p–
χ a( – a) – 
a=
p–
χ  a( – a) – p (p – )
a=
+ p (p – )(p – ) – p (p – )
p–
p–

– p – p p +  – 
χ a( – a) – 
χ  a( – a)
a=
a=
= (p – ) p – p – p –  + (p – ) τ  (χ ) + τ  (χ  ) .
(.)
So if |p – , then combining (.) and (.) we can deduce the asymptotic formula


p p– p– p– 

ma + na ma + na e
e
=+
p
p
m= a=
m= a=
= p – p – p +  τ  (χ ) + τ  (χ  ) .
Now our theorem follows from (.) and (.).
(.)
Zhu and Han Journal of Inequalities and Applications 2014, 2014:116
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by MZ and DH. All authors contributed equally to the writing of this paper. All
authors read and approved the final manuscript.
Author details
1
School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi, P.R. China. 2 Department of Mathematics, Northwest
University, Xi’an, Shaanxi, P.R. China.
Acknowledgements
The authors would like to thank the referee for his very helpful and detailed comments, which have significantly
improved the presentation of this paper. This work is supported by the P.S.F. (2013JZ001) and N.S.F. (11371291) of
P.R. China, and the SF of Education of Shaanxi province (12JK0877).
RETRACTED 3rd SEPTEMBER 2014 doi:10.1186/1029-242X-2014-344
Received: 7 February 2014 Accepted: 6 March 2014 Published: 19 March 2014
References
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3. Cochrane, T, Zheng, Z: Upper bounds on a two-term exponential sums. Sci. China Ser. A 44, 1003-1015 (2001)
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5. Cochrane, T, Pinner, C: Using Stepanov’s method for exponential sums involving rational functions. J. Number Theory
116, 270-292 (2006)
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