The Petersson–Knopp Identity for the Homogeneous Dedekind Sums

Journal of Number Theory  NT1928 journal of number theory 57, 223230 (1996) article no. 0045

The PeterssonKnopp Identity for the Homogeneous Dedekind Sums Zhiyong Zheng* Department of Mathematics, Zhongshan University, Guangzhou 510275, People's Republic of China Communicated by Alan C. Woods Received July 26, 1994; revised November 2, 1994

The homogeneous Dedekind sum is defined by S(a, b, q)= : r mod q

ar q

br q

\\ ++\\ ++

.

This paper shows that d

:

:

d

: S

d | n r1 =1 r2 =1

\

n n a+r 1 q, b+r 2 q, dq =n_(n) S(a, b, q). d d

+

It is the generalization of Knopp's identity for homogeneous Dedekind sums.  1996 Academic Press, Inc.

1. Introduction In [6], Knopp derived the following arithmetical identity d

:

: S

d | n r=1

n

\d a+rq, dq+ =_(n) S(a, q).

(1)

Here S(a, q) is the inhomogeneous Dedekind sum given by S(a, q)= : r mod q

r q

ar q

\\ ++\\ ++ ,

where ((x))=x&[x]& 12 if x{integer, ((x))=0 otherwise, and _(n) is the sum of the positive divisors of n. Knopp's identity is valid for arbitrary integers a and q with q>0, and his derivation uses the functional equation for Dedekind's eta-function '({) * Research supported by the National Science Foundation of China.

223 0022-314X96 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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ZHIYONG ZHENG

together with properties of Hecke operators. Elementary proofs of Knopp's identity have been given by several authors, for example, Goldberg [3], Parson [7] and Zheng [9]. Parson and Rosen [8] have extended Knopp's identity to the generalized Dedekind sums. Identities for sums of Dedekind type have been given by Apostol and Vu [2]. The purpose of this paper is to extend Knopp's identity to homogeneous Dedekind sums. To state our results, let a, b and q be integers with q>0. The homogeneous Dedekind sum is defined by (cf. [4]) ar q

r mod q

Theorem 1. d

:

:

br q

\\ ++\\ ++ .

S(a, b, q)= :

(2)

For any positive integer n, we have d

: S

d | n r1 =1 r2 =1

n

n

\d a+r q, d b+r q, dq+ =n_(n) S(a, b, q), 1

2

(3)

where _(n)= d | n d is the sum of the positive divisors of n. To state the more generalized result, we let B r (x) (r=0, 1, ...) be the Bernoulli polynomials given by  zr ze xz = : B r (x) z e &1 r=0 r!

(|z| <2?).

(4)

Following [5], for any non-negative integers : and ;, we define the generalized homogenous Dedekind sums by S:, ; (a, b, q)= :

P:

r mod q

ar br P; , q q

\ + \ +

(5)

where Pr (x)=Br (x&[x]) is the periodic extension into R of the Bernoulli polynomial Br (x) on [0, 1]. We have Theorem 2.

For any non-negative integers : and ;, d

: d :+;&2 : d|n

d

: S:, ;

r1 =1 r2 =1

n

n

\d a+r q, d b+r q, dq+ 1

2

=n_ :+;&1 (n) S:, ; (a, b, q),

(6)

where _ r (n)= d | n d r. The method of proving the above theorems is completely elementary.

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PETERSSONKNOPP IDENTITY

2. Proof of the Theorem To prove Theorem 1, we need the following lemmas. Lemma 1. If a, q are integers with q>0, for any real x, we have : r mod q

ar

qx

\\x+ q ++ =(a, q) \\(a, q)++ ,

(7)

where (a, q) denotes the greatest common divisor of a and q. Proof.

If (a, q)=1, it is easily seen that : r mod q

ar

\\x+ q ++ =

: r mod q

r

\\x+q++ =((qx)).

If (a, q)>1, let a 1 =a(a, q), q 1 =q(a, q), then, we have : r mod q

ra 1

\\x+ q ++ =(a, q) 1

: r mod q1

a1 r

\\x+ q ++ 1

=(a, q)((q 1 x)) The lemma follows. Lemma 2. For any positive integers k, we have (i) (ii) (iii) Proof.

S(ka, b, kq)=(b, k) S(a, b(b, k), q) S(a, kb, kq)=(a, k) S(a(a, k), b, q) S(ka, kb, kq)=kS(a, b, q). By Lemma 1 and (2), let r=sq+t, we have kq

ar

br

\\ q ++\\kq++ at bt bs =: : \\ q ++\\qk + k ++ at bt =(b, k) : \\ q ++\\ q(b, k)++ b =(b, k) S a, \ (b, k) , q+ .

S(ka, b, kq)= :

r=1

k&1

q

s=0 t=1

q

t=1

This proves (i) and in the same way we have (ii). If we replace b by kb in (i), then (iii) follows immediately.

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ZHIYONG ZHENG

Remark. Comparing (iii) with (2) of [4], we conclude the latter is not correct. Now, we can prove Theorem 1 by the use of Lemma 1 and Lemma 2. From (2), we have d

:

d

d | n r1 =1 r2 =1

d&1

\

n n a+r 1 q, b+r 2 q, dq d d

+ nar rr nbr rr : : : \\d q+ d ++\\d q + d ++ nar nbr : (d, r) \\d(r, d) q++\\d(r, d) q++ nar nbr : m : \\ dq ++\\ dq ++

: S

:

d&1

dq

1

=:

2

2

2

d | n r1 =0 r2 =0 r=1 dq

2

=:

d | n r=1

dqm

2

=:

d|n m|d

r=1 (r, dm)=1

dqmt

natr

nbtr

\\ dq ++\\ dq ++ na nb dq = : : m : +(t) S \ m , m , mt+ na nb = : m +(t) S \ m , m , sq+ na nb = : m S \ m , m , sq+ : +(t) na nb = : m S \ m , m , sq+ : m 2 : +(t) :

=:

d|n m|d

t | dm

r=1

2

d|n m|d

t | dm

2

mts | n

2

ms | n

t | nms

2

ms=n

=n_(n) S(a, b, q). This completes the proof of Theorem 1. To prove Theorem 2, we have to modify Lemma 1 and Lemma 2 as follows. Lemma 3. If m, a, q are integers with m0 and q>0, then for any real x, we have : r mod q

\

Pm x+

ar qx =(a, q) m q 1&m Pm . q (a, q)

+

\ +

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PETERSSONKNOPP IDENTITY

Proof. The multiplication formula for Bernoulli polynomials is known (cf. Apostol [1, Chap. 12. Exercise 15]): q&1

r

\ q+ =q

: B m x+ r=0

1&m

B m (qx).

(9)

It follows that r

\ q+ =q

Pm x+

: r mod q

1&m

Pm (qx).

(10)

If (a, q)=1, then we have

\

Pm x+

: r mod q

ar r = : Pm x+ =q 1&m Pm (qx). q q r mod q

+

\ +

If (a, q)>1, let a 1 =a(a, q), q 1 =q(a, q). We also have :

\

Pm x+

r mod q

ar =(a, q) q

+

r mod q1

=(a, q) q

\

Pm x+

: 1&m 1

a1 r q1

+

Pm (q 1 x)

=(a, q) m q 1&m Pm

qx

\(a, q)+ .

This proves Lemma 3. Lemma 4. For any non-negative integers : and ;, and any positive integers k, we have (i)

S:, ; (ak, b, qk)=(b, k) ; k 1&; S:, ; (a, b(b, k), q)

(ii) (iii)

S:, ; (a, bk, qk)=(a, k) : k 1&: S:, ; (a(k, a), b, q) S:, ; (ak, bk, qk)=kS:, ; (a, b, q).

Proof.

Let r=sq+t. By Lemma 3, it follows that qk

ar

br

\ q + P \qk+ bt bs at =: : P \ qk+ k + P \ q + at bt =(b, k) k : P \ q + P \ (b, k) q+ b =(b, k) k S \a, (b, k) , q+ .

S:, ; (ak, b, qk)= : P:

;

r=1 q

k&1

;

:

t=1 s=0

q

;

1&;

:

;

t=1

;

1&;

:, ;

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ZHIYONG ZHENG

So, (i) is true. In the same way, we have (ii). If we replace b by kb in (i), (iii) follows immediately. This completes the proof of Lemma 4. Now, we prove Theorem 2 by Lemma 3 and Lemma 4. From (5), we have d

d

: d :+;&2 : d|n

\

: S:, ;

r1 =1 r2 =1 d

n na +r 1 q, b+r 2 q, dq d d

d

dq

+

rr 1

nar

nbr

rr 2

\ d q + d + P \d q + d + nar nbr = : : (r, d) P \d(r, d) q+ P \ d(r, d) q+ nar nbr =: : m : P P \ dq + \ dq + natr abtr =: : m : +(t) : P P \ dq + \ dq + na nb dq =: : m : +(t) S \ m , m , mt+ na nb = : m +(t) S \ m , m , sq+ na nb = : m S \ m , m , sq+ : +(t) na nb = : m S \ m , m , sq+ = : d :+;&2 : d|n

:

: P:

;

2

2

r1 =1 r2 =1 r=1

dq

:+;

:

;

d | n r=1

dqm

:+;

:

d|n m|d

;

r=1 (r, dm)=1

dqmt

:+;

:

d|n m|d

t | dm

;

r=1

:+;

:, ;

d|n m|d

t | dm

:+;

:, ;

mts | n

:+;

:, ;

ms | n

t | nms

:+;

:, ;

ms=n

=n_ :+;&1 (n) S:, ; (a, b, q). This completes the proof of Theorem 2.

3. The Further Results Following Apostol and Vu [2], we consider the homogeneous sums of Dedekind type, which are defined by f (a, b, q)= : r mod q

A

ar br B , q q

\ + \ +

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PETERSSONKNOPP IDENTITY

where A(x) and B(x) are given functions defined for rational values of x. Also, each satisfies a multiplication formula of the form r

\ k+ =k

F x+

: r mod k

V(F)

F(kx),

(12)

for every integer k>0 and every rational x, where the constant V(F) depends on F but not on k or x. We refer to such a sum f (a, b, q) as being of homogeneous Dedekind type (V(A), V(B)). It is easy to see that the S(a, b, q) of Theorem 1 is of type (0, 0) and the S:, ; (a, b, q) of Theorem 2 is of type (1&:, 1&;). To generalize, we have Theorem 3. If f (a, b, q) is of homogeneous Dedekind type of (V(A), V(B)), then for every positive integer n, d

: d &V(A)&V(B) : d|n

d

: f

r1 =1 r2 =1

nb

n

\d a+r q, d +r q, dq+ 1

2

=n_ * (n) f (a, b, q), where *=1&V(A)&V(B). Proof. The method is similar to that in Theorem 1 and Theorem 2. We omit the details.

Acknowledgment The author expresses his thanks to the referee for some valuable suggestions.

References 1. T. M. Apostol, ``Introduction to Analytic Number Theory,'' Springer-Verlag, New York HeidelbergBerlin, 1976. 2. T. M. Apostol and Thiennu H. Vu, Identities for sums of Dedekind type, J. Number Theory 14 (1982), 391396. 3. L. A. Goldberg, An elementary proof of Knopp's theorem on Dedekind sums, J. Number Theory 12 (1980), 541542. 4. R. R. Hall and M. N. Huxley, Dedekind sums and continued fractions, Acta Arith. LXIII (1993), 7990. 5. R. R. Hall and J. C. Wilson, On reciprocity formula for inhomogeneous and homogeneous Dedekind sums, Math. Proc. Camb. Phil. Soc. 114 (1993), 924. 6. M. I. Knopp, Hecke operators and an identity for Dedekind sums, J. Number Theory 12 (1980), 29.

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7. L. A. Parson, Dedekind sums and Hecke operators, Math. Proc. Cambridge. Philos. Soc. 88 (1980), 1114. 8. L. A. Parson and K. Rosen, Hecke operators and Lambert series, Math. Scand. 49 (1981), 514. 9. Zhiyong Zheng, On a theorem of Dedekind sums, Acta Math. Sinica 37 (1994), 690694. 10. Zhiyong Zheng, Some properties of Dedekind sums, Ann. Math. China A 16 (1995), 8186.

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