IJMMS 31:12 (2002) 745–750
PII. S0161171202201246
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE FIRST POWER MEAN OF L-FUNCTIONS WITH THE WEIGHT
OF GENERAL KLOOSTERMAN SUMS
ZHANG WENPENG
Received 20 January 2002
The main purpose of this paper is using the estimates for character sums and the analytic
method to study the first power mean of Dirichlet L-functions with the weight of general
Kloosterman sums, and give an interesting asymptotic formula.
2000 Mathematics Subject Classification: 11L05, 11M20.
1. Introduction. Let q ≥ 2 be an integer, χ denotes a Dirichlet character modulo q.
For any integers m and n, we define the general Kloosterman sums S(m, n, χ, q) as
follows:
S(m, n, χ, q) =
q
a=1
χ(a)e
ma + nā
,
q
(1.1)
where ā denotes the inverse of a modulo q and e(y) = e2π iy . This summation is very
important, because it is a generalization of the classical Kloosterman sums. Many authors had studied the properties of S(m, n, χ, q). For instance, Chowla [1] and Malyšev
[3] obtained a sharper upper bound estimation for S(m, n, χ, q). That is,
S(m, n, χ, p) (m, n, p)1/2 p 1/2+ ,
(1.2)
where p is a prime, is any fixed positive number, and (m, n, p) denotes the greatest
common divisor of m, n, and p. But for an arbitrary composite number q, we do not
know how large |S(m, n, χ, q)| is. In fact the value of |S(m, n, χ, q)| is quite irregular
if q is not a prime. The main purpose of this paper is to obtain some good distribution
properties of |S(m, n, χ, q)| in some weight mean value problems. For convenience,
in this paper we always suppose q ≥ 3 be an integer and L(s, χ) denotes the Dirichlet
L-function corresponding to character χ mod q. Then we can use the estimates for
character sums and the analytic method to prove the following main result.
Theorem 1.1. For any integers m and n with (mn, q) = 1, we have the asymptotic
formula
S(m, n, χ, q)2 · L(1, χ) = C · φ2 (q) + O q3/2 · d2 (q) · ln2 q ,
(1.3)
χ≠χ0
where


2
2
2
2
4
2n

1
2
n

C=
1 + 2 2 + 4 4 + · · · + 2n 2n + · · · 
4
·
p
4
·
p
4
·
p
p
(1.4)
746
ZHANG WENPENG
is a constant, χ≠χ0 denotes the summation over all nonprincipal characters modulo q,
p with (p, q) = 1, d(q) is the divisor function,
p denotes the product over all primes
2
.
=
(2n)!/(n!)
φ(q) is the Euler function, and 2n
n
For general integer k ≥ 2, whether there exists an asymptotic formula for
S(m, n, χ, q)2k L(1, χ)
(1.5)
χ≠χ0
is an unsolved problem.
2. Some lemmas. In order to complete the proof of Theorem 1.1, we need the following lemmas.
Lemma 2.1. For any integer q ≥ 3, we have the estimate
d1/2
q/d−1 s=1
d|q
χ(sd + 1)L(1, χ) = O q · d(q) · ln2 q .
(2.1)
χ≠χ0
Proof. Let N = q3/2 , χ be a nonprincipal character mod q and A(χ, y) =
N<n≤y χ(n). Then by Abel identity and Pólya-Vinogradov inequality, we have
χ(n) +∞ A(χ, y)
+
dy
n
y2
N
n≤N
χ(n)
ln q
+O
.
=
n
q
n≤N
(2.2)
χ(n) ln q
L(1, χ) = +O
.
n q
n≤N
(2.3)
L(1, χ) =
So that
On the other hand, let r (n) be a multiplicative function defining by
r pα =
2α
α
4α
,
r (1) = 1,
(2.4)
where p is a prime and α is any positive integer. For the function r (n), it is easy to
prove that
n
= 1,
d
d|n
χ(nm)r (m)r (n)
χ(n)r (n) 2
=
n
mn
n≤N
m≤N n≤N
r (d) · r
=
χ(n)
+
n
n≤N
N<n≤N 2
χ(n)r (n, N)
,
n
(2.5)
(2.6)
ON THE FIRST POWER MEAN OF L-FUNCTIONS . . .
747
where
r (n, N) =
r (d) · r
d|n
d, n/d≤N
n
.
d
(2.7)
Note that the triangle inequalities
|a| − |b| ≤ |a ± b|,
|a ± b| ≤ |a| + |b|,
(2.8)
from (2.3), (2.6), Cauchy inequality, and the orthogonality relationship for character
sums

φ(q),
χ(n) =
0,
χ mod q

q
φ(q),
χ(a) =
0,
a=1
if n ≡ 1 mod q;
otherwise,
if χ = χ0 ;
otherwise,
we have
d1/2
q/d−1 χ(sd + 1)L(1, χ)
s=1
d|q
=
1/2
d
d|q
χ≠χ0
q/d−1 χ(n) + O ln q χ(sd + 1)
n χ≠χ0
n≤N
s=1
χ(n) + O q · d(q) · ln q
χ(sd + 1)
n
χ
n≤N
s=1
d|q


q/d−1 χ(n) χ(n)r (n) 2 
1/2

−
d
χ(sd + 1) n
n
χ
n≤N
n≤N
s=1
d|q
2 q/d−1 χ(n)r (n) χ(sd + 1)
+ q · d(q) · ln q
+ d1/2
n
χ
n≤N
s=1
d|q
 2 1/2
q  χ(n) χ(n)r (n, N) 
1/2 √
−
d ·
·
n
n
d
χ
2
n≤N
d|q
=
d1/2
q/d−1 n≤N
+ φ(q)
d1/2
s=1
d|q


q · d(q) ·  χ
+ φ(q)
d|q
q/d−1
m≤N n≤N
(sd+1)m≡n mod q
N<n≤N 2
2 1/2
χ(n)r (n, N) 
+ q · d(q) · ln q
n
r (m)r (n)
mn
m≤N n≤N
d1/2
m≡n mod d
m≠n
r (m)r (n)
+ q · d(q) · ln q
mn
(2.9)
748
ZHANG WENPENG
1/2

 
q3/2 · d(q) · 

N<m≤N 2 N<n≤N 2
r (m, N) · r (n, N) 



mn
+ q · d(q) · ln2 q
m≡n mod q
2
q · d(q) · ln q.
(2.10)
This proves Lemma 2.1.
Lemma 2.2. For any integer q ≥ 3, we have the asymptotic formula
L(1, χ) = C · φ(q) + O q1/2 · ln q ,
(2.11)
χ≠χ0
where
2
2
2


2
4
2n
∞
r 2 (n)
1
2
n
1 +
=
+
+ · · · + 2n 2n + · · · 
C=
n2
42 · p 2 44 · p 4
4 ·p
p
n=1
(2.12)
is an absolute constant.
Proof. Let N = q3/2 . Then from (2.3), (2.6), the orthogonality relationship for character sums, and the method of proving Lemma 2.1, we have
χ(n) L(1, χ) =
+ O(ln q)
n χ≠χ0
χ≠χ0 n≤N
2
χ(n)r (n) + O(ln q)
= n
χ
n≤N


χ(n)r (n) 2
χ(n) 
−
+ 
n n
χ
n≤N
n≤N
r (m) · r (n)
+ O(ln q)
= φ(q) ·
mn
m≤N n≤N
m≡n mod q

χ(n) · r (n, N) 
+ O n
χ N<n≤N 2

= φ(q) ·

∞
n=1
(n,q)=1
r 2 (n)
+ O ln2 q
2
n
  1/2  + O q
χ
= φ(q) ·
∞
n=1
(n,q)=1
N<n≤N 2

2 1/2
χ(n) · r (n, N)  

n
r 2 (n)
+ O q1/2 · ln q
n2
= C · φ(q) + O q1/2 · ln q .
This proves Lemma 2.2.
(2.13)
ON THE FIRST POWER MEAN OF L-FUNCTIONS . . .
749
Lemma 2.3. Let m, n, and q be integers with q ≥ 3. Then we have the estimates
S(m, n, q) =
where
a
q
ma + nā
e
(m, n, q)1/2 q1/2 d(q),
q
a=1
(2.14)
denotes the summation over all a such that (a, q) = 1.
Proof. See [2].
3. Proof of Theorem 1.1. In this section, we will complete the proof of Theorem 1.1.
First we have
L(1, χ) · S(m, n, χ, q)2
χ≠χ0
=
q
q
e
r =1 s=1
(r − s)m + (r̄ − s̄)n
q
χ(r s̄)L(1, χ)
χ≠χ0
q
q
r (1 − s̄)m + r̄ (1 − s)n e
χ(s)L(1, χ)
=
q
χ≠χ0
r =1 s=1
q
q
r (1 − s̄)m + r̄ (1 − s)n L(1, χ) +
e
χ(s)L(1, χ).
= φ(q)
q
χ≠χ0
χ≠χ0
s=2 r =1
(3.1)
From (3.1) and Lemmas 2.2 and 2.3 we get
L(1, χ) · S(m, n, χ, q)2
χ≠χ0
= C · φ2 (q) + O q3/2 · ln q


q
1/2 1/2
L(1, χ).
+ O q d(q)
χ(s)
(s − 1)m, (s̄ − 1)n, q
χ≠χ0
s=2
(3.2)
Note that (m, q) = (n, q) = (s, q) = 1, and (1 − s, q) = (ss̄ − s, q) = (s(s̄ − 1), q) = (s̄ −
1, q), so we have (s − 1, s̄ − 1, q) = (s − 1, q). Thus from (3.2) and Lemma 2.1 we obtain
the asymptotic formula
L(1, χ) · S(m, n, χ, q)2
χ≠χ0
= C · φ2 (q) + O q3/2 · ln q


q
1/2
1/2
L(1, χ)
(s − 1, q) χ(s)
+ O q d(q)
χ≠χ0
s=2
= C · φ2 (q) + O q3/2 · ln q


q/d−1
1/2
1/2 

d χ(td + 1) L(1, χ) + O q d(q)
χ≠χ0
d|q t=1
= C · φ2 (q) + O q3/2 · d2 (q) · ln2 q .
This completes the proof of the theorem.
(3.3)
750
ZHANG WENPENG
Acknowledgment. This work is supported by the National Natural Science Foundation of China (NSFC) and the Shaanxi Province Natural Science Foundation of China
(PNSF).
References
[1]
[2]
[3]
S. Chowla, On Kloosterman’s sum, Norske Vid. Selsk. Forh. (Trondheim) 40 (1967), 70–72.
T. Estermann, On Kloosterman’s sum, Mathematika 8 (1961), 83–86.
A. V. Malyšev, A generalization of Kloosterman sums and their estimates, Vestnik
Leningrad. Univ. 15 (1960), no. 13, 59–75 (Russian).
Zhang Wenpeng: Research Center for Basic Science, Xi’an Jiaotong University, Xi’an,
Shaanxi, China
E-mail address: wpzhang@nwu.edu.cn