IJMMS 32:1 (2002) 47–55
PII. S0161171202006877
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON A SUM ANALOGOUS TO DEDEKIND SUM AND ITS MEAN
SQUARE VALUE FORMULA
ZHANG WENPENG
Received 1 March 2001
The main purpose of this paper is using the mean value theorem of Dirichlet L-functions
to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.
2000 Mathematics Subject Classification: 11N37, 11M20.
1. Introduction. For a positive integer k and an arbitrary integer h, the classical
Dedekind sum S(h, k) is defined by
S(h, k) =
k ah
a
a=1
k
k
,
(1.1)
where

1

x − [x] − ,
2
(x) =

0,
if x is not an integer;
(1.2)
if x is an integer.
The various properties of S(h, k) were investigated by many authors. For example,
Carlitz [1] obtained a reciprocity theorem of S(h, k). Conrey et al. [2] studied the mean
value distribution of S(h, k), and first proved the following important asymptotic
formula:
k
2m
S(h, k)
2m = fm (k) k
+ O k9/5 + k2m−1+1/(m+1) log3 k ,
12
h=1
where
h
(1.3)
denotes the summation over all h such that (k, h) = 1, and
∞
fm (n)
ζ 2 (2m) ζ(s + 4m − 1)
=2
·
ζ(s).
s
n
ζ(4m)
ζ 2 (s + 2m)
n=1
(1.4)
The author [4] improved the error term of (1.3) for m = 1. In October, 2000, Todd
Cochrane (personal communication) introduced a sum analogous to Dedekind sum as
follows:
C(h, k) =
k a
ah
,
k
k
a=1
(1.5)
48
ZHANG WENPENG
where a defined by equation aa ≡ 1 mod k. Then he suggested to study the arithmetical properties and mean value distribution properties of C(h, k). About first problem,
we have not made any progress at present. But for the second problem, we use the
estimates for character sums and the mean value theorem of Dirichlet L-functions to
prove the following main conclusion.
Theorem 1.1. Let k be any integer with k > 2. Then we have the asymptotic formula
(p + 1)2 / p 2 + 1 + 1/p 3α
5
4 ln k
2
φ (k)
,
C (h, k) =
+
O
k
exp
144
1 + 1/p + 1/p 2
ln ln k
h=1
p α k
k
2
where exp(y) = ey , φ(k) is Euler function,
divisors of k with p α |k and p α+1 k.
p α k
(1.6)
denotes the product over all prime
It seems that our methods are useless for mean value
k
h=1 C
2m
(h, k) with m > 1.
2. Some lemmas. To complete the proof of Theorem 1.1, we need the following
lemmas.
Lemma 2.1. Let integer k ≥ 3 and (h, k) = 1. Then
S(h, k) =
1 d2
π 2 k d|k φ(d)
2
χ(h)
L(1, χ)
,
(2.1)
χ mod d
χ(−1)=−1
where χ denotes a Dirichlet character modulo d with χ(−1) = −1, and L(s, χ) denotes
the Dirichlet L-function corresponding to χ.
Proof. See [3].
Lemma 2.2. Let k be any integer with k > 2. Then we have the identity
k
C (h, k) =
µ(d)
S a,
S(a, k),
d
a=1
h=1
d|k
k
2
k
(2.2)
where µ(d) is Möbius function.
Proof. From the definition of S(h, k) and C(h, k), we have
k
h=1
2
C (h, k) =
k
h=1


k ah
a
a=1
k
k
2

k bh
a
b
ah
=
.
k
k
k
k
a=1 b=1
h=1
k
k
(2.3)
ON A SUM ANALOGOUS TO DEDEKIND SUM AND ITS MEAN SQUARE . . .
49
Since (ab, k) = 1, so if h round through a complete residue system modulo k, then
bh also round through a complete residue system modulo k. Therefore, note that the
identities
k
b=1
=
µ(d)
d|k
k/d
,
S(a, k) = S(a, k),
(2.4)
b=1
we have
k
k h
b
abh
k
k
k
k
a=1 b=1
h=1
k k
a
b
=
S ab, k
k
k
a=1 b=1
C 2 (h, k) =
h=1
=
k k
a
k k
a
b
S(ab, k)
k
k
a=1 b=1
k k/d ab
b
=
µ(d)
S(a, k)
k/d
k/d
a=1 b=1
d|k
=
k
k
S a,
S(a, k)
d
a=1
µ(d)
d|k
=
(2.5)
k
k
S a,
S(a, k).
d
a=1
µ(d)
d|k
This proves Lemma 2.2.
0
Lemma 2.3. Let u and v be integers with (u, v) = d ≥ 2, χu
the principal character
0
mod u, and χv the principal character mod v. Then we have the asymptotic formula
L 1, χχ 0 2 L 1, χχ 0 2
u
χ mod d
χ(−1)=−1
5π 4
φ(d)
=
144
v
2
p 2 − 1 /p 2 p 2 + 1
φ(d)
3 ln m
+
O
exp
,
2
2
d
ln ln m
p|d p / p − 1
(2.6)
p|uv
where p|n denotes the product over all prime divisors of n, (u, v)denotes the greatest
common divisor of u and v, and m = max(u, v).
0
Proof. Let r (n) = t|n χu
(t)χv0 (n/t), χ an odd character mod d. Then for parameter N ≥ d, applying Abel’s identity we have
∞
χ(n)r (n)
0
L 1, χχv0 =
L 1, χχu
n
n=1
=
1≤n≤N
χ(n)r (n)
+
n
∞
N
A(y, χ)
dy,
y2
(2.7)
50
ZHANG WENPENG
where A(y, χ) =
N<n≤y
χ(n)r (n). Note that the partition identities
A(y, χ) =
√
n≤ y
+
−
−
√
m≤ y
√
n≤ N
√
m≤ N
χ(m)χv0 (m)
m≤N/n
χ(m)χv0 (m)
√
n≤ y
+
0
χ(n)χu
(n)
n≤y/m

χ(m)χv0 (m)
0
χ(n)χu
(n)
−
χ(m)χv0 (m)
m≤y/n

0
χ(n)χu
(n)
√
n≤ N
0
χ(n)χu
(n)
n≤N/m

0
χ(n)χu
(n)

0
χ(n)χu
(n)
√
n≤ y
√
n≤ N
(2.8)

χ(n)χv0 (n)

χ(n)χv0 (n).
Applying Cauchy inequality and the estimates for character sums
χ≠χ0
N≤n≤M
2
χ(n)
=
χ≠χ0
N≤n≤M≤N+d
= φ(d)
N≤n≤M≤N+d
≤
2
χ(n)
χ0 (n) − N≤n≤M≤N+d
2
χ0 (n)
(2.9)
φ2 (d)
4
and note that the identities
N≤n≤M
0
χ(n)χu
(n) =
µ(d)χ(d)
d|u
χ(n),
N/d≤n≤M/d
µ(d)
=
1 + µ(p)
= 2ω(u) ,
(2.10)
p|u
d|u
where ω(u) denotes the number of all different prime divisors of u. We have
A(y, χ)
2 y
√
n≤ y χ mod d
χ(−1)=−1
χ mod d
χ(−1)=−1
+ y
+
2
0
χ(m)χu
(m)
m≤y/n
√
m≤ y χ mod d
χ(−1)=−1
χ mod d
χ(−1)=−1
2
χ(n)χv0 (n)
n≤y/m
(2.11)
2 2
0
0
χ(n)χu (n)
× χ(n)χv (n)
√
√
n≤ y
yφ2 (d)2ω(u)+ω(v) .
n≤ y
51
ON A SUM ANALOGOUS TO DEDEKIND SUM AND ITS MEAN SQUARE . . .
Thus from (2.11) and Cauchy inequality we get
χ mod d
χ(−1)=−1
2 ∞ A(y, χ)
∞ ∞
1
dy ≤
2
2
2
N
y
N N y z








 ∞ 1 




 N y2 
χ mod d
χ(−1)=−1
2
1/2
χ mod d
χ(−1)=−1

A(y, χ)
· A(z, χ)

 dydz


A(y, χ)
2 




dy 

φ2 (d) ω(u)+ω(v)
.
2
N
(2.12)
Note that for (ab, d) = 1, from the orthogonality relation for character sums modulo
d we have

1


φ(d),
if a ≡ b mod d;




2
χ(a)χ(b) = − 1 φ(d), if a ≡ −b mod d;


2

χ mod d



χ(−1)=−1
0,
otherwise.
(2.13)
So that
1≤n≤N
χ mod d
χ(−1)=−1
=
2
χ(n)r (n) n
r (a)r (b) 1
r (a)r (b)
1
φ(d)
− φ(d)
2
ab
2
ab
1≤a,b≤N
1≤a,b≤N
(ab,d)=1
a≡b(d)
(ab,d)=1
a≡−b(d)


N [N/d]
τ(b)τ(d + b)
r (a)
2
1


= φ(d)
+ O φ(d)
2
a2
(d + b)b
1≤a≤N
b=1
=1
(a,d)=1
+ O φ(d)
+ O φ(d)
d−1
τ(a)τ(d − a)
a(d − a)
a=1
1≤a≤N (1+a/d)≤≤N/d
τ(a)τ(d − a)
a(d − a)
∞
r (n)
2
φ(d)
ln N
1
+O
exp
,
= φ(d)
2
n2
d
ln ln N
n=1
(n,d)=1
(2.14)
52
ZHANG WENPENG
where τ(n) is the divisor function and r (n) ≤ τ(n) exp((1 + ) ln 2 ln n/ln ln n),
1≤n≤N
χ mod d
χ(−1)=−1
χ(n)r (n)
n
∞
N
A(y, χ)
dy
y2

(ln N)2
∞
N
1 


y2 

χ mod d
χ(−1)=−1

A(y, χ)

 dy

(2.15)
φ3/2 (d)(ln N)2 N −1/2 2ω(u)+ω(v) .
Taking parameter N = d3 and note the identity
r (n)
2
∞
n=1
(n,d)=1
n2
=
5π 4
72
p|uv
2
2
p − 1 /p 2 p 2 + 1
.
2
2
p|d p / p − 1
(2.16)
From (2.11), (2.12), (2.14), and (2.15) we obtain
L 1, χχ 0 2 L 1, χχ 0 2
u
χ mod d
χ(−1)=−1
5π 4
φ(d)
=
144
v
(2.17)
2
2
p − 1 /p 2 p 2 + 1
φ(d)
3 ln m
+O
exp
.
2
2
d
ln ln m
p|d p / p − 1
p|uv
This proves Lemma 2.3.
Lemma 2.4. Let p be a prime, and let α, β be nonnegative integers with β ≥ α. Then
we have the identity
d1 |p β d2 |p α
d2 d2
1 2 φ(d)
φ d1 φ d2
2
2
p1 − 1 /p12 p12 + 1
2
2
p1 |d p1 / p1 − 1
p1 |d1 d2
2
2
2
p − 1 p 2α p β − p α
1 + 1/p − 1/p 3α+1
,
+
= p 3α
1 + 1/p + 1/p 2
(p − 1)2 p 2 + 1
where d = (d1 , d2 ) denotes the greatest common divisors of d1 and d2 .
Proof. Note that d = (d1 , d2 ), we have
d1 |p α d2 |p α
d2 d2
1 2 φ(d)
φ d1 φ d2
α α
2
2
p1 − 1 /p12 p12 + 1
2
2
p1 |d p1 / p1 − 1
p1 |d1 d2
p 2u+2v
φ pu , pv
=
u φ pv
φ
p
u=0 v=0
2
2
p1 − 1 /p12 p12 + 1
2
2
p1 |(p u ,p v ) p1 / p1 − 1
p1 |p u+v
(2.18)
ON A SUM ANALOGOUS TO DEDEKIND SUM AND ITS MEAN SQUARE . . .
= 1+2
+2
53
2
2
2
3
α
p −1
p −1
p 2β
p 4β
+
φ pβ p2 p2 + 1
φ pβ p4 p2 + 1
β=1
β=1
α
3
α
p 2β p 2γ p 2 − 1
φ pγ p4 p2 + 1
β=1 γ=β+1
α−1
2
3
3α
α
p − 1 p2 − 1
p − 1 p2 − 1
+
(p − 1)2 p 2 + 1
p 3 − 1 (p − 1) p 2 + 1
2
3
α−1
p 3β p α−β − 1
p −1
+2
2
p2 p2 + 1
p −1
β=1
= 1+2
α
3α
2
3
p − 1 p2 − 1
p − 1 p2 − 1
= 1+2
+ 3
(p − 1)2 p 2 + 1
p − 1 (p − 1) p 2 + 1
2
3
3α−2
− pα p2 − 1
p p 3α−3 − 1 p 2 − 1
p
−
2
+2
(p − 1)2 p 2 + 1
p 3 − 1 (p − 1)2 p 2 + 1
1 2
1 2
1 −1
1
p 4α
1 − 2 − 3α+1 1 −
= α 1 − 3
φ p
p
p
p
p
2
−1 1
1
1
1
= p 3α 1 + + 2
1+
− 3α+1 ,
p p
p
p
2
β
α
p1 |pu+v p12 − 1 /p12 p12 + 1
p 2u+2v
φ pu , pv
2
2
φ pu φ pv
p1 |(p u ,p v ) p1 / p1 − 1
u=0 v=α+1
(2.19)
(2.20)
3
β−α−1 j+α+2 2
α
p j+α+2 p 2 − 1 2
p
p −1
2i
+
p
p − 1 p2 p2 + 1
p − 1 p4 p2 + 1
j=0
i=1
j=0
β−α−1
=
2
2
2
3
p −1
p −1
p 2 p 2α − 1 p α+2 p β−α − 1
p α+2 p β−α − 1
+
=
(p − 1)2
p2 p2 + 1
p2 − 1
(p − 1)2
p4 p2 + 1
2
2
p − 1 p 2α p β − p α
=
.
(2.21)
(p − 1)2 p 2 + 1
Now combining (2.19) and (2.20) we have
2
p1 |d1 d2 p12 − 1 /p12 p12 + 1
d2 d2
1 2 φ d1 , d2
2
2
φ d1 φ d2
p1 |(d1 ,d2 ) p1 / p1 − 1
β
d |p α
1
d2 |p
α α
p 2u+2v
φ pu , pv
=
u φ pv
φ
p
u=0 v=0
+
α
β
2
2
p1 − 1 /p12 p12 + 1
2
2
p1 |(p u ,p v ) p1 / p1 − 1
p1 |p u+v
2
2
p1 − 1 /p12 p12 + 1
2
2
p1 |(p u ,p v ) p1 / p1 − 1
2
2
p − 1 p 2α p β − p α
.
+
(p − 1)2 p 2 + 1
p 2u+2v
φ pu , pv
u φ pv
φ
p
u=0 v=α+1
= p 3α
(1 + 1/p)2 − 1/p 3α+1
1 + 1/p + 1/p 2
p1 |p u+v
(2.22)
This proves Lemma 2.4.
54
ZHANG WENPENG
3. Proof of the theorem. In this section, we complete the proof of Theorem 1.1.
Let k be an integer with k ≥ 3. Then applying Lemmas 2.1 and 2.2 we have
k
k
S(a, k)
C (h, k) =
µ(d)
S a,
d
a=1
h=1
d|k

k
u2
 d
µ(d)
=
 2
π k u|k/d φ(u)
a=1
d|k
k
2

 1 v2
× 2
π k v|k φ(v)
1
=
k2 π 4
µ(d)d
d|k
×

2 
χ(a)
L(1, χ)

χ mod u
χ(−1)=−1

2 
χ(a)
L(1, χ)

(3.1)
χ mod v
χ(−1)=−1
u2 v 2
φ(u)φ(v)
u|k/d v|k
k
2 2
χ1 (a)χ2 (a)
L 1, χ1 L 1, χ2 .
χ1 mod u
χ2 mod v a=1
χ1 (−1)=−1 χ2 (−1)=−1
For each χ1 mod u, it is clear that there exists one and only one k1 |u with a unique
0
0
primitive character χk11 mod k1 such that χ1 = χk11 χu
, here χu
denotes the principal
character mod u. Similarly, we also have χ2 = χk22 χv0 , here k2 |v and χk22 is a primitive
character mod k2 . Note that u|k and v|k, from the orthogonality of characters we have
k
k
1
χk1 (a)χk0 (a) χk22 (a)χk0 (a)
χ1 (a)χ2 (a) =
a=1
a=1

φ(k), if k1 = k2 , χ 1 = χ 2 ;
k1
k2
=
0,
otherwise.
(3.2)
Let d1 = (u, v). If k1 = k2 and χk11 = χk22 , then χk11 χd0 1 is also a character mod d1 . So
from (3.1), (3.2), and Lemma 2.3 we have
k
C 2 (h, k) =
h=1
=
u2 v 2
φ(k) L 1, χχ 0 2 L 1, χχ 0 2
µ(d)d
u
v
2
4
k π d|k
φ(u)φ(v) χ mod(u,v)
u|k/d v|k
φ(k) µ(d)du2 v 2
k2 π 4 d|k u|k/d v|k φ(u)φ(v)
χ(−1)=−1
5π 4 φ (u, v)
144
3 ln k
+ O exp
ln ln k
=
5φ(k) µ(d)du2 v 2 φ (u, v)
144k2 d|k u|k/d v|k φ(u)φ(v)
+ O k exp
4 ln k
ln ln k
2 2 2 2 p −1 /p p +1
2
2
p|(u,v) p / p −1
p|uv
2
2
p − 1 /p 2 p 2 + 1
2
2
p|(u,v) p / p − 1
p|uv
.
(3.3)
55
ON A SUM ANALOGOUS TO DEDEKIND SUM AND ITS MEAN SQUARE . . .
Since φ(n) and µ(n) are multiplicative functions, so from the multiplicative properties of these functions, (3.3) and Lemma 2.4 and note that the identities (for any
multiplicative functions f (u) and g(v))
µ(d)d
d|k
p
3α
f (u)g(v) =
p α k
u|k/d v|k
f (u)g(v) − p
u|p α v|p α
f (u)g(v) ,
u|p α−1 v|p α
2
2
2
2
1 + 1/p − 1/p 3α+1
− 1/p 3α−2
p − 1 p 2α−2 p − p α−1
3α−3 1 + 1/p
−p p
+
1+1/p+1/p 2
1 + 1/p + 1/p 2
(p − 1)2 p 2 + 1
=
1
p 3α (1 − 1/p) (p + 1)2
+
,
1 + 1/p + 1/p 2 p 2 + 1
p 3α
(3.4)
we have
k
C 2 (h, k) =
h=1
d|k
=
k
k
S(a, k)
S a,
d
a=1
µ(d)
5 φ(k) 144 k2 pα k
µ(d)d
u|p α /d v|p α
d|p α
uv
φ (u, v)
φ(u)φ(v)
2
2
p1 − 1 /p12 p12 + 1
2
2
p1 |(u,v) p1 / p1 − 1
p1 |uv
×
4 ln k
+ O k exp
ln ln k
=
(3.5)
(p + 1)2 / p 2 + 1 + 1/p 3α
5
4 ln k
φ2 (k)
.
+
O
k
exp
144
1 + 1/p + 1/p 2
ln ln k
p α ||k
This completes the proof of Theorem 1.1.
Acknowledgment. This work was supported by the National Natural Science
Foundation of China (NSFC) and the Shaanxi Province Natural Science Foundation of
China (PNSF).
References
[1]
[2]
[3]
[4]
L. Carlitz, A reciprocity theorem of Dedekind sums, Pacific J. Math. 3 (1953), 523–527.
J. B. Conrey, E. Fransen, R. Klein, and C. Scott, Mean values of Dedekind sums, J. Number
Theory 56 (1996), no. 2, 214–226.
W. Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordeaux 8 (1996),
no. 2, 429–442.
, A note on the mean square value of the Dedekind sums, Acta Math. Hungar. 86
(2000), no. 4, 275–289.
Zhang Wenpeng: Research Center for Basic Science, Xi’an Jiaotong University, Xi’an,
Shaanxi, China