Guo et al. Advances in Difference Equations 2014, 2014:103
http://www.advancesindifferenceequations.com/content/2014/1/103
RESEARCH
Open Access
On the fourth power mean of the general kth
Kloosterman sums
Xiaoyan Guo1* , Guohua Geng2 and Xiaowei Pan3
*
Correspondence:
[email protected]
1
School of Mathematics, Northwest
University, Xi’an, Shaanxi, P.R. China
Full list of author information is
available at the end of the article
Abstract
Let q > 2 be an integer, and let χ be a Dirichlet character modulo q. For integers m, n
and k, the general kth Kloosterman sum S(m, n, k, χ ; q) is defined by
q
k
k
S(m, n, k, χ ; q) = a=1 χ (a)e( ma q+na ), where denotes the summation over all a
with (a, q) = 1, e(y) = e2π iy , and a is the inverse of a modulo q such that aa ≡ 1 mod q
≤ q. In this paper we further study the fourth power mean
and
1 ≤ a
q
4
m=1 |S(m, n, k, χ ; q)| , and we give some identities.
χ mod q
MSC: 11F20
Keywords: general kth Kloosterman sum; fourth power mean; identity
1 Introduction
Let q >  be an integer, and let χ be a Dirichlet character modulo q. For arbitrary integers
m and n, the general Kloosterman sum S(m, n, χ ; q) is defined by
S(m, n, χ ; q) =
q
a=
ma + na
,
χ (a)e
q
where denotes the summation over all a with (a, q) = , e(y) = eπ iy , and a is the inverse
of a modulo q such that aa ≡  mod q and  ≤ a ≤ q. For q = p a prime, Chowla [] and
Malyshev [] proved an upper bound:
S(m, n, χ ; p) (m, n, p)/ p/+ ,
where (m, n, p) is the great common divisor of m, n and p.
For general integer q > , we do not know how large |S(m, n, χ ; q)| is. However,
|S(m, n, χ ; q)| enjoys good value distribution properties. For fixed integer n with (n, q) = ,
Zhang [] showed the identity
q
S(m, n, χ ; q) = φ  (q)q d(q)
–
χ mod q m=
pα q
pα– –  α – pα–

·
+
,
α +  pα (p – ) (α + )pα
where d(q) is the divisor function, φ(q) is the Euler function, and
uct over all prime divisors p of q with pα | q and pα+ q.
pα q
denotes the prod-
©2014 Guo et al.; 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.
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Page 2 of 7
For integers m, n and k, the general kth Kloosterman sum S(m, n, k, χ ; q) is defined by
S(m, n, k, χ ; q) =
q
a=
k
ma + nak
.
χ (a)e
q
Suppose that (nk, q) = . Liu and Zhang [] proved the identity:
q
S(m, n, k, χ ; q)
χ mod q m=

= φ (q)q


(k, p – )
pα q
α (pα – )
(p – )
+
.
α–+
–
(k, p – )p pα pα (p – )
In this paper we further consider the situation (k, q) > . Our result is a generalization
of [].
Theorem . Let p be an odd prime, n be any integer. Let α and k be positive integers with
d = (k, p – ) and (k, pα– ) = pδ . Then we have
α
p
S m, n, k, χ ; pα 
χ mod pα m=
⎧
d
 α α+δ
  α α+δ– α–δ–
⎪
(p
– ), α–
≤ δ ≤ α – ;
⎪

⎨ dφ (p )p ( – φ(p) ) + d φ (p )p
d

α
α+δ


α
α+δ–
δ+
= dφ (p )p ( – φ(p) ) + d φ (p )p
(p – )
⎪
⎪
⎩ + d φ  (pα )pα+δ– (p – )(α – δ – ),
 ≤ δ ≤ α–
.

From Theorem . we immediately get the following corollary.
Corollary . Let p be an odd prime, n be any integer. Let α and k be positive integers.
Assume that one of the following conditions holds:
() α = ;
() α =  and (k, pα– ) > ;
() α ≥  and (k, pα– ) = pα– .
Then we have
pα
S m, n, k, χ ; pα  = φ  pα pα k, φ pα
–
χ mod pα m=
d
.
φ(p)
Theorem . Let q >  be an odd number, n be an integer with (n, q) = , and let k be a
positive integer. Then we have
q
S(m, n, k, χ ; q)
χ mod q m=
=
pα q
(k,pα– )=pδ
α– ≤δ≤α–

α α+δ
(k, p – )
(k, p – )φ p p
–
φ(p)

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+ (k, p – ) φ pα pα+δ– pα–δ– – 
 
×
pα q
(k,pα– )=pδ
(k, p – )
(k, p – )φ  pα pα+δ  –
φ(p)
≤δ≤ α–

+ (k, p – ) φ  pα pα+δ– pδ+ – 
+ (k, p – ) φ  pα pα+δ– (p – )(α – δ – ) .
From Theorem . we can get the following corollary.
Corollary . Let q >  be an odd number, n be an integer with (n, q) = , and let k be a
positive integer. Assume that one of the following conditions holds:
() q is square-free;
() q = p · · · pr and p · · · pr | k;
α
() q = ri= pi i and q | k ri= pi .
αi ≥
Then we have
q
(k, p – )
S(m, n, k, χ ; q) = ω(q) φ  (q)q k, φ(q)
.
–
φ(p)
pα q
χ mod q m=
2 Some lemmas
To complete the proof of theorems, we need the following lemmas.
Lemma . Let p be an odd prime, and let α and k be positive integers with (k, pα– ) = pδ .
Write d = (k, p – ). Then we have
⎧
d
α δ
 α+δ– α–δ–
⎪
(p
– ), α–
≤ δ ≤ α – ;
⎪ dφ(p )p ( – φ(p) ) + d p

⎨
d
α
δ

α+δ–
δ+
 = dφ(p )p ( – φ(p) ) + d p
(p – )
⎪
⎪
a= b=
⎩ + d pα+δ– (p – )(α – δ – ),
 ≤ δ ≤ α–
.

pα |(ak –)(bk –)
pα
pα
Proof For α = , we get
α
α
p
p
=
a= b=
pα |(ak –)(bk –)
p– p–
a= b=
p|(ak –)(bk –)
d
.
 = d(p – ) – d = dφ(p)  –
φ(p)

Now we assume that α ≥ . It is not hard to show that
α
α
p
p
α
=
p
α
α α
p

β= γ = a=
b=
β+γ ≥α pβ ak – pγ bk –
a= b=
pα |(ak –)(bk –)
α
=
p
α
β=
α
p
α
a= γ =α–β b=
pβ ak –
pγ bk –
α
=
p
α
β=
α
p
a=
b=
pβ ak – pα–β |bk –

(.)
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α
=
α
p
p
α
+
a=
b=
pak – pα |bk –
p
α– β=
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α
p
α
+
a=
b=
pβ ak – pα–β |bk –
α
p
p

a=
b=
pα |ak –
= φ pα – pα– k, φ(p) · k, φ pα + k, φ pα φ pα
+
α–
α–β p
k, φ pβ – pα–β– k, φ pβ+ · pβ k, φ pα–β
β=
α–
=p
φ(p) – k, φ(p) · k, pα– · k, φ(p)
α–
 β– β α–β– + pα– k, φ(p)
p k, p
– k, p · k, p
β=
= dφ pα k, pα–  –
+ d pα–
d
φ(p)
α–
β– β α–β– p k, p
– k, p · k, p
.
(.)
β=
If (k, pα– ) = , by (.) we have
α
α
p
p
α
 = dφ p
a= b=
pα |(ak –)(bk –)
d
+ d φ pα (α – ).
–
φ(p)
(.)
If (k, pα– ) = pα– , then
α
 = dφ pα pα–  –
α
p
p
a= b=
pα |(ak –)(bk –)
d
.
φ(p)
(.)
On the other hand, for α =  and (k, pα– ) > , we have
α
 = dφ pα k, pα–  –
α
p
p
a= b=
pα |(ak –)(bk –)
d
.
φ(p)
Next we suppose that α ≥  and (k, pα– ) = pδ with  ≤ δ ≤ α – . Then
α
α
p
p

a= b=
pα |(ak –)(bk –)
α
= dφ p
α–
k, p
= dφ pα pδ  –
α–
β– β α–β– d
+ d pα–
–
p k, p
– k, p · k, p
φ(p)
β=
α–
α–β– d
+ d φ pα pδ
k, p
.
φ(p)
β=δ+
(.)
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Noting that
α–
α–δ–
k, pα–β– =
k, pu
u=
β=δ+
α–δ–
pu ,
α – δ –  ≤ δ,
α–δ– δ
u
u= p +
u=δ+ p , α – δ –  ≥ δ + 
u=
δ
=
=
⎧ α–δ–
–
⎨p
,
δ≥
⎩
δ≤
p–
pδ+ –
p–
+ (α – δ – )pδ ,
α–
,

α–
.

Therefore
α
α
p
p

a= b=
pα |(ak –)(bk –)
⎧
d
α–
α δ
 α+δ– α–δ–
⎪
(p
– ),
≤ δ ≤ α – ;
⎪

⎨ dφ(p )p ( – φ(p) ) + d p
d
α
δ
= dφ(p )p ( – φ(p) )
⎪
⎪
⎩ + d pα+δ– (pδ+ – ) + d pα+δ– (p – )(α – δ – ),  ≤ δ ≤ α– .

(.)
Now combining (.), (.)-(.) we immediately get
α
α
p
p

a= b=
pα |(ak –)(bk –)
⎧
d
α–
α δ
 α+δ– α–δ–
⎪
(p
– ),
≤ δ ≤ α – ;
⎪

⎨ dφ(p )p ( – φ(p) ) + d p
d
α
δ
= dφ(p )p ( – φ(p) )
⎪
⎪
⎩ + d pα+δ– (pδ+ – ) + d pα+δ– (p – )(α – δ – ),  ≤ δ ≤ α– ,

for α ≥ .
Lemma . Let n, k, k , k be integers with (n, k k ) = (k , k ) = . Then for any character
χ mod k k , there exist integers n and n with (n , k ) = (n , k ) =  such that
n ≡ n k + n k (mod k k ),
and for these integers we have
S(m, n, k, χ ; k k ) = S(mk  , n k , k, χ ; k ) · S(mk  , n k , k, χ ; k ),
where χ = χ χ with χ mod k and χ mod k .
Proof This is Lemma . of [].
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3 Proof of the theorems
First we prove Theorem .. Let p be an odd prime, and let α and k be positive integers.
Assume that n is any integer. We have
k
pα
k
pα
ma + nak mbk + nb
S m, n, k, χ ; pα  =
χ (a)e
χ (b)e –
pα
pα
a=
b=
k k
k
mb (a – ) + nb (ak – )
=
χ (a)e
.
pα
a=
pα
pα
b=
Then from the orthogonality relation for characters and the trigonometric identity we get
α
p
S m, n, k, χ ; pα 
χ mod pα m=
α
α
=
α
p
p
p
χ mod pα m= a=
pα
×
α
α
=φ p
d=
pα
=φ p
k
mdk (ck – ) + nd (ck – )
χ (c)e –
pα
pα
c=
α
=φ p
b=
k k
k
mb (a – ) + nb (ak – )
χ (a)e
pα
pα
pα
a=
b=
d= m=
pα
pα
pα
a=
b=
d= m=
pα
pα
pα
pα
e
pα
b=
e
pα
a=
pα
e
d= m=
k
k
m(ak – )(bk – dk ) + n(ak – )(b – d )
pα
k
k
mdk (ak – )(bk – ) + nd (ak – )(b – )
pα
k
k
mdk (ak – )(bk – ) + nd ak b (ak – )(bk – )
pα
α
p
= φ pα pα
.

a= b=
pα |(ak –)(bk –)
Write d = (k, p – ) and (k, pα– ) = pδ . By Lemma . we immediately have
α
p
S m, n, k, χ ; pα 
χ mod pα m=
⎧
d
 α α+δ
  α α+δ– α–δ–
⎪
(p
– ), α–
≤ δ ≤ α – ;
⎪

⎨ dφ (p )p ( – φ(p) ) + d φ (p )p
d

α
α+δ


α
α+δ–
δ+
= dφ (p )p ( – φ(p) ) + d φ (p )p
(p – )
⎪
⎪
⎩ + d φ  (pα )pα+δ– (p – )(α – δ – ),
 ≤ δ ≤ α–
.

This completes the proof of Theorem ..
Now we prove Theorem .. Let q >  be an odd number, n be an integer with (n, q) = ,
and let k be a positive integer. Write
q=
r
i=
α
pi i
and
m=
r
mi q
α
i=
pi i
.
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By Lemma . and Theorem . we have
q
S(m, n, k, χ ; q)
χ mod q m=
=
r
=
i=
i
i
αi
pα q
(k,pα– )=pδ
α α+δ
(k, p – )
(k, p – )φ p p
–
φ(p)

α– ≤δ≤α–

+ (k, p – ) φ  pα pα+δ– pα–δ– – 
×
i
α
χi mod pi i mi =
i
pi 
S mi , ni qα , k, χi ; pαi i
i
p
=
αi
α
χi mod pi i mi =
i=
r
pi 
q
q
αi S mi qα
,
k,
χ
;
p
,
n
i αi
i i
α
pi pi
p
pα q
(k,pα– )=pδ
α+δ
(k, p – )
(k, p – )φ p p
–
φ(p)

α
≤δ≤ α–

+ (k, p – ) φ  pα pα+δ– pδ+ – 
+ (k, p – ) φ  pα pα+δ– (p – )(α – δ – ) .
This proves Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1
School of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China. 2 School of Information, Northwest University,
Xi’an, Shaanxi, P.R. China. 3 Department of Health Management, Xi’an Medical University, Xi’an, Shaanxi, P.R. China.
Acknowledgements
This work is supported by the P.E.D. (2013JK0561) and N.S.F. (11371291) of P.R. China.
Received: 24 February 2014 Accepted: 16 March 2014 Published: 04 Apr 2014
References
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2. Malyshev, AV: A generalization of Kloosterman sums and their estimates. Vestn. Leningr. Univ. 15, 59-75 (1960)
(Russian)
3. Zhang, W: On the general Kloosterman sum and its fourth power mean. J. Number Theory 104, 156-161 (2004)
4. Liu, HY, Zhang, WP: On the general k-th Kloosterman sums and its fourth power mean. Chin. Ann. Math., Ser. B 25,
97-102 (2004)
10.1186/1687-1847-2014-103
Cite this article as: Guo et al.: On the fourth power mean of the general kth Kloosterman sums. Advances in Difference
Equations 2014, 2014:103