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An elliptic analogue of generalized Dedekind–Rademacher sums
Journal of Number Theory 128 (2008) 1060–1073 www.elsevier.com/locate/jnt
An elliptic analogue of generalized Dedekind–Rademacher sums Tomoya Machide Department of Mathematics, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan Received 12 November 2006; revised 16 April 2007 Available online 5 June 2007 Communicated by Matthias Beck
Abstract In this paper we introduce an elliptic analogue of the generalized Dedekind–Rademacher sums which satisfy reciprocity laws. In these sums, Kronecker’s double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of S. Fukuhara and N. Yui concerning the elliptic Apostol– Dedekind sums. We also mention a relation between the generating function of Kronecker’s double series and that of the (Debye) elliptic polylogarithms studied by A. Levin. © 2007 Elsevier Inc. All rights reserved. MSC: 11F20; 11B68 Keywords: Dedekind sums; Generalized Dedekind–Rademacher sums; Reciprocity laws; Elliptic analogue; Kronecker’s double series; Elliptic Apostol–Dedekind sums
1. Introduction After pioneering works by Dedekind [6], much attention has been directed to generalizations of Dedekind sums and their reciprocity laws (see [4,22] for a good picture of previously defined generalizations of Dedekind sums).
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Let a, b and c be positive integers, and x, y and z real numbers. U. Halbritter [11] and R.R. Hall et al. [12] have defined the generalized Dedekind–Rademacher sums Sm,n xa yb zc by Sm,n
a x
b y
c z
:=
j (mod c)
j +z j +z − x B˜ n b −y , B˜ m a c c
(1)
and have proved their reciprocity laws. Here B˜ m (x) is the mth Bernoulli function defined as follows: We denote the Bernoulli polynomials by Bm (x). ∞
Bm (x) exξ = ξ m−1 . ξ e −1 m! m=0
Bm := Bm (0) is the mth Bernoulli number. Let {x} be the fractional part of a real number x. Then the Bernoulli functions B˜ m (x) are defined by 0 B˜ m (x) = Bm ({x})
(m = 1, x ∈ Z), (otherwise).
As special cases, these sums involve the generalizations of Dedekind sums introduced by T.M. Apostol [1], L. Carlitz [5], M. Mikolás [20], U. Dieter [7], C. Meyer [19], H. Rademacher [21], and L. Takács [24]. Note that so-called Apostol–Dedekind sums [1] are expressed by S1,n 10 b0 0c . Moreover, various elliptic analogues of the classical Dedekind sums have been studied by several authors [2,3,9,10,13,14,23]. R. Sczech introduced the elliptic Dedekind sums as an elliptic analogue of the classical Dedekind sums. A. Bayad [2,3] and S. Egami [9] studied the multiple elliptic Dedekind sums as an elliptic analogue of Zagier’s multiple Dedekind sums [26]. Recently, S. Fukuhara and N. Yui [10] have introduced the elliptic Apostol–Dedekind sums as an elliptic analogue of the classical Apostol–Dedekind sums. However their elliptic analogue did not involve an elliptic analogue of the classical Bernoulli functions. So they have raised the problem to find a true analogue in the elliptic world of the classical Apostol–Dedekind sums which involve these functions. In this paper we answer to the above problem. We define an elliptic analogue of the generalized Dedekind–Rademacher sums (elliptic Dedekind–Rademacher sums) which involve an elliptic analogue of the classical Bernoulli functions. For the definition of this elliptic analogue, we make use of Kronecker’s double series [25]. E.V. Ivashkevich et al. [15, Section 3.1] and K. Katayama [17] noted that these series can be considered as an elliptic generalization of the classical Bernoulli functions. The principal aim of this paper is to establish the reciprocity laws for the elliptic Dedekind– Rademacher sums (Theorem 5). As a corollary, we reproduce the reciprocity laws [12] for the classical generalized Dedekind–Rademacher sums (Theorem 12). We also mention a relation between the generating function of Kronecker’s double series and that of the (Debye) elliptic polylogarithms studied by A. Levin [18] in order to enforce the validity of our elliptic generalization (see Section 2). The paper is organized as follows: In Section 2 we review Kronecker’s double series and their basic properties. A relation between the generating function of Kronecker’s double series and that
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of the (Debye) elliptic polylogarithms is also discussed. In Section 3 we define elliptic Dedekind– Rademacher sums, and obtain their reciprocity laws (Theorem 5). The reciprocity laws [12] for the classical generalized Dedekind–Rademacher sums are reproduced by the degeneration of the elliptic Dedekind–Rademacher sums (Theorem 12). 2. Kronecker’s double series In this section we review Kronecker’s double series and some of their basic properties. We also mention a relation between the generating function of Kronecker’s double series and that of the (Debye) elliptic polylogarithms studied by A. Levin [18]. Let τ be in the upper half-plane. We shall use the following notations: e(x) := e2πix , q := e(τ ), and Jacobi’s theta function 1 1 2 1 1 m+ x+ e τ + m+ 2 2 2 2 m∈Z ∞ 1 x x −e − 1 − e(−x)q m 1 − e(x)q m 1 − q m . = iq 8 e 2 2
θ (x; τ ) :=
(2)
m=1
We note that θ (x; τ ) has the quasi periodicity: θ (x + 1; τ ) = −θ (x; τ ),
τ θ (x + τ ; τ ) = −e − − x θ (x; τ ). 2
(3)
For x, ξ ∈ C \ Z + τ Z, we define the function F (x, ξ ; τ ) as follows: F (x, ξ ; τ ) := where θ (x; τ ) =
∂ ∂x θ (x; τ ).
F (x, ξ ; τ ) = 2πi
θ (0; τ )θ (x + ξ ; τ ) , θ (x; τ )θ (ξ ; τ )
By virtue of (2) we have x+ξ e( x+ξ 2 ) − e(− 2 )
(e( x2 ) − e(− x2 ))(e( ξ2 ) − e(− ξ2 )) ∞ (1 − e(−x − ξ )q m )(1 − e(x + ξ )q m )(1 − q m )2 × ∞ m=1 . m m m m m=1 (1 − e(−x)q )(1 − e(x)q )(1 − e(−ξ )q )(1 − e(ξ )q )
(4)
Thus, for fixed ξ ∈ C \ Z + τ Z, the function F (x, ξ ; τ ) with respect to x is meromorphic with only simple poles on the lattice Z + τ Z. In addition, it satisfies the following properties by (3): F (x + 1, ξ ; τ ) = F (x, ξ ; τ ),
F (x + τ, ξ ; τ ) = e(−ξ )F (x, ξ ; τ ),
Res F (x, ξ ; τ ) = e(−n ξ ) (n, n ∈ Z).
x=n τ +n
(5) (6)
Remark 1. S. Egami [9] and S. Fukuhara and N. Yui [10] used the function ϕ(τ, z) = 1 z 1 2πi F ( 2πi , 2 ; τ ) for their elliptic analogues. A. Bayad also used the following function for his or her elliptic analogue (see [2, p.34]):
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
D[ω1 ,ω2 ] (z; φ) = where τ =
ω1 ω2
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φ z φ 1 z Im( ω2 ) F e , ;τ , ω2 ω2 Im τ ω2 ω2
and z, φ ∈ / ω1 Z + ω2 Z.
Let us introduce Kronecker’s double series. Set F (x , x; ξ ; τ ) := e(xξ )F (−x + xτ, ξ ; τ ). We define functions Bm (x , x; τ ) by F (x , x; ξ ; τ ) =
∞ Bm (x , x; τ ) (2πi)m ξ m−1 . m!
(7)
m=0
Namely F (x , x; ξ ; τ ) gives the generating function of Bm (x , x; τ ). It is easily seen from (5) that Bm (x , x; τ ) have the following periodicity: Bm (x + 1, x; τ ) = Bm (x , x + 1; τ ) = Bm (x , x; τ ).
(8)
In view of the expansion (10) and (11) below, Bm (x , x; τ ) are called Kronecker’s double series. We will give an explicit form of the mth Kronecker’s double series Bm (x , x; τ ). The function F (x, ξ ; τ ) has the following expression ([25], [16, p. 446]):
∞
∞
qj qj e(−j ξ ) − e(j ξ ) j e(x) − q e(−x) − q j j =1 j =1 1 1 + +1 |Im x|, |Im ξ | < Im τ . + e(x) − 1 e(ξ ) − 1
F (x, ξ ; τ ) = 2πi
So one can see from (7) that
Bm (x , x; τ ) = m
∞ (x − j )m−1 j =1
−
∞
(x + j )
j =1
m−1
e(−xτ )q j e(−x ) − e(−xτ )q j
e(xτ )q j m−1 e(−x + xτ ) + x e(x ) − e(xτ )q j e(−x + xτ ) − 1
+ Bm (x). (9)
e(−x +xτ ) We remark that B1 (0, 0; τ ) can not be defined since the function e(−x +xτ )−1 becomes infinity if x = x = 0. Originally, Kronecker’s double series have been introduced in the following way. Let x and x be real numbers with −1 < x < 0. Kronecker proved the following equation [25].
F (x , x; ξ ; τ ) = −
e e(n x + nx) , −ξ + n τ + n
(10)
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where
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
e
denotes Eisenstein summation [25], i.e., e
= lim
N
lim
N →∞ N →∞
N
.
n =−N n=−N
So we have ⎧ 1 ⎪ ⎪ ⎨ e e(n x + nx) m! Bm (x , x; τ ) = − ⎪ (2πi)m (n τ + n)m ⎪ ⎩ (n ,n)
(m = 0), (m 1).
(11)
=(0,0)
The series in (10) and (11) were studied by Kronecker [25]. As noted in [15], Kronecker’s double series degenerate into the classical Bernoulli functions. We reconstruct this statement and give its proof for reader’s convenience. Proposition 2. Let x and x be real numbers with x ∈ / Z. Then ⎧ ⎨ 1 1 + e(x ) = i cot(πx ) lim Bm (x , x; τ ) = 2 1 − e(x ) 2 ⎩ τ →i∞ B˜ m (x)
(m = 1, x ∈ Z),
(12)
(otherwise),
especially Re lim Bm (x , x; τ ) = B˜ m (x). τ →i∞
(13)
Here Re w means the real part of a complex number w. Proof. The proposition is trivial if m = 0. Suppose that m 1. One obtains from (8) that Bm x , {x}; τ = Bm (x , x; τ ), where {x} means the real part of x. So it is sufficient to prove the proposition when 0 x < 1. Then we have lim e(xτ )q j = lim e(−xτ )q j = 0 (j ∈ Z1 ),
τ →i∞
τ →i∞
(14)
and ⎧ 1 ⎨ + xτ ) e(−x m−1 lim x = 1 − e(x ) τ →i∞ e(−x + xτ ) − 1 ⎩ 0 Eqs. (9), (14) and (15) induce (12) since B1 = −1/2.
2
(m = 1, x = 0), (otherwise).
(15)
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The generating function F (x , x; ξ ; τ ) of Kronecker’s double series is related to that of the (Debye) elliptic polylogarithms studied by A. Levin [18] in the following sense. Set i∞ e(−xt) Λ(ξ ; −2πix) := 2πi ξ e(−t)−1 dt. The Debye polylogarithms Λm (ξ ) are defined by [18]: i∞ 2πi ξ
∞
e(−xt) dt = Λm+1 (ξ )(2πi)m+1 (−x)m . e(−t) − 1
(16)
m=0
Namely Λ(ξ ; −2πix) gives the generating function of Λm (ξ ). Since ∞
Bm (x) e(xξ ) ∂ Λ(−ξ ; −2πix) = 2πi = (2πi)m ξ m−1 , ∂ξ e(ξ ) − 1 m!
(17)
m=0
∂ ∂ξ Λ(−ξ ; −2πix)
becomes the generating function of the classical Bernoulli functions (polynomials). On the other hand, A. Levin has shown that the modified generating function Λ(ξ, τ ; x , x) of the (Debye) elliptic polylogarithms satisfies the following equation (see [18, Proposition 3.1]): ∂ Λ(ξ, τ ; −2πix , −2πix) = F (x , x; ξ ; τ ). ∂ξ
(18)
Comparing (17) and (18), we can consider that F (x , x; ξ ; τ ) is an elliptic analogue of the generating function of the classical Bernoulli functions (polynomials). In this point of view Bm (x , x; τ ) become an elliptic analogue of the classical Bernoulli functions since F (x , x; ξ ; τ ) is the generating function of Bm (x , x; τ ). Remark 3. Let Lim (z) denote the Euler polylogarithms, i.e., Lim (z) :=
∞ zn . nm n=1
One of the basic properties of the Euler polylogarithms is the iterated integral representation. We can formally consider that (16) is based on it in the following trick: 2πi
e(xξ ) dξ = 2πi e(ξ ) − 1
e(xξ )Li0 e(−ξ ) dξ
= −e(xξ )Li1 e(−ξ ) + 2πix = ···
= −e(xξ ) =−
∞
m=0
∞
k=0
e(xξ )Li1 e(−ξ ) dξ
Lim+1 e(−ξ ) x m
m=0 m
Lik+1
(2πi)−k ξ m−k e(−ξ ) (2πix)m (m − k)!
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=
∞
Λm+1 (−ξ )(2πi)m+1 (−x)m .
m=0
Here we assume that the series is formal, and use [18, Proposition 1.1(a)] for the last equation. We wish that this formal calculation gives us a key to an elliptic analogue of the classical Euler polylogarithms and their iterated integral representation. 3. Elliptic Dedekind–Rademacher sums In this section we define elliptic Dedekind–Rademacher sums, and obtain their reciprocity laws. We also reproduce the reciprocity laws [12] for the classical generalized Dedekind– Rademacher sums by the degeneration of the elliptic Dedekind–Rademacher sums. To simplify the notations, we begin with the following lemma. Lemma 4. Let a, b be positive integers, and x, y real numbers. Let a, b be the greatest common divisor of a and b. Then the following statements are equivalent: (i) ay − bx ∈ / a, bZ, (ii) a j +y − x ∈ / Z (j = 0, . . . , b − 1), b (iii) ( xa + a1 Z) ∩ ( yb + b1 Z) = ∅. Proof. We can prove this lemma by simple calculation, so we omit the proof.
2
Let a, a , b, b , c, c be positive integers, and x, x , y, y , z, z real numbers. Suppose that / a , c Z, a z − c x ∈
b z − c y ∈ / b , c Z.
(19)
c ) := ((a , a), (b , b), (c , c)), ( x , y , z ) := ((x , x), (y , y), (z , z)). Set ( a , b, We define the elliptic Dedekind–Rademacher sums as follows: τ Sm,n
a x
b y
1 c := z c
Bm a
j (mod c) j (mod c )
j
a + z j +z − x; τ − x , a c c a
b j + z j +z − y; τ . × Bn b − y , b c c b
(20)
It follows from (19) and Lemma 4(ii) that B1 (0, 0; τ ) does not appear in (20). So this sum is well-defined. If m = 1 and n = 1, or if az − cx ∈ / a, cZ, bz − cy ∈ / b, cZ, then we see from (12) and Lemma 4(ii) that lim
τ →i∞
τ Sm,n
a x
b y
c z
= Sm,n
a x
b y
c z
.
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a b c τ except the above case is troubleHowever to obtain the explicit forms of limτ →i∞ Sm,n x y z some because the following formula forces us to consider many cases. i 1 ˜ B1 (x , 0; τ ) = cot(πx ) = 0 = B1 (0) x ∈ / +Z . 2 2 In addition the explicit forms are not necessary for reproducing the reciprocity laws of the classical generalized Dedekind–Rademacher sums from the elliptic ones; it is needed that the real part a b c τ of limτ →i∞ Sm,n (see the proof of Theorem 12). So we omit their explicit forms. x y z Our principal aim of this paper is to give the reciprocity laws for the elliptic Dedekind– Rademacher sums. As R.R. Hall et al. have noted in [12], these laws mix various pairs of indices (m, n), so they are conveniently stated in terms of the generating function ⎛ ⎞ m−1 n−1 ∞ a b c X Y 1 τ a b c τ⎝ ⎠ S . S x y z := m!n! m,n x y z a b X Y Z m,n=0 Here X and Y are non-zero variables and the variable Z, which does not appear explicitly on the right-hand side of the definition, is defined by −X − Y . Because of (7) and (20), the generating function of the elliptic Dedekind–Rademacher sums satisfies ⎛
⎞ a b c (2πi)2 Sτ ⎝ x y z ⎠ 2πiX 2πiY 2πiZ X a j + z 1 j +z − x; ; τ = F a − x , a c c c a a j (mod c) j (mod c )
Y b j + z j +z × F b − y; ; τ . − y , b c c b b
(21)
The reciprocity laws for the elliptic Dedekind–Rademacher sums are the following: Theorem 5. Let X, Y, Z be variables with X + Y + Z = 0, and a, a , b, b , c, c positive integers, and x, y, z real numbers. Let x , y and z be real numbers such that a y − b x ∈ / a , b Z,
a z − c x ∈ / a , c Z,
b z − c y ∈ / b , c Z.
c ) := ((a , a), (b , b), (c , c)), ( Set ( a , b, x , y , z ) := ((x , x), (y , y), (z , z)). Then we have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a b c b c a c a b Sτ ⎝ x y z ⎠ + Sτ ⎝ y z x ⎠ + Sτ ⎝ z x y ⎠ = 0. X Y Z Y Z X Z X Y Here the left is over the three terms obtained by cyclic permutation of the columns of
(22)
(23) a
b c x y z . XY Z
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The strategy of our proof, which is the same method employed by A. Bayad [3] and S. Egami [9], is to apply the residue theorem. Proof of Theorem 5. Set wa = −
x x + τ, a a
wb = −
y y + τ, b b
wc = −
z z + τ. c c
We define the function G(α) as follows:
X a Y b Z c G(α) = F a (α − wa ), ; τ F b (α − wb ), ; τ F c (α − wc ), ; τ . a a b b c c
It follows from (5) and X + Y + Z = 0 that G(α + 1) = G(α + τ ) = G(α), namely G(α) is a doubly periodic function. Since the function F (x, ξ ; τ ) with respect to x has the simple poles on the lattice Z + τ Z, the poles of G(α) are on the following lattices: wa +
1 τ Z + Z, a a
wb +
1 τ Z + Z, b b
wc +
1 τ Z + Z. c c
By virtue of the condition (22) and Lemma 4(iii), we have wi +
1 τ 1 τ Z ∩ w Z =∅ Z + + Z + j i i j j
i, j = a, b, c (i = j ) .
This implies that the order of every pole of G(α) is equal to one. Because the function G(α) is doubly periodic, the sum of the residues of G(α) at its poles in any period parallelogram is equal to zero. So it follows that
0=
s=a,b,c j (mod s) j (mod s )
Res
α=ws −j /s +(j/s)τ
G(α).
Consider the case s = c. By virtue of (5) and (6), we have
Res
j (mod c) j (mod c )
1 = c
α=wc −j /c +(j/c)τ
j (mod c) j (mod c )
G(α)
j +z a X a −Z j +z F − a −x τ, ; τ e j −x + a c c c a a a
j +z b Y b j +z ×F − b −y τ, ; τ −y + b c c b b b 1 xX yY zZ X a j +z j +z = e + + − x; ; τ F a − x ,a c a b c c c a a j (mod c) j (mod c )
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j + z j +z Y b × F b − y , b − y; ; τ c c b b ⎛ a b zZ xX yY + + Sτ ⎝ x = (2πi)2 e y a b c 2πiX 2πiY
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⎞ c z ⎠ . 2πiZ
Since one can calculate the other cases in the same way, we obtain (23).
2
Remark 6. As R.R. Hall et al. have remarked in [12, Section 4], the left-hand side of (23) is invariant under all permutations of the columns since Sτ is symmetric in its first two columns. We can also see this fact since the function G(α) is invariant under all permutations of the a b c columns of x y z . XY Z
Remark 7. Let k , k be positive integers, and let ( ak , b k , c k ) = (a , ka), (b , kb), (c , kc) , a k , b k , c k = (k a , a), (k b , b), (k c , c) . Then we have τ Sm,n
τ Sm,n
a k x
b k y
a k
b k
c k
x
y
z
c k z
1
τ
k = kSm,n
k τ = Sm,n
a x a x
c , z b c . y z
b y
These equations allow us to reduce (23) to the case that the integers a, b and c (respectively a , b and c ) have no common factor. We introduce the generating function of the classical generalized Dedekind–Rademacher sums: ⎛ ⎞ m−1 n−1 ∞ a b c Y 1 X a b c S ⎝ x y z ⎠ := Sm,n . x y z m!n! a b X Y Z m,n=0 To reproduce the reciprocity laws for these sums from (23) (Theorem 12 bellow), the cotangent sum studied by U. Dieter [8] is introduced: j +z 1 j +z −x cot π b −y , c(a, b, c; x, y, z) := cot π a j (mod c) c c c j +z where j (mod c) means excluding j such that a j +z c − x ∈ Z or b c − y ∈ Z. We need the following lemmas and proposition for Theorem 12:
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Lemma 8. Let a, b, c be positive integers, and x, y, z real numbers. Then the following statements are equivalent: (i) (x, y, z) ∈ (a, b, c)R + Z3 . (ii) There is an integer j0 such that a j0c+z − x, b j0c+z − y ∈ Z. Proof. Suppose that (i) holds. Then there are integers j0 , m, n and a real number r such that (x, y, z) = (a, b, c)r + (−m, −n, −j0 ). So we have r=
m + x n + y j0 + z = = . a b c
Therefore one gets a
j0 + z − x = m, c
b
j0 + z − y = n. c
Hence (ii) follows. The reverse is also proved in a similar way.
2
Remark 9. Since the statement (i) is invariant under all permutations of the columns of the statement (ii) is too. We shall implicitly use this fact in the proof of Theorem 12.
a b c xyz
,
Lemma 10. Let a, b and c be positive integers with no common factor, and let m be the least c c common multiple of a,c and b,c . Then we have m = c. Proof. For any prime number p, we define a nonnegative integer p (a) by factorization of prime numbers: a=
p p (a) ,
where the product is extended over all prime numbers p. Then we have c = p p (c)−min{ p (a), p (c)} , a, c
c = p p (c)−min{ p (b), p (c)} . b, c
Thus m=
p max{ p (c)−min{ p (a), p (c)}, p (c)−min{ p (b), p (c)}} .
Let p be a prime number. It is seen that max p (c) − min p (a), p (c) , p (c) − min p (b), p (c) = p (c) − min p (a), p (b), p (c) . Since a, b, c have no common factor, min{ p (a), p (b), p (c)} is equal to zero. So we get m = c. 2
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
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Proposition 11. Let a, a , b, b , c, c be positive integers, and x, x , y, y , z, z real numbers. Suppose that a, b, c have no common factor, and that a z − c x ∈ / a , c Z,
b z − c y ∈ / b , c Z.
(24)
c ) := ((a , a), (b , b), (c , c)), ( Set ( a , b, x , y , z ) := ((x , x), (y , y), (z , z)). Let Re w denote the real part of a complex number w. Then we have Re lim
τ →i∞
τ Sm,n
b y
a x
c z
⎧ a b c ⎪ ⎪ S 1,1 x y z ⎪ ⎪ ⎪ ⎨ = − 14 c(a , b , c ; x , y , z ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Sm,n a b c x y z
(m = n = 1, (x, y, z) ∈ (a, b, c)R + Z3 ),
(25)
(otherwise).
Proof. We remark that Sm,n xa yb zc are real numbers since a, b, c, x, y, z are real numbers, and that cot w is a real number if w is too. Suppose that m = n = 1 and (x, y, z) ∈ (a, b, c)R + Z3 . By Lemma 8, there is an integer j0 such that a j0c+z − x, b j0c+z − y ∈ Z. Let j be an integer. The condition a
j +z j +z − x, b −y ∈Z c c
(26)
is equivalent to j ≡ j0
c j ≡ j0 mod . b, c
c mod , a, c
By virtue of Lemma 10, (26) is also equivalent to j ≡ j0
(mod c).
Thus we see from B˜ 1 (0) = 0 and the above remark that τ Re lim S1,1 τ →i∞
= S1,1
a x
a x
b y
b y
c z
c z
−
1 4c
j (mod c )
j + z j +z cot π b . cot π a − x − y c c
The second term in the last formula is equal to − 14 c(a , b , c ; x , y , z ) by virtue of the condition (24) and Lemma 4(ii), so this case is proved. The other case can be also obtained by attention to the above remark. 2 Now we are in a position to reproduce the reciprocity laws [12] of the classical generalized Dedekind–Rademacher sums.
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Theorem 12. (See [12].) Let X, Y, Z be variables with X + Y + Z = 0, and a, b, c positive integers with no common factor, and x, y, z real numbers. Then we have ⎛
a b S⎝ x y X Y 1 −4 = 0
⎞ ⎛ c b z ⎠ + S⎝ y Z Y
c z Z
⎞ ⎛ a c x ⎠ + S⎝ z X Z
a x X
⎞ b y⎠ Y
((x, y, z) ∈ (a, b, c)R + Z3 ), (otherwise).
(27)
Proof. We may assume that X, Y and Z are real numbers. Suppose that (x, y, z) ∈ (a, b, c)R + Z3 . When τ tends to i∞ in (23), an equation is obtained. By directing our attention to the real part of the equation, we can get by virtue of (25) ⎛
a S⎝ x X −
b y Y
⎞ ⎛ c b z ⎠ + S⎝ y Z Y
c z Z
⎞ ⎛ a c x ⎠ + S⎝ z X Z
a x X
⎞ b y⎠ Y
1 c(a , b , c ; x , y , z ) + c(b , c , a ; y , z , x ) + c(c , a , b ; z , x , y ) = 0, 4
where x , y and z are real numbers such that / a , b Z, a y − b x ∈
a z − c x ∈ / a , c Z,
b z − c y ∈ / b , c Z.
(28)
In [8, Theorem 2.3] the reciprocity law of c(a , b , c ; x , y , z ) has been given: c(a , b , c ; x , y , z ) + c(b , c , a ; y , z , x ) + c(c , a , b ; z , x , y ) = −1.
(29)
2
Thus this case is proved. The other case is also proved the same way. Remark 13. Let k be a positive integer. As in Remark 7, one gets Sm,n
ka x
kb y
kc z
= kSm,n
a x
b y
c z
.
This allows us to remove the condition in Theorem 12 that the integers a, b and c have no common factor. Remark 14. The theorem in [12, Section 4] is incorrect, i.e., the right-hand side of (7) in it should be multiplied by −1. The cause is that the proposition in [12, Section 3] has a small error. Remark 15. In [8, Theorem 2.3] it is supposed that a , b , c are pairwise relatively prime. But, under the condition (28), this supposition is not necessary. The reasons are that (29) is obtained by [8, Eq. (2,5)] and [8, Lemma 2.2] (additive formula for cotangent function), and that [8, Eq. (2,5)] holds for any positive integers a, b, c.
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Acknowledgments The author thanks Professors Youichi Shibukawa and Hiroshi Yamashita for their encouragement. He expresses his gratitude to Professor Kimio Ueno for suggesting him the equation (23), and to Professor Michitomo Nishizawa for his valuable advice. He is also grateful to Professor Yumiko Hironaka for her comments on the preliminary version, and to the referees for improving this paper. References [1] T.M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950) 147–157. [2] A. Bayad, Sommes de Dedekind elliptiques et formes de Jacobi, Ann. Inst. Fourier (Grenoble) 51 (2001) 29–42. [3] A. Bayad, Sommes elliptiques multiples d’Apostol–Dedekind–Zagier, C. R. Math. Acad. Sci. Paris, Ser. I 339 (2004) 457–462. [4] M. Beck, Dedekind cotangent sums, Acta Arith. 109 (2003) 109–130. [5] L. Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953) 513–522. [6] R. Dedekind, Erläuterungen zu den Fragmenten XXVIII, in: B. Riemann’s Gesammelte mathematische Werke, second ed., Teubner, Leipzig, 1892, pp. 466–478. [7] U. Dieter, Das Verhalten der kleinschen Funktionen log σg,h (w1 , w2 ) gegenüber Modultransformationen und verallgemeinerte Dedekindsche Summen, J. Reine Angew. Math. 201 (1959) 37–70. [8] U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Theory 18 (1984) 289–305. [9] S. Egami, An elliptic analogue of multiple Dedekind sums, Compos. Math. 99 (1995) 99–103. [10] S. Fukuhara, N. Yui, Elliptic Apostol sums and their reciprocity laws, Trans. Amer. Math. Soc. 356 (10) (2004) 4237–4245. [11] U. Halbritter, Some new reciprocity formulas for generalized Dedekind sums, Results Math. 8 (1985) 21–46. [12] R.R. Hall, J.C. Wilson, D. Zagier, Reciprocity formulae for general Dedekind–Rademacher sums, Acta Arith. 73 (1995) 389–396. [13] H. Ito, On a property of elliptic Dedekind sums, J. Number Theory 27 (1987) 17–21. [14] H. Ito, A density result for elliptic Dedekind sums, Acta Arith. 112 (2004) 199–208. [15] E.V. Ivashkevich, N.Sh. Izmailianand, C. Hu, Kronecker’s double series and exact asymptotic expansions for free models of statistical mechanics on torus, J. Phys. A: Math. Gen. 35 (2002) 5543–5561. [16] C. Jordan, Fonctions Elliptiques, Springer, Seideberg, 1981. [17] K. Katayama, On the values of Eisenstein series, Tokyo J. Math. 1 (1) (1978) 157–188. [18] A. Levin, Elliptic polylogarithms: An analytic theory, Compos. Math. 106 (3) (1997) 267–282. [19] C. Meyer, Über einige Anwendungen Dedekindscher Summen, J. Reine Angew. Math. 198 (1957) 143–203. [20] M. Mikolás, On certain sums generating the Dedekind sums and their reciprocity laws, Pacific J. Math. 7 (1957) 1167–1178. [21] H. Rademacher, Some remarks on certain generalized Dedekind sums, Acta Arith. 9 (1964) 97–105. [22] H. Rademacher, E. Grosswald, Dedekind Sums, Carus Math. Monogr., vol. 16, Mathematical Association of America, Washington, DC, 1972. [23] R. Sczech, Dedekindsummen mit elliptischen Funktionen, Invent. Math. 76 (1984) 523–551. [24] L. Takács, On generalized Dedekind sums, J. Number Theory 11 (1979) 264–272. [25] A. Weil, Elliptic Functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin/New York, 1976. [26] D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973) 149–172.
An elliptic analogue of generalized Dedekind–Rademacher sums Tomoya Machide Department of Mathematics, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan Received 12 November 2006; revised 16 April 2007 Available online 5 June 2007 Communicated by Matthias Beck
Abstract In this paper we introduce an elliptic analogue of the generalized Dedekind–Rademacher sums which satisfy reciprocity laws. In these sums, Kronecker’s double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of S. Fukuhara and N. Yui concerning the elliptic Apostol– Dedekind sums. We also mention a relation between the generating function of Kronecker’s double series and that of the (Debye) elliptic polylogarithms studied by A. Levin. © 2007 Elsevier Inc. All rights reserved. MSC: 11F20; 11B68 Keywords: Dedekind sums; Generalized Dedekind–Rademacher sums; Reciprocity laws; Elliptic analogue; Kronecker’s double series; Elliptic Apostol–Dedekind sums
1. Introduction After pioneering works by Dedekind [6], much attention has been directed to generalizations of Dedekind sums and their reciprocity laws (see [4,22] for a good picture of previously defined generalizations of Dedekind sums).
E-mail address: [email protected]. 0022-314X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2007.05.004
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
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Let a, b and c be positive integers, and x, y and z real numbers. U. Halbritter [11] and R.R. Hall et al. [12] have defined the generalized Dedekind–Rademacher sums Sm,n xa yb zc by Sm,n
a x
b y
c z
:=
j (mod c)
j +z j +z − x B˜ n b −y , B˜ m a c c
(1)
and have proved their reciprocity laws. Here B˜ m (x) is the mth Bernoulli function defined as follows: We denote the Bernoulli polynomials by Bm (x). ∞
Bm (x) exξ = ξ m−1 . ξ e −1 m! m=0
Bm := Bm (0) is the mth Bernoulli number. Let {x} be the fractional part of a real number x. Then the Bernoulli functions B˜ m (x) are defined by 0 B˜ m (x) = Bm ({x})
(m = 1, x ∈ Z), (otherwise).
As special cases, these sums involve the generalizations of Dedekind sums introduced by T.M. Apostol [1], L. Carlitz [5], M. Mikolás [20], U. Dieter [7], C. Meyer [19], H. Rademacher [21], and L. Takács [24]. Note that so-called Apostol–Dedekind sums [1] are expressed by S1,n 10 b0 0c . Moreover, various elliptic analogues of the classical Dedekind sums have been studied by several authors [2,3,9,10,13,14,23]. R. Sczech introduced the elliptic Dedekind sums as an elliptic analogue of the classical Dedekind sums. A. Bayad [2,3] and S. Egami [9] studied the multiple elliptic Dedekind sums as an elliptic analogue of Zagier’s multiple Dedekind sums [26]. Recently, S. Fukuhara and N. Yui [10] have introduced the elliptic Apostol–Dedekind sums as an elliptic analogue of the classical Apostol–Dedekind sums. However their elliptic analogue did not involve an elliptic analogue of the classical Bernoulli functions. So they have raised the problem to find a true analogue in the elliptic world of the classical Apostol–Dedekind sums which involve these functions. In this paper we answer to the above problem. We define an elliptic analogue of the generalized Dedekind–Rademacher sums (elliptic Dedekind–Rademacher sums) which involve an elliptic analogue of the classical Bernoulli functions. For the definition of this elliptic analogue, we make use of Kronecker’s double series [25]. E.V. Ivashkevich et al. [15, Section 3.1] and K. Katayama [17] noted that these series can be considered as an elliptic generalization of the classical Bernoulli functions. The principal aim of this paper is to establish the reciprocity laws for the elliptic Dedekind– Rademacher sums (Theorem 5). As a corollary, we reproduce the reciprocity laws [12] for the classical generalized Dedekind–Rademacher sums (Theorem 12). We also mention a relation between the generating function of Kronecker’s double series and that of the (Debye) elliptic polylogarithms studied by A. Levin [18] in order to enforce the validity of our elliptic generalization (see Section 2). The paper is organized as follows: In Section 2 we review Kronecker’s double series and their basic properties. A relation between the generating function of Kronecker’s double series and that
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of the (Debye) elliptic polylogarithms is also discussed. In Section 3 we define elliptic Dedekind– Rademacher sums, and obtain their reciprocity laws (Theorem 5). The reciprocity laws [12] for the classical generalized Dedekind–Rademacher sums are reproduced by the degeneration of the elliptic Dedekind–Rademacher sums (Theorem 12). 2. Kronecker’s double series In this section we review Kronecker’s double series and some of their basic properties. We also mention a relation between the generating function of Kronecker’s double series and that of the (Debye) elliptic polylogarithms studied by A. Levin [18]. Let τ be in the upper half-plane. We shall use the following notations: e(x) := e2πix , q := e(τ ), and Jacobi’s theta function 1 1 2 1 1 m+ x+ e τ + m+ 2 2 2 2 m∈Z ∞ 1 x x −e − 1 − e(−x)q m 1 − e(x)q m 1 − q m . = iq 8 e 2 2
θ (x; τ ) :=
(2)
m=1
We note that θ (x; τ ) has the quasi periodicity: θ (x + 1; τ ) = −θ (x; τ ),
τ θ (x + τ ; τ ) = −e − − x θ (x; τ ). 2
(3)
For x, ξ ∈ C \ Z + τ Z, we define the function F (x, ξ ; τ ) as follows: F (x, ξ ; τ ) := where θ (x; τ ) =
∂ ∂x θ (x; τ ).
F (x, ξ ; τ ) = 2πi
θ (0; τ )θ (x + ξ ; τ ) , θ (x; τ )θ (ξ ; τ )
By virtue of (2) we have x+ξ e( x+ξ 2 ) − e(− 2 )
(e( x2 ) − e(− x2 ))(e( ξ2 ) − e(− ξ2 )) ∞ (1 − e(−x − ξ )q m )(1 − e(x + ξ )q m )(1 − q m )2 × ∞ m=1 . m m m m m=1 (1 − e(−x)q )(1 − e(x)q )(1 − e(−ξ )q )(1 − e(ξ )q )
(4)
Thus, for fixed ξ ∈ C \ Z + τ Z, the function F (x, ξ ; τ ) with respect to x is meromorphic with only simple poles on the lattice Z + τ Z. In addition, it satisfies the following properties by (3): F (x + 1, ξ ; τ ) = F (x, ξ ; τ ),
F (x + τ, ξ ; τ ) = e(−ξ )F (x, ξ ; τ ),
Res F (x, ξ ; τ ) = e(−n ξ ) (n, n ∈ Z).
x=n τ +n
(5) (6)
Remark 1. S. Egami [9] and S. Fukuhara and N. Yui [10] used the function ϕ(τ, z) = 1 z 1 2πi F ( 2πi , 2 ; τ ) for their elliptic analogues. A. Bayad also used the following function for his or her elliptic analogue (see [2, p.34]):
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
D[ω1 ,ω2 ] (z; φ) = where τ =
ω1 ω2
1063
φ z φ 1 z Im( ω2 ) F e , ;τ , ω2 ω2 Im τ ω2 ω2
and z, φ ∈ / ω1 Z + ω2 Z.
Let us introduce Kronecker’s double series. Set F (x , x; ξ ; τ ) := e(xξ )F (−x + xτ, ξ ; τ ). We define functions Bm (x , x; τ ) by F (x , x; ξ ; τ ) =
∞ Bm (x , x; τ ) (2πi)m ξ m−1 . m!
(7)
m=0
Namely F (x , x; ξ ; τ ) gives the generating function of Bm (x , x; τ ). It is easily seen from (5) that Bm (x , x; τ ) have the following periodicity: Bm (x + 1, x; τ ) = Bm (x , x + 1; τ ) = Bm (x , x; τ ).
(8)
In view of the expansion (10) and (11) below, Bm (x , x; τ ) are called Kronecker’s double series. We will give an explicit form of the mth Kronecker’s double series Bm (x , x; τ ). The function F (x, ξ ; τ ) has the following expression ([25], [16, p. 446]):
∞
∞
qj qj e(−j ξ ) − e(j ξ ) j e(x) − q e(−x) − q j j =1 j =1 1 1 + +1 |Im x|, |Im ξ | < Im τ . + e(x) − 1 e(ξ ) − 1
F (x, ξ ; τ ) = 2πi
So one can see from (7) that
Bm (x , x; τ ) = m
∞ (x − j )m−1 j =1
−
∞
(x + j )
j =1
m−1
e(−xτ )q j e(−x ) − e(−xτ )q j
e(xτ )q j m−1 e(−x + xτ ) + x e(x ) − e(xτ )q j e(−x + xτ ) − 1
+ Bm (x). (9)
e(−x +xτ ) We remark that B1 (0, 0; τ ) can not be defined since the function e(−x +xτ )−1 becomes infinity if x = x = 0. Originally, Kronecker’s double series have been introduced in the following way. Let x and x be real numbers with −1 < x < 0. Kronecker proved the following equation [25].
F (x , x; ξ ; τ ) = −
e e(n x + nx) , −ξ + n τ + n
(10)
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where
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
e
denotes Eisenstein summation [25], i.e., e
= lim
N
lim
N →∞ N →∞
N
.
n =−N n=−N
So we have ⎧ 1 ⎪ ⎪ ⎨ e e(n x + nx) m! Bm (x , x; τ ) = − ⎪ (2πi)m (n τ + n)m ⎪ ⎩ (n ,n)
(m = 0), (m 1).
(11)
=(0,0)
The series in (10) and (11) were studied by Kronecker [25]. As noted in [15], Kronecker’s double series degenerate into the classical Bernoulli functions. We reconstruct this statement and give its proof for reader’s convenience. Proposition 2. Let x and x be real numbers with x ∈ / Z. Then ⎧ ⎨ 1 1 + e(x ) = i cot(πx ) lim Bm (x , x; τ ) = 2 1 − e(x ) 2 ⎩ τ →i∞ B˜ m (x)
(m = 1, x ∈ Z),
(12)
(otherwise),
especially Re lim Bm (x , x; τ ) = B˜ m (x). τ →i∞
(13)
Here Re w means the real part of a complex number w. Proof. The proposition is trivial if m = 0. Suppose that m 1. One obtains from (8) that Bm x , {x}; τ = Bm (x , x; τ ), where {x} means the real part of x. So it is sufficient to prove the proposition when 0 x < 1. Then we have lim e(xτ )q j = lim e(−xτ )q j = 0 (j ∈ Z1 ),
τ →i∞
τ →i∞
(14)
and ⎧ 1 ⎨ + xτ ) e(−x m−1 lim x = 1 − e(x ) τ →i∞ e(−x + xτ ) − 1 ⎩ 0 Eqs. (9), (14) and (15) induce (12) since B1 = −1/2.
2
(m = 1, x = 0), (otherwise).
(15)
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The generating function F (x , x; ξ ; τ ) of Kronecker’s double series is related to that of the (Debye) elliptic polylogarithms studied by A. Levin [18] in the following sense. Set i∞ e(−xt) Λ(ξ ; −2πix) := 2πi ξ e(−t)−1 dt. The Debye polylogarithms Λm (ξ ) are defined by [18]: i∞ 2πi ξ
∞
e(−xt) dt = Λm+1 (ξ )(2πi)m+1 (−x)m . e(−t) − 1
(16)
m=0
Namely Λ(ξ ; −2πix) gives the generating function of Λm (ξ ). Since ∞
Bm (x) e(xξ ) ∂ Λ(−ξ ; −2πix) = 2πi = (2πi)m ξ m−1 , ∂ξ e(ξ ) − 1 m!
(17)
m=0
∂ ∂ξ Λ(−ξ ; −2πix)
becomes the generating function of the classical Bernoulli functions (polynomials). On the other hand, A. Levin has shown that the modified generating function Λ(ξ, τ ; x , x) of the (Debye) elliptic polylogarithms satisfies the following equation (see [18, Proposition 3.1]): ∂ Λ(ξ, τ ; −2πix , −2πix) = F (x , x; ξ ; τ ). ∂ξ
(18)
Comparing (17) and (18), we can consider that F (x , x; ξ ; τ ) is an elliptic analogue of the generating function of the classical Bernoulli functions (polynomials). In this point of view Bm (x , x; τ ) become an elliptic analogue of the classical Bernoulli functions since F (x , x; ξ ; τ ) is the generating function of Bm (x , x; τ ). Remark 3. Let Lim (z) denote the Euler polylogarithms, i.e., Lim (z) :=
∞ zn . nm n=1
One of the basic properties of the Euler polylogarithms is the iterated integral representation. We can formally consider that (16) is based on it in the following trick: 2πi
e(xξ ) dξ = 2πi e(ξ ) − 1
e(xξ )Li0 e(−ξ ) dξ
= −e(xξ )Li1 e(−ξ ) + 2πix = ···
= −e(xξ ) =−
∞
m=0
∞
k=0
e(xξ )Li1 e(−ξ ) dξ
Lim+1 e(−ξ ) x m
m=0 m
Lik+1
(2πi)−k ξ m−k e(−ξ ) (2πix)m (m − k)!
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T. Machide / Journal of Number Theory 128 (2008) 1060–1073
=
∞
Λm+1 (−ξ )(2πi)m+1 (−x)m .
m=0
Here we assume that the series is formal, and use [18, Proposition 1.1(a)] for the last equation. We wish that this formal calculation gives us a key to an elliptic analogue of the classical Euler polylogarithms and their iterated integral representation. 3. Elliptic Dedekind–Rademacher sums In this section we define elliptic Dedekind–Rademacher sums, and obtain their reciprocity laws. We also reproduce the reciprocity laws [12] for the classical generalized Dedekind– Rademacher sums by the degeneration of the elliptic Dedekind–Rademacher sums. To simplify the notations, we begin with the following lemma. Lemma 4. Let a, b be positive integers, and x, y real numbers. Let a, b be the greatest common divisor of a and b. Then the following statements are equivalent: (i) ay − bx ∈ / a, bZ, (ii) a j +y − x ∈ / Z (j = 0, . . . , b − 1), b (iii) ( xa + a1 Z) ∩ ( yb + b1 Z) = ∅. Proof. We can prove this lemma by simple calculation, so we omit the proof.
2
Let a, a , b, b , c, c be positive integers, and x, x , y, y , z, z real numbers. Suppose that / a , c Z, a z − c x ∈
b z − c y ∈ / b , c Z.
(19)
c ) := ((a , a), (b , b), (c , c)), ( x , y , z ) := ((x , x), (y , y), (z , z)). Set ( a , b, We define the elliptic Dedekind–Rademacher sums as follows: τ Sm,n
a x
b y
1 c := z c
Bm a
j (mod c) j (mod c )
j
a + z j +z − x; τ − x , a c c a
b j + z j +z − y; τ . × Bn b − y , b c c b
(20)
It follows from (19) and Lemma 4(ii) that B1 (0, 0; τ ) does not appear in (20). So this sum is well-defined. If m = 1 and n = 1, or if az − cx ∈ / a, cZ, bz − cy ∈ / b, cZ, then we see from (12) and Lemma 4(ii) that lim
τ →i∞
τ Sm,n
a x
b y
c z
= Sm,n
a x
b y
c z
.
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
1067
a b c τ except the above case is troubleHowever to obtain the explicit forms of limτ →i∞ Sm,n x y z some because the following formula forces us to consider many cases. i 1 ˜ B1 (x , 0; τ ) = cot(πx ) = 0 = B1 (0) x ∈ / +Z . 2 2 In addition the explicit forms are not necessary for reproducing the reciprocity laws of the classical generalized Dedekind–Rademacher sums from the elliptic ones; it is needed that the real part a b c τ of limτ →i∞ Sm,n (see the proof of Theorem 12). So we omit their explicit forms. x y z Our principal aim of this paper is to give the reciprocity laws for the elliptic Dedekind– Rademacher sums. As R.R. Hall et al. have noted in [12], these laws mix various pairs of indices (m, n), so they are conveniently stated in terms of the generating function ⎛ ⎞ m−1 n−1 ∞ a b c X Y 1 τ a b c τ⎝ ⎠ S . S x y z := m!n! m,n x y z a b X Y Z m,n=0 Here X and Y are non-zero variables and the variable Z, which does not appear explicitly on the right-hand side of the definition, is defined by −X − Y . Because of (7) and (20), the generating function of the elliptic Dedekind–Rademacher sums satisfies ⎛
⎞ a b c (2πi)2 Sτ ⎝ x y z ⎠ 2πiX 2πiY 2πiZ X a j + z 1 j +z − x; ; τ = F a − x , a c c c a a j (mod c) j (mod c )
Y b j + z j +z × F b − y; ; τ . − y , b c c b b
(21)
The reciprocity laws for the elliptic Dedekind–Rademacher sums are the following: Theorem 5. Let X, Y, Z be variables with X + Y + Z = 0, and a, a , b, b , c, c positive integers, and x, y, z real numbers. Let x , y and z be real numbers such that a y − b x ∈ / a , b Z,
a z − c x ∈ / a , c Z,
b z − c y ∈ / b , c Z.
c ) := ((a , a), (b , b), (c , c)), ( Set ( a , b, x , y , z ) := ((x , x), (y , y), (z , z)). Then we have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a b c b c a c a b Sτ ⎝ x y z ⎠ + Sτ ⎝ y z x ⎠ + Sτ ⎝ z x y ⎠ = 0. X Y Z Y Z X Z X Y Here the left is over the three terms obtained by cyclic permutation of the columns of
(22)
(23) a
b c x y z . XY Z
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T. Machide / Journal of Number Theory 128 (2008) 1060–1073
The strategy of our proof, which is the same method employed by A. Bayad [3] and S. Egami [9], is to apply the residue theorem. Proof of Theorem 5. Set wa = −
x x + τ, a a
wb = −
y y + τ, b b
wc = −
z z + τ. c c
We define the function G(α) as follows:
X a Y b Z c G(α) = F a (α − wa ), ; τ F b (α − wb ), ; τ F c (α − wc ), ; τ . a a b b c c
It follows from (5) and X + Y + Z = 0 that G(α + 1) = G(α + τ ) = G(α), namely G(α) is a doubly periodic function. Since the function F (x, ξ ; τ ) with respect to x has the simple poles on the lattice Z + τ Z, the poles of G(α) are on the following lattices: wa +
1 τ Z + Z, a a
wb +
1 τ Z + Z, b b
wc +
1 τ Z + Z. c c
By virtue of the condition (22) and Lemma 4(iii), we have wi +
1 τ 1 τ Z ∩ w Z =∅ Z + + Z + j i i j j
i, j = a, b, c (i = j ) .
This implies that the order of every pole of G(α) is equal to one. Because the function G(α) is doubly periodic, the sum of the residues of G(α) at its poles in any period parallelogram is equal to zero. So it follows that
0=
s=a,b,c j (mod s) j (mod s )
Res
α=ws −j /s +(j/s)τ
G(α).
Consider the case s = c. By virtue of (5) and (6), we have
Res
j (mod c) j (mod c )
1 = c
α=wc −j /c +(j/c)τ
j (mod c) j (mod c )
G(α)
j +z a X a −Z j +z F − a −x τ, ; τ e j −x + a c c c a a a
j +z b Y b j +z ×F − b −y τ, ; τ −y + b c c b b b 1 xX yY zZ X a j +z j +z = e + + − x; ; τ F a − x ,a c a b c c c a a j (mod c) j (mod c )
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
j + z j +z Y b × F b − y , b − y; ; τ c c b b ⎛ a b zZ xX yY + + Sτ ⎝ x = (2πi)2 e y a b c 2πiX 2πiY
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⎞ c z ⎠ . 2πiZ
Since one can calculate the other cases in the same way, we obtain (23).
2
Remark 6. As R.R. Hall et al. have remarked in [12, Section 4], the left-hand side of (23) is invariant under all permutations of the columns since Sτ is symmetric in its first two columns. We can also see this fact since the function G(α) is invariant under all permutations of the a b c columns of x y z . XY Z
Remark 7. Let k , k be positive integers, and let ( ak , b k , c k ) = (a , ka), (b , kb), (c , kc) , a k , b k , c k = (k a , a), (k b , b), (k c , c) . Then we have τ Sm,n
τ Sm,n
a k x
b k y
a k
b k
c k
x
y
z
c k z
1
τ
k = kSm,n
k τ = Sm,n
a x a x
c , z b c . y z
b y
These equations allow us to reduce (23) to the case that the integers a, b and c (respectively a , b and c ) have no common factor. We introduce the generating function of the classical generalized Dedekind–Rademacher sums: ⎛ ⎞ m−1 n−1 ∞ a b c Y 1 X a b c S ⎝ x y z ⎠ := Sm,n . x y z m!n! a b X Y Z m,n=0 To reproduce the reciprocity laws for these sums from (23) (Theorem 12 bellow), the cotangent sum studied by U. Dieter [8] is introduced: j +z 1 j +z −x cot π b −y , c(a, b, c; x, y, z) := cot π a j (mod c) c c c j +z where j (mod c) means excluding j such that a j +z c − x ∈ Z or b c − y ∈ Z. We need the following lemmas and proposition for Theorem 12:
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Lemma 8. Let a, b, c be positive integers, and x, y, z real numbers. Then the following statements are equivalent: (i) (x, y, z) ∈ (a, b, c)R + Z3 . (ii) There is an integer j0 such that a j0c+z − x, b j0c+z − y ∈ Z. Proof. Suppose that (i) holds. Then there are integers j0 , m, n and a real number r such that (x, y, z) = (a, b, c)r + (−m, −n, −j0 ). So we have r=
m + x n + y j0 + z = = . a b c
Therefore one gets a
j0 + z − x = m, c
b
j0 + z − y = n. c
Hence (ii) follows. The reverse is also proved in a similar way.
2
Remark 9. Since the statement (i) is invariant under all permutations of the columns of the statement (ii) is too. We shall implicitly use this fact in the proof of Theorem 12.
a b c xyz
,
Lemma 10. Let a, b and c be positive integers with no common factor, and let m be the least c c common multiple of a,c and b,c . Then we have m = c. Proof. For any prime number p, we define a nonnegative integer p (a) by factorization of prime numbers: a=
p p (a) ,
where the product is extended over all prime numbers p. Then we have c = p p (c)−min{ p (a), p (c)} , a, c
c = p p (c)−min{ p (b), p (c)} . b, c
Thus m=
p max{ p (c)−min{ p (a), p (c)}, p (c)−min{ p (b), p (c)}} .
Let p be a prime number. It is seen that max p (c) − min p (a), p (c) , p (c) − min p (b), p (c) = p (c) − min p (a), p (b), p (c) . Since a, b, c have no common factor, min{ p (a), p (b), p (c)} is equal to zero. So we get m = c. 2
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
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Proposition 11. Let a, a , b, b , c, c be positive integers, and x, x , y, y , z, z real numbers. Suppose that a, b, c have no common factor, and that a z − c x ∈ / a , c Z,
b z − c y ∈ / b , c Z.
(24)
c ) := ((a , a), (b , b), (c , c)), ( Set ( a , b, x , y , z ) := ((x , x), (y , y), (z , z)). Let Re w denote the real part of a complex number w. Then we have Re lim
τ →i∞
τ Sm,n
b y
a x
c z
⎧ a b c ⎪ ⎪ S 1,1 x y z ⎪ ⎪ ⎪ ⎨ = − 14 c(a , b , c ; x , y , z ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Sm,n a b c x y z
(m = n = 1, (x, y, z) ∈ (a, b, c)R + Z3 ),
(25)
(otherwise).
Proof. We remark that Sm,n xa yb zc are real numbers since a, b, c, x, y, z are real numbers, and that cot w is a real number if w is too. Suppose that m = n = 1 and (x, y, z) ∈ (a, b, c)R + Z3 . By Lemma 8, there is an integer j0 such that a j0c+z − x, b j0c+z − y ∈ Z. Let j be an integer. The condition a
j +z j +z − x, b −y ∈Z c c
(26)
is equivalent to j ≡ j0
c j ≡ j0 mod . b, c
c mod , a, c
By virtue of Lemma 10, (26) is also equivalent to j ≡ j0
(mod c).
Thus we see from B˜ 1 (0) = 0 and the above remark that τ Re lim S1,1 τ →i∞
= S1,1
a x
a x
b y
b y
c z
c z
−
1 4c
j (mod c )
j + z j +z cot π b . cot π a − x − y c c
The second term in the last formula is equal to − 14 c(a , b , c ; x , y , z ) by virtue of the condition (24) and Lemma 4(ii), so this case is proved. The other case can be also obtained by attention to the above remark. 2 Now we are in a position to reproduce the reciprocity laws [12] of the classical generalized Dedekind–Rademacher sums.
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T. Machide / Journal of Number Theory 128 (2008) 1060–1073
Theorem 12. (See [12].) Let X, Y, Z be variables with X + Y + Z = 0, and a, b, c positive integers with no common factor, and x, y, z real numbers. Then we have ⎛
a b S⎝ x y X Y 1 −4 = 0
⎞ ⎛ c b z ⎠ + S⎝ y Z Y
c z Z
⎞ ⎛ a c x ⎠ + S⎝ z X Z
a x X
⎞ b y⎠ Y
((x, y, z) ∈ (a, b, c)R + Z3 ), (otherwise).
(27)
Proof. We may assume that X, Y and Z are real numbers. Suppose that (x, y, z) ∈ (a, b, c)R + Z3 . When τ tends to i∞ in (23), an equation is obtained. By directing our attention to the real part of the equation, we can get by virtue of (25) ⎛
a S⎝ x X −
b y Y
⎞ ⎛ c b z ⎠ + S⎝ y Z Y
c z Z
⎞ ⎛ a c x ⎠ + S⎝ z X Z
a x X
⎞ b y⎠ Y
1 c(a , b , c ; x , y , z ) + c(b , c , a ; y , z , x ) + c(c , a , b ; z , x , y ) = 0, 4
where x , y and z are real numbers such that / a , b Z, a y − b x ∈
a z − c x ∈ / a , c Z,
b z − c y ∈ / b , c Z.
(28)
In [8, Theorem 2.3] the reciprocity law of c(a , b , c ; x , y , z ) has been given: c(a , b , c ; x , y , z ) + c(b , c , a ; y , z , x ) + c(c , a , b ; z , x , y ) = −1.
(29)
2
Thus this case is proved. The other case is also proved the same way. Remark 13. Let k be a positive integer. As in Remark 7, one gets Sm,n
ka x
kb y
kc z
= kSm,n
a x
b y
c z
.
This allows us to remove the condition in Theorem 12 that the integers a, b and c have no common factor. Remark 14. The theorem in [12, Section 4] is incorrect, i.e., the right-hand side of (7) in it should be multiplied by −1. The cause is that the proposition in [12, Section 3] has a small error. Remark 15. In [8, Theorem 2.3] it is supposed that a , b , c are pairwise relatively prime. But, under the condition (28), this supposition is not necessary. The reasons are that (29) is obtained by [8, Eq. (2,5)] and [8, Lemma 2.2] (additive formula for cotangent function), and that [8, Eq. (2,5)] holds for any positive integers a, b, c.
T. Machide / Journal of Number Theory 128 (2008) 1060–1073
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Acknowledgments The author thanks Professors Youichi Shibukawa and Hiroshi Yamashita for their encouragement. He expresses his gratitude to Professor Kimio Ueno for suggesting him the equation (23), and to Professor Michitomo Nishizawa for his valuable advice. He is also grateful to Professor Yumiko Hironaka for her comments on the preliminary version, and to the referees for improving this paper. References [1] T.M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950) 147–157. [2] A. Bayad, Sommes de Dedekind elliptiques et formes de Jacobi, Ann. Inst. Fourier (Grenoble) 51 (2001) 29–42. [3] A. Bayad, Sommes elliptiques multiples d’Apostol–Dedekind–Zagier, C. R. Math. Acad. Sci. Paris, Ser. I 339 (2004) 457–462. [4] M. Beck, Dedekind cotangent sums, Acta Arith. 109 (2003) 109–130. [5] L. Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953) 513–522. [6] R. Dedekind, Erläuterungen zu den Fragmenten XXVIII, in: B. Riemann’s Gesammelte mathematische Werke, second ed., Teubner, Leipzig, 1892, pp. 466–478. [7] U. Dieter, Das Verhalten der kleinschen Funktionen log σg,h (w1 , w2 ) gegenüber Modultransformationen und verallgemeinerte Dedekindsche Summen, J. Reine Angew. Math. 201 (1959) 37–70. [8] U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Theory 18 (1984) 289–305. [9] S. Egami, An elliptic analogue of multiple Dedekind sums, Compos. Math. 99 (1995) 99–103. [10] S. Fukuhara, N. Yui, Elliptic Apostol sums and their reciprocity laws, Trans. Amer. Math. Soc. 356 (10) (2004) 4237–4245. [11] U. Halbritter, Some new reciprocity formulas for generalized Dedekind sums, Results Math. 8 (1985) 21–46. [12] R.R. Hall, J.C. Wilson, D. Zagier, Reciprocity formulae for general Dedekind–Rademacher sums, Acta Arith. 73 (1995) 389–396. [13] H. Ito, On a property of elliptic Dedekind sums, J. Number Theory 27 (1987) 17–21. [14] H. Ito, A density result for elliptic Dedekind sums, Acta Arith. 112 (2004) 199–208. [15] E.V. Ivashkevich, N.Sh. Izmailianand, C. Hu, Kronecker’s double series and exact asymptotic expansions for free models of statistical mechanics on torus, J. Phys. A: Math. Gen. 35 (2002) 5543–5561. [16] C. Jordan, Fonctions Elliptiques, Springer, Seideberg, 1981. [17] K. Katayama, On the values of Eisenstein series, Tokyo J. Math. 1 (1) (1978) 157–188. [18] A. Levin, Elliptic polylogarithms: An analytic theory, Compos. Math. 106 (3) (1997) 267–282. [19] C. Meyer, Über einige Anwendungen Dedekindscher Summen, J. Reine Angew. Math. 198 (1957) 143–203. [20] M. Mikolás, On certain sums generating the Dedekind sums and their reciprocity laws, Pacific J. Math. 7 (1957) 1167–1178. [21] H. Rademacher, Some remarks on certain generalized Dedekind sums, Acta Arith. 9 (1964) 97–105. [22] H. Rademacher, E. Grosswald, Dedekind Sums, Carus Math. Monogr., vol. 16, Mathematical Association of America, Washington, DC, 1972. [23] R. Sczech, Dedekindsummen mit elliptischen Funktionen, Invent. Math. 76 (1984) 523–551. [24] L. Takács, On generalized Dedekind sums, J. Number Theory 11 (1979) 264–272. [25] A. Weil, Elliptic Functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin/New York, 1976. [26] D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973) 149–172.