THE RAMANUJAN JOURNAL, 5, 227–236, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. 

On the Mean Value of the Square of a Generalized Dedekind Sum∗ XIE MIN AND ZHANG WENPENG Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China Received April 13, 2000; Accepted August 16, 2001

Abstract. The main purpose of this paper is to use the mean value theorem of the Dirichlet L-functions to study the distribution property of a generalized Dedekind sum, and give a sharper mean square value formula. Key words:

the general Dedekind sums, mean value, asymptotic formula

2000 Mathematics Subject Classification:

1.

Primary—11N37

Introduction

Let q be a positive integer, and let the integers h and n be arbitrary. The general Dedekind sum S(h, n, q) is defined by S(h, n, q) =

q  a=1

    a ¯ ah Bn B¯ n q q

where B¯ n (x) =



Bn (x − [x]), if x is not an integer; 0, if x is an integer.

Bn (x) is the Bernoulli polynomial, B¯ n (x) defined on the interval 0 < x ≤ 1 is the n-th Bernoulli periodic function. Some arithmetical properties of S(h, n, q) can be found in Apostol [2] and Carlitz [3]. For n = 1, S(h, 1, q) = S(h, q) is the classical Dedekind sum. It plays such a prominent role in the study of the modular forms theory that it has attracted many experts in number theory (see references [4–7]). In this paper, we use the estimates for the character sums and the mean value theorem of Dirichlet L-functions to study the distribution property of the general Dedekind sum S(h, n, q), and give a mean square value formula. That is, we prove the following asymptotic formulas: ∗ This

work is supported by the National Natural Science Foundation of P.R. China.

228

MIN AND WENPENG

Theorem.

Let k be any integer with k ≥ 3.

i) If n is an odd number, then k 

|S(h, n, k)|2 =

h=1

 × 1−

1 p (4n−1)α

 ( p 2n − 1)2 2(n!)4 ζ 4 (2n) kφ(k) 2n−1 4n 4 π ζ (4n) ( p − 1)( p 4n−1 − 1) p α k     p( p 2n−1 − 1)2 4 ln k 2−n + O k exp ; ( p 2n − 1)2 ln ln k

ii) If n is an even number, then k 

|S(h, n, k)|2

h=1

   ( p 2n − 1)2 p( p 2n−1 − 1)2 1 2(n!)4 ζ 4 (2n) kφ(k) 1 − 42n−1 π 4n ζ (4n) ( p − 1)( p 4n−1 − 1) p (4n−1)α ( p 2n − 1)2 p α k  −1    2(n!)4 ζ 4 (n)  1 1 1 2 − 2n−1 4n k 1 − 2n−1 1 − (2n−1)α 1− n 4 π p p p p α k    (n!)4 ζ 4 (n) 4 ln k + 2(n−1) 4n φ(k) + O k 2−n exp , 4 π ln ln k

where h denotes the summation over all h such that (k, h) = 1, pα k denotes the product over all prime divisors of k with p α | k and p α+1 | k, exp(y) = e y , and φ(k) is the Euler function. =

2.

Some lemmas

To complete the proof of the theorem, we need the following lemmas. Lemma 1. identities

Let q ≥ 3 be an integer. Then for any integer h with (h, q) = 1, we have the

i) if n is an odd number, S(h, n, q) =

 d 2n (n!)2 4n−1 q 2n−1 π 2n d|q φ(d)



χ (h)|L(n, χ )|2 ;

χ mod d χ (−1)=−1

ii) if n is an even number, S(h, n, q) =

(n!)2 4n−1 q 2n−1 π 2n

 d 2n φ(d) d|q

 χ mod d χ (−1)=1

χ (h)|L(n, χ )|2 −

(n!)2 4n−1 π 2n

ζ 2 (n),

where χ denotes a Dirichlet character modulo d, L(n, χ ) denotes the Dirichlet L-function corresponding to χ , φ(d) denotes the Euler function, and ζ (s) is the Riemann zeta-function.

229

MEAN VALUE OF THE SQUARE OF A GENERALIZED DEDEKIND SUM

✷

Proof: (See reference [8]).

Lemma 2. Let u and v be integers with (u, v) = d ≥ 2, χu0 be the principal character mod u, χv0 be the principal character mod v. Then we have the asymptotic formula 

i)



 

  L s, χ χ 0 2  L s, χ χ 0 2 v

u

χ mod d χ(−1)=−1

ζ 4 (2s) = φ(d) 2ζ (4s)



( p 2s −1)2 p|uv p 2s ( p 2s +1)

p 2s p|d p 2s −1



  φ(d) 3 ln m exp , ds ln ln m



  φ(d) 3 ln m exp , ds ln ln m

+O

 

 

  L s, χ χ 0 2  L s, χ χ 0 2

ii)

v

u

χ mod d χ(−1)=1

ζ 4 (2s) = φ(d) 2ζ (4s)



( p 2s −1)2 p|uv p 2s ( p 2s +1)

p 2s p|d p 2s −1

+O

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). Proof: i) Let r (n) = t|n χu0 (t)χv0 (n/t) and let χ be an odd character mod d. Then for parameter N ≥ d, we apply Abel’s identity to deduce that ∞



 χ (n)r (n) L s, χ χu0 L s, χ χv0 = ns n=1  ∞  χ (n)r (n) A(y, χ ) = + s dy, s n y s+1 N 1≤n≤N



where A(y, χ ) = A(y, χ ) =

N
 √

−

√

χ (n)χu0 (n)

n≤ N

− +

 √





n≤ y

 √

n≤ N



χ (m)χv0 (m) +

m≤y/n





χ (n)r (n). Note the partition identity

χ (n)χu0 (n)

n≤ y

(1)

√

m≤ y



m≤N/n



χ (m)χv0 (m)

√

−

√



χ (n)χv0 (n)

n≤ y





m≤ N



χ (n)χu0 (n) χ (n)χu0 (n)

χ (m)χv0 (m)

 √

n≤ N

 χ (n)χv0 (n) .



χ (n)χu0 (n)

n≤y/m

χ (m)χv0 (m)



n≤N/m

χ (n)χu0 (n)

230

MIN AND WENPENG

We apply the Cauchy inequality and estimates for character sums to deduce 2 2             χ(n) = χ (n)       χ =χ0 N ≤n≤M χ =χ0 N ≤n≤M≤N +d  2     φ 2 (d)   = φ(d) χ0 (n) −  χ0 (n) ≤ .  N ≤n≤M≤N +d  4 N ≤n≤M≤N +d Note that the identities 



χ (n)χu0 (n) =

N ≤n≤M

and

µ(d)χ (d)

d|u



|µ(d)| =





χ (n)

N/d≤n≤M/d

(1 + |µ( p)|) = 2ω(u) ,

p|u

d|u

where ω(u) denotes the number of all different prime divisors of u. We then have  2   √      2 0 |A(y, χ )|  y χ (m)χu (m)    √ m≤y/n χ mod d n≤ y χ mod d χ(−1)=−1

χ (−1)=−1

+

+

√

y





√ m≤ y χ mod d χ (−1)=−1

 χ mod d χ(−1)=−1

 2      0 χ (n)χv (n)   n≤y/m 

 2  2           0 0 χ (n)χu (n) ·  χ (n)χv (n)   n≤√ y   n≤√ y 

 yφ 2 (d)2ω(u)+ω(v) .

(2)

Thus from (1) and the Cauchy inequality we get    ∞ A(y, χ ) 2 s dy   y s+1 N χ mod d χ(−1)=−1

 ≤ s2

∞





N

∞ N

   s2  



1   y s+1 z s+1

  χ mod d χ (−1)=−1

 ∞ N

y

1  s+1 

 χ mod d χ(−1)=−1

 |A(y, χ )| · |A(z, χ )| dy dz 12

2

 φ 2 (d) ω(u)+ω(v)  |A(y, χ )|2  dy   2 .  N 2s−1

(3)

MEAN VALUE OF THE SQUARE OF A GENERALIZED DEDEKIND SUM

231

Note that for (ab, d) = 1, from the orthogonality relation for character sums modulo d we have 1 if a ≡ b mod d;   2 φ(d),  χ (a)χ¯ (b) = − 12 φ(d), if a ≡ −b mod d;   χ mod d 0, otherwise. χ(−1)=−1 So that  χ mod d χ(−1)=−1

=

    χ (n)r (n) 2      1≤n≤N  ns

1 φ(d) 2



r (a)r (b) 1 − φ(d) a s bs 2

1≤a,b≤N (ab,d)=1 a≡b(d)

 1≤a,b≤N (ab,d)=1 a≡−b(d)

r (a)r (b) a s bs

  /d] N [N    |r (a)|2 τ (b)τ (d + b) 1 = φ(d) + O φ(d) 2 a 2s (d + b)s bs 1≤a≤N b=1 =1 

(a,d)=1

+ O φ(d)

d−1  τ (a)τ (d − a)

a s (d − a)s

a=1





 τ (a)τ (d − a) + O φ(d) a s (d − a)s 1≤a≤N (1+a/d)≤≤N/d    ∞  1 |r (n)|2 φ(d) ln N = φ(d) + O exp , 2 n 2s ds ln ln N n=1 



(4)

(n,d)=1

ln 2 ln n where τ (n) is the divisor function and r (n) ≤ τ (n)  exp( (1+) ). ln ln n     ∞   χ (n)r (n) A(y, χ ) s dy s n y s+1 N χ mod d 1≤n≤N χ(−1)=−1

 N −s+2



 ∞ N

1   y s+1

  χ mod d χ (−1)=−1

 |A(y, χ¯ )| dy

 φ 2 (d)N −2s+ 2 2ω(u)+ω(v) . 3

5

(5)

Set the parameter N = d 3 and note the identity ∞  |r (n)|2 ζ 4 (2s) = 2s n ζ (4s) n=1

(n,d)=1



( p 2s −1)2 p|uv p 2s ( p 2s +1)

p 2s p|d p 2s −1

.

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MIN AND WENPENG

From (1), (3), (4) and (5) we obtain  

 

  L s, χ χ 0 2  L s, χ χ 0 2 u v χ mod d χ(−1)=−1

ζ 4 (2s) = φ(d) 2ζ (4s)



( p 2s −1)2 p|uv p 2s ( p 2s +1)

p 2s p|d p 2s −1



  φ(d) 3 ln m +O exp . ds ln ln m

This proves the first part of Lemma 2. ii) Note that for (ab, d) = 1, 1   2 φ(d), if a ≡ b mod d;  χ (a)χ¯ (b) = 12 φ(d), if a ≡ −b mod d;   χ mod d 0, otherwise. χ(−1)=1 By the same methods from the first part of the proof we conclude part two of the lemma. ✷ Lemma 3.

Let q be an integer with q ≥ 3. Then we have the identity  d1 |q d2 |q

=q

d12s d22s φ(d) φ(d1 )φ(d2 )

4s−1

 p α q



( p 2s −1)2 p|d1 d2 p 2s ( p 2s +1)

p 2s p|d p 2s −1  2

 ( p 2s − 1) p( p 2s−1 − 1)2 1 , 1 − (4s−1)α ( p − 1)( p 4s−1 − 1) p ( p 2s − 1)2

where d = (d1 , d2 ) denotes the greatest common divisor of d1 and d2 . Proof: Since φ(n) is a multiplicative function, we can assume q = p α , a power of prime p. Note that φ( p n ) = p n (1 − 1p ) and if d = (d1 , d2 ), we have  

d12s d22s

d 1 | p α d2 | p α

φ(d1 )φ(d2 )

=1+2

φ(d)

2

( p12s −1) p1 |d1 d2 p 2s ( p 2s +1) 1 1

p12s p1 |d p12s −1

α α   p 2sβ ( p 2s − 1)2 p 4sβ ( p 2s − 1)3 + β 2s 2s φ( p ) p ( p + 1) β=1 φ( p β ) p 4s ( p 2s + 1) β=1

α−1  α  p 2sβ p 2sγ ( p 2s − 1)3 φ( p γ ) p 4s ( p 2s + 1) β=1 γ =β+1  (2s−1)α    (4s−1)α p − 1 ( p 2s − 1)2 − 1 ( p 2s − 1)3 p =1+2 + ( p − 1)( p 2s + 1)( p 2s−1 − 1) ( p 4s−1 − 1)( p − 1)( p 2s + 1)

+2

233

MEAN VALUE OF THE SQUARE OF A GENERALIZED DEDEKIND SUM



 p (4s−1)α−2s − p (2s−1)α ( p 2s − 1)2 +2 ( p − 1)( p 2s−1 − 1)( p 2s + 1)   p 2s−1 p (4s−1)(α−1) − 1 ( p 2s − 1)3 − 2 4s−1 (p − 1)( p − 1)( p 2s−1 − 1)( p 2s + 1)   ( p 2s − 1)2 p( p 2s−1 − 1)2 1 (4s−1)α = p . 1 − (4s−1)α ( p − 1)( p 4s−1 − 1) p ( p 2s − 1)2 ✷

This completes the proof of Lemma 3. 3.

Proof of the theorem

In this section we complete the proof of the theorem. i) Let n be an odd number, m be an integer with m ≥ 3. Then from i) of Lemma 1 we have m 

|S(h, n, m)|2

h=1

 m   =  h=1

=

2 (n!) 4n−1 m 2n−1 π 2n 2

 d 2n φ(d) d|m

 χ mod d χ (−1)=−1

 χ (h)|L(n, χ )|2 

  d 2n d 2n (n!)4 1 2 42(n−1) m 4n−2 π 4n d1 |m d2 |m φ(d1 )φ(d2 ) ×





m 

χ1 (h)χ2 (h)|L(n, χ1 )|2 |L(n, χ2 )|2 .

(6)

χ1 mod d1 χ2 mod d2 h=1 χ1 (−1)=−1 χ2 (−1)=−1

For each χ1 mod d1 , there exists one and only one q1 | d1 with a unique primitive character χq11 mod q1 such that χ1 = χq11 χd01 . Here χd01 denotes the principal character mod d1 . Similarly, we also have χ2 = χq22 χd02 and q2 | d2 , and χq22 is a primitive character mod q2 . Note that since d1 |m and d2 |m, from the orthogonality of characters we have m 

χ1 (h)χ2 (h) =

h=1

m     1 χq1 (h)χm0 (h) χq22 (h)χm0 (h) h=1

 =

φ(m), 0,

if q1 = q2 and χq11 = χq22 ; otherwise.

(7)

234

MIN AND WENPENG

Let d = (d1 , d2 ). If q1 = q2 and χq11 = χq22 , then χq11 χd0 is also a character mod d. So from (6), (7), Lemmas 2 and 3 we have m  |S(h, n, m)|2 h=1

=

=

(n!)4 φ(m)   d12n d22n 42(n−1) m 4n−2 π 4n d1 |m d2 |m φ(d1 )φ(d2 )





 

  L n, χ χ 0 2  L n, χ χ 0 2

χ mod d χ (−1)=−1

d1

d2

(n!)4 φ(m)   d12n d22n 42(n−1) m 4n−2 π 4n d1 |m d2 |m φ(d1 )φ(d2 )  

   ( p 2n −1)2  ζ 4 (2n) 2n 2n φ(d) 3 ln m p|d1 d2 p ( p +1) × +O exp φ(d)

p 2n  2ζ (4n)  dn ln ln m p|d 2n p −1

2(n!) φ(m)ζ (2n)   d12n d22n = 2n−1 4n−2 4n 4 m π ζ (4n) d1 |m d2 |m φ(d1 )φ(d2 )

   ( p 2n −1)2 4 ln m p|d1 d2 p 2n ( p 2n +1) 2−n × φ(d)

+ O m exp p 2n ln ln m p|d 2n 4

4

p −1

=

 ζ (2n) ( p 2n − 1)2 mφ(m) ζ (4n) ( p − 1)( p 4n−1 − 1) p α ||m      1 p( p 2n−1 − 1)2 4 ln m 2−n × 1 − (4n−1)α + O m . exp p ( p 2n − 1)2 ln ln m 4

2(n!)

4

42n−1 π 4n

This completes the proof of i) in the theorem. ii) Let n be an even number, m be an integer with m ≥ 3. Then from ii) of Lemma 1 we have m  |S(h, n, m)|2 h=1

 m   =  h=1

 m   =  h=1

−

m  h=1 m 

2 (n!) 4n−1 m 2n−1 π 2n

 d 2n φ(d) d|m

(n!) 4n−1 m 2n−1 π 2n

 d 2n φ(d) d|m

2

2

χ (h)|L(n, χ )|2 −

χ mod d χ(−1)=1

 χ mod d χ(−1)=1

2(n!)4 ζ 2 (n)  d 2n 42(n−1) m 2n−1 π 4n d|m φ(d)

(n!)4 ζ 4 (n) 2(n−1) π 4n 4 h=1 ≡ E1 + E2 + E3. +



2  χ (h)|L(n, χ )|2 

 χ mod d χ (−1)=1

χ (h)|L(n, χ )|2

(n!)  ζ 2 (n) 4n−1 π 2n 2

MEAN VALUE OF THE SQUARE OF A GENERALIZED DEDEKIND SUM

235

Next we discuss E 1 , E 2 and E 3 . With the methods already applied we deduce E1 =

 ( p 2n − 1)2 2(n!)4 ζ 4 (2n) mφ(m) 42n−1 π 4n ζ (4n) ( p − 1)( p 4n−1 − 1) p α m ! "    1 p( p 2n−1 − 1)2 4 ln m 2−n × 1 − (4n−1)α +O m exp . p ( p 2n − 1)2 ln ln m

(8)

From the orthogonality relation for character sums  d  φ(d), if χ = χd0 , χ (h) = 0, if χ = χd0 h=1 we have E2 = −

=− =−

2(n!)4 ζ 2 (n)  d 2n 2(n−1) 4 m 2n−1 π 4n d|m φ(d) 2(n!)4 ζ 2 (n) 42(n−1) m 2n−1 π 4n

 χ mod d χ (−1)=1

h m  χ (h)|L(n, χ )|2 d h=1

 d 2n m 

2 φ(d) L n, χd0  φ(d) d d|m

2(n!)4 ζ 4 (n)



42(n−1) m 2(n−1) π 4n

d|m

d 2n−1



(1 − p −n )2 .

p|d

From the methods of Lemma 3 we have      p (2n−1)α − 1 ( p n − 1)2 d 2n−1 (1 − p −n )2 = p( p 2n−1 − 1) p α m d|m p|d −1     1 1 1 2 2n−1 =m 1 − 2n−1 1 − (2n−1)α 1− n . p p p p α m So that E2 = −

 −1    1 1 1 2 2(n!)4 ζ 4 (n)  1 − 1 − 1 − m 42(n−1) π 4n p 2n−1 p (2n−1)α pn p α m

(9)

and  (n!)4 (n!)4 ζ 4 (n) ζ 4 (n) 1 = 2(n−1) 4n φ(m). 2(n−1) 4n 4 π 4 π h=1 m

E3 =

(10)

From (8), (9), and (10) we can obtain ii) in the theorem. This completes the proof of the theorem. ✷

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MIN AND WENPENG

Acknowledgment The authors express their gratitude to the referee for his very helpful and detailed comments. References 1. 2. 3. 4. 5. 6. 7. 8.

T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. T.M. Apostol, “Theorems on generalized Dedekind sums,” Pacific Journal of Mathematics 2 (1950) 1–9. L. Carlitz, “Some theorems the generalized Dedekind sums,” Pacific Journal of Mathematics 3 (1953) 513–522. J.B. Conrey, E. Fransen, R. Klein, and C. Scott, “Mean values of Dedekind sums,” J. Number Theory 56 (1996) 214–226. L.J. Mordell, “The reciprocity formula for Dedekind sums,” Amer. J. Math. 73 (1951) 593–598. W. Zhang, “On the mean values of Dedekind sums,” Journal de Theorie des Nombres 8 (1996) 429–442. W. Zhang, “A note on the mean square values of Dedekind sums,” Acta Mathematica Hungarica 86 (2000) 275–289. W. Zhang, “Some identities about general Dedekind sums and Dirichlet L - function,” Acta Mathematica Sinica 44 (2001) 269–272.