Wang and Li Journal of Inequalities and Applications 2014, 2014:44
http://www.journalofinequalitiesandapplications.com/content/2014/1/44
RESEARCH
Open Access
A hybrid mean value involving a new sum
and Kloosterman sums
Xiaohan Wang1 and Xiaoxue Li2*
*
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
In this paper, we introduce a new sum, analogous to Cochrane sums, and use
elementary and analytic methods to study the hybrid mean value problem involving
this sum and Kloosterman sums, and we give an interesting asymptotic formula for it.
MSC: Primary 11L40; 11F20
Keywords: new sum analogous to Cochrane sums; Kloosterman sums; hybrid mean
value; asymptotic formula
1 Introduction
Let q be a natural number and h an integer with (h, q) = . The Cochrane sum C(h, q) is
defined by
C(h, q) =
q ah
a
,
q
q
a=
where
x – [x] –  , if x is not an integer;
(x) =
,
if x is an integer,
q
a is defined by aa ≡  mod q, and a= denotes the summation over all  ≤ a ≤ q such
that (a, q) = .
This sum was introduced by Professor Todd Cochrane, and it has been studied by many
authors, and one obtained many interesting results. Related works can be found in [–]
and [–]. For example, Zhang [] studied the hybrid mean value properties of Cochrane
sums and Kloosterman sums and proved that for any prime p > , we have the asymptotic
formula
p–
h=
 ln p
– 
,
K(h, ; p)C(h, p) =
p + O p · exp
π 
ln ln p
where exp(y) = ey ,
K(m, n; q) =
q
ma + na
e
q
a=
denotes the Kloosterman sum, and e(y) = eπ iy .
©2014 Wang and Li; 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.
Wang and Li Journal of Inequalities and Applications 2014, 2014:44
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Now for any odd prime p >  and integer h with (h, q) = , we define another sum analogous to the Cochrane sum as follows:
p–
   a h
a
.
C (h, p) =
p
p
a=
The main purpose of this paper is using elementary and analytic methods to study
the hybrid mean value problem involving C (h, p) and the Kloosterman sum and give an
asymptotic formula for it. That is, we shall prove the following.
Theorem Let p be an odd prime. Then we have the asymptotic formula
p–
h=
–
 ln p

/
.
K(h, ; p)C (h, p) =
· p + O p · exp
π 
ln ln p
For general odd number q > , whether there exists an asymptotic formula for
q
K(h, ; q)C (h, q)
h=
is still an open problem.
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 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 integer a with (a, p) = , we have the identity
C(a, p) =
–
π  (p – )
χ (a)τ  (χ )L (, χ ),
χ mod p
χ(–)=–
where χ runs through the Dirichlet characters mod p with χ (–) = –, and τ (χ ) =
p–
a
a= χ (a)e( p ) denotes the classical Gauss sum corresponding to χ .
Proof See reference [].
Lemma  Let p be an odd prime and h an integer with (h, p) = . We define C (h, p) as
follows:
C (h, p) =
p– ah
a
a
.
p
p
p
a=
Then we have the identity
⎧   ⎨– π  p– χ mod p χ (h)τ (χ )τ (χχ )L(, χ)L(, χχ ), if p ≡  mod ;
χ(–)=–
C (h, p) =
⎩
,
if p ≡  mod ,
where χ = ( p∗ ) denotes the Legendre symbol.
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Proof From the orthogonality relation for characters modp we have
C (h, p) =
p– p– hb
a
 a χ (ab)
p –  a=
p χ mod p
p
p
b=
p– p–

a
hb
a
.
=
χ (a)
χ (b)
p –  χ mod p a= p
p
p
()
b=
If χ = χ is the principal character modp, then we have
p–
χ (b)
b=
hb
p
=
p– b
p
b=
=
p– b
b=
p
–


= .
()
If χ is an even character mod p (that is, χ is a non-principal character and χ (–) = ), then
we have
p–
χ (b)
b=
hb
p
= χ (h)
p–
χ (b)
b=
b 
–
p 
p–
p–
p–
 

= χ (h)
χ (b)b –
χ (b) = χ (h)
χ (b)b.
p

p
b=
b=
()
b=
Since
p–
χ (b)b =
b=
p–
χ (p – b)(p – b) =
b=
=p
p–
p–
χ (b)(p – b)
b=
χ (b) –
b=
p–
b=
χ (b)b = –
p–
χ (b)b,
b=
so that
p–
χ (b)b = .
b=
From this identity, (), and () we know that if χ is an even character modp, then
p–
χ (b)
b=
hb
p
= .
()
If χ is an odd character mod p and p ≡  mod . Let χ = ( p∗ ) denote the Legendre symbol,
then χχ must be an even character modp, so we have
p– a
a=
p
a
= .
p
χ (a)
()
Wang and Li Journal of Inequalities and Applications 2014, 2014:44
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If χ is an odd character mod p and p ≡  mod , then χχ must be an odd character
modp, so we have
p– a
p
a=
p– a
 a
=
χ (a)a
p
p a= p
χ (a)
=
i
τ (χχ )L(, χχ )
π
()
and
p–
χ (b)
b=
hb
p
χ (h) i
χ (b)b = χ (h)τ (χ )L(, χ).
p
π
p–
=
()
b=
Combining (), (), ()-() we may immediately deduce the identity
C (h, p) =
p– p– hb
a
 a χ (ab)
p –  a=
p χ mod p
p
p
b=
p– p–

a
hb
a
=
χ (a)
χ (b)
p –  χ mod p a= p
p
p
b=
=
⎧   ⎨– π  p– χ mod p χ (h)τ (χ )τ (χχ )L(, χ)L(, χχ ), if p ≡  mod ;
⎩
χ(–)=–
if p ≡  mod .
,
This proves Lemma .
Lemma  Let p >  be an odd prime, then we have the asymptotic formula
 ln p

.
L (, χ ) = p + O exp

ln ln p
χ mod p

χ(–)=–
Proof For any non-principal character χ mod p, applying Abel’s identity (see Theorem .
of []) we have


L (, χ ) =
p
χ (n)d(n)
n=
n
∞
+
p
A(y, χ )
dy,
y
()
where d(n) denotes the divisor function, and A(y, χ ) =
p <n≤y χ (n)d(n).

From [] we know that for any real number y > p , we have the estimate
A(y, χ ) yφ  (p).
χ mod p
χ(–)=–
()
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From () and () we can deduce that
∞
p
 d(n) |A(χ, y)|
L (, χ ) =
χ (n) + O
dy

 n= n χ mod p
y
χ mod p
χ mod p p

χ(–)=–
χ(–)=–
p

p–
=

χ(–)=–
n=
n≡ mod p
d(n) p – 
–
n

p
n=
n≡– mod p
d(n)
+ O()
n
p
d(pl ± )

 ln p

= p + O exp
,
= p+O

l

ln ln p
l=
ln n
where we have used the estimate d(n) exp( (+)
), >  is any fixed real number. This
ln ln n
proves Lemma .
Lemma  Let p >  be a prime with p ≡  mod . Then we have the estimates
p–  ln p
.
=
O
p
·
exp
χ
(a)L(,
χ
)L(,
χχ
)
 ln ln p
a= χ mod p
χ(–)=–
Proof From the method of proving Lemma  we have
χ (a)L(, χ)L(, χχ )
χ mod p
χ(–)=–
p
=

 n=
d|n χ (d)
n
χ (an) – χ (–an) + O()
χ mod p
 ln p
d(a)
.
· p + O exp
=O
a
ln ln p
()
So applying () we may immediately deduce the estimate
p– χ
(a)L(,
χ)L(,
χχ
)
 a= χ mod p
χ(–)=–
p–
p–
d(a)
 ln p
 ln p
.
p +O
= O p · exp
=O
exp
a
ln ln p
ln ln p
a=
a=
This proves Lemma .
3 Proof of the theorem
In this section, we shall use the lemmas to complete the proof of our theorem. For any
prime p > , note that the identities
p–
h=
χ (h)K(h, ; p) =
p– p–
p
hb
b b
= τ (χ )
= τ  (χ )
e
χ (h)e
χ (b)e
p
p
p
b=
h=
b=
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and
p– ah



a
a
= C(h, p) + C (h, p).
+
C (h, p) =
 a=
p
p
p


If p ≡  mod , then from Lemma , Lemma , and the properties of the Gauss sum τ (χ )
we have
p–
K(h, ; p)C (h, p)
h=
=
p–
–
χ (h)K(h, ; p)τ  (χ )L (, χ )
π  (p – ) χ mod p
h=
χ(–)=–
+
p–
–
χ (h)K(h, ; p)τ (χ )τ (χχ )(, χ )(, χχ )
π  (p – ) χ mod p
h=
χ(–)=–
=
+
=
–
π  (p – )
χ mod p
χ(–)=–
–
π  (p – )
τ  (χ )τ (χ )τ (χχ )L(, χ)L(, χχ )
χ mod p
χ(–)=–
p
π  (p – )
–
τ  (χ )τ  (χ )L (, χ )
τ (χ )τ (χχ )L(, χ )L(, χχ )
χ mod p
χ(–)=–
p
L (, χ ).
π  (p – ) χ mod p
()
χ(–)=–
Note that |τ (χ )| =
τ (χ )τ (χχ ) =
√
p and that we have the identity
p– p–
χ (a)χ (b)χ (b)e
a= b=
=
p–
χ (a)
p–
a=
= τ (χ )
b=
p–
a+b
p
ab + b
χ (b)e
p
= τ (χ )
χ (a)χ (a + ),
a=
so from Lemma  we have
τ (χ )τ (χχ )L(, χ)L(, χχ )
χ mod p
χ(–)=–
=
√
p–
χ (a)χ (a + )L(, χ)L(, χχ )
p·
χ mod p a=
χ(–)=–
p–
χ (a)χ (a + )
a=
()
Wang and Li Journal of Inequalities and Applications 2014, 2014:44
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≤
√
Page 7 of 7
p– p·
χ (a)L(, χ)L(, χχ )
a= χ mod p
χ(–)=–
 ln p
= O p · exp
.
ln ln p


()
Combining (), (), and Lemma  we can deduce the asymptotic formula
p–
K(h, ; p)C (h, p) = –
h=

 ln p


 · exp
.
·
p
+
O
p
π 
ln ln p
()
If p ≡  mod , then from Lemma  we know that C (h, p) =  C(h, p), so from Lemma 
and the method of proving () we also have
p–
h=
K(h, ; p)C (h, p) = –
 p
L (, χ )
π  p –  χ mod p
χ(–)=–
 ln p




.
= –  · p + O p · exp
π
ln ln p
()
Now the theorem follows from the asymptotic formulas () and ().
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XW carried out the proofs of the lemmas, XL carried out the part of Introduction, XW and XL carried out the theorem’s
proof. All authors read and approved the final manuscript.
Author details
1
Economy and Finance School, Xi’an International Studies 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 referees for their 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.
Received: 12 December 2013 Accepted: 10 January 2014 Published: 27 Jan 2014
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10.1186/1029-242X-2014-44
Cite this article as: Wang and Li: A hybrid mean value involving a new sum and Kloosterman sums. Journal of
Inequalities and Applications 2014, 2014:44