ELLIPTIC APOSTOL SUMS AND THEIR RECIPROCITY LAWS 1. Introduction Let p and q be ...

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ArticleelectronicallypublishedonMay10,2004
ELLIPTIC APOSTOL SUMS AND THEIR RECIPROCITY LAWS
SHINJIFUKUHARAANDNORIKOYUI
Abstract. Weintroduce anellipticanalogue oftheApostolsums,whichwe
callellipticApostolsums. Thesesumsarede(cid:12)nedbymeansofcertainelliptic
functions withacomplex parameter (cid:28) havingpositiveimaginarypart. When
(cid:28) !i1,these ellipticApostolsumsrepresentthewell-knownApostolgener-
alizedDedekindsums. AlsotheseellipticApostol sumsaremodularformsin
thevariable(cid:28). Weobtainareciprocitylawforthesesums,whichgivesriseto
new relationsbetween certainmodularforms(ofonevariable).
1. Introduction
Let p and q be relatively prime positive integers. First let us recall the de(cid:12)ni-
tion of Apostol’s generalized Dedekind sums s (q;p) from [1], which we will call
k
throughout this paper the Apostol sums: for a positive integer k,
pX(cid:0)1
(cid:22) (cid:22)q
s (q;p):= B(cid:22) ( ):
k k
p p
(cid:22)=1
Here B(cid:22) (x) denotes the k-th Bernoulli function. That is, B(cid:22) (x) is given by the
k k
Fourier expansion
X+1 e2(cid:25)imx
B(cid:22) (x):=(cid:0)k! :
k (2(cid:25)im)k
m=(cid:0)1
m6=0
Itiswellknownthatfor0(cid:20)x<1,B(cid:22) (x)reducestothek-thBernoullipolynomial.
k
We denote by B the k-th Bernoulli number.
k
If k is even, s (q;p)is trivial. If k is odd, a reciprocitylaw for the Apostol sums
k
was obtained by Apostol [1, p. 149]:
p2n(cid:0)2s2n(cid:0)81(q;p)+q2n(cid:0)2s2n(cid:0)1(p;q) 9
(1.1) = 1 <Xn (2n(cid:0)1)!B2jB2n(cid:0)2jp2jq2n(cid:0)2j + (2n(cid:0)1)B2n= (n>1):
pq : (2j)!(2n(cid:0)2j)! 2n ;
j=0
ReceivedbytheeditorsSeptember 30,2002and,inrevisedform,August7,2003.
2000 Mathematics Subject Classi(cid:12)cation. Primary11F20;Secondary 33E05,11F11.
Keywordsandphrases. GeneralizedDedekindsums(Apostolsums),ellipticfunctions,elliptic
Apostolsums,modularforms,reciprocitylaws.
The(cid:12)rstauthorwaspartiallysupportedbyGrant-in-AidforScienti(cid:12)cResearch(C)12640089,
MinistryofEducation,Sciences, SportsandCulture,Japan.
Thesecondauthor waspartiallysupportedbyaResearchGrantfromNSERC,Canada.
(cid:13)c2004 American Mathematical Society
4237
4238 SHINJIFUKUHARAANDNORIKOYUI
Thereareanumberofdi(cid:11)erentproofsfortheApostol’sreciprocitylaw(1.1)(cf.
[1, 2, 6, 16]). However, we believe that our approach [12] that makes use of the
following trigonometric identity (1.2) provides an elegant proof.
Forrelativelyprimepositiveintegerspandq,andforz 2C,wehave([7,Theorem
2.4], [11, 12])
pX(cid:0)1 Xq(cid:0)1
1 (cid:22)q(cid:25) (cid:22)(cid:25) 1 (cid:22)p(cid:25) (cid:22)(cid:25)
cot( )cot(z+ )+ cot( )cot(z+ )
p p p q q q
(1.2) (cid:22)=1 (cid:22)=1
1
=(cid:0)cot(pz)cot(qz)+ csc2(z)(cid:0)1:
pq
We are grateful to the referee for pointing out that this trigonometric sum is
nothing but a special case of a reciprocity formula for the so-called Dedekind{
Rademacher sums (which are Dedekind sums with two congruence parameters (z
and 0 in (1.2))). Introduction of these congruence parameters into the classical
Dedekind sums may be viewed as unifying all Apostol sums into a single sum. In
fact,weexpandbothsidesof (1.2)intoLaurentseriesinz andinadditionweapply
(cid:12)niteFouriertransformstothem. Thencomparingthecoe(cid:14)cientsofz2n(cid:0)2 ofboth
sides, we arriveat the reciprocitylaw for the Apostol sums (1.1) in the case n>1.
For n = 1, this formula yields the reciprocity law for the classical Dedekind sums
[18, p. 18]:
1 Xp(cid:0)1 (cid:22)q(cid:25) (cid:22)(cid:25) 1 Xq(cid:0)1 (cid:22)p(cid:25) (cid:22)(cid:25) p2+q2+1(cid:0)3pq
(1.3) cot( )cot( )+ cot( )cot( )= :
p p p q q q 3pq
(cid:22)=1 (cid:22)=1
Inviewofthefactthat(1.2)givesrisetothe Apostolreciprocitylaw (1.1),wemay
regardthe identity (1.2) as a \generating function" for the reciprocity law (1.1).
In this paper,we willde(cid:12)ne anelliptic analogueofthe Apostolsums (De(cid:12)nition
2.1),usingWeierstrass}and(cid:16)-functions. Itisde(cid:12)nedasa(cid:12)nitesumoftermswhere
everyterm is the productoftwo functions relatedto elliptic functions (Weierstrass
} and (cid:16)-functions). Its very de(cid:12)nition depends in an essentialwayon the parity of
two relatively prime positive integers p and q (see Remark 2.1, De(cid:12)nition 2.1 and
Remark2.2). Thenwewillinvestigatepropertiesofthese sums. Firstweobtainan
elliptic analogue of the identity (1.2) (Theorem 2.1). (This new identity will serve
as a \generating function" for a reciprocity law.) Then we establish a reciprocity
law for these elliptic analogues (Theorem 2.2).
Inthemid80’s,ellipticanaloguesoftheclassicalDedekindsums(so-calledellip-
tic Dedekind sums) were studied by several authors (see [19, 14]). Earlier, higher-
dimensional(ormultiple)DedekindsumswerediscussedinthepaperofZagier[20].
As anelliptic analogueof Zagier’smultiple Dedekind sums, higher-dimensional(or
multiple) elliptic Dedekind sums wereintroducedandinvestigated,ratherrecently,
byEgami[8],andalsobyBayad[3]. Unfortunatelythereciprocitylawsobtainedby
Egami ([8, Theorem 1]) and Bayad ([3, Theorem 2.2]) for elliptic Dedekind sums
were incorrect, i.e., the right-hand sides of their formulae should be divided by
a (cid:1)(cid:1)(cid:1)a . (In the proof of Lemma 3.1 below (see also Remark 3.1), the reader will
0 n
(cid:12)nd a reason why their errors occurred.)
Underthiscircumstance,itwouldbedesirabletoestablishourresults(Theorem
2.1 and Theorem 2.2) independently from the papers of Egami and Bayad, and
indeed, this is one of our purposes of the present paper. The strategy of our proof
ELLIPTIC APOSTOL SUMS AND THEIR RECIPROCITY LAWS 4239
for Theorem 2.1 is to apply Liouville’s theorem (that any bounded entire function
onCmustbe constant),anditdi(cid:11)ers fromthe methodemployedbyEgami[8]and
Bayad[3] (who used the residue theorem).
We also calculate the limit of the elliptic reciprocity laws (2.3) and (2.4) at i1.
We show that the limit formula indeed coincides with the reciprocity law (1.2) for
Apostol sums. (This fact may justify our terminology \elliptic Apostol sums".)
Finallywewillbrie(cid:13)yinterpretelliptic Apostolsums intermsofmodularforms,
andobservethatthey giverisetosomenew relationsbetweenmodularforms. (We
refer the reader who are interested in modular forms aspects of elliptic Dedekind
sums to the paper of Bayad[3], where the author has carriedout investigations on
elliptic Dedekind sums and their reciprocity laws in terms of Jacobi forms.)
We should mention that the elliptic analogues of the Apostol sums treated in
thispaperarestrictlyspeakinganaloguesandgeneralizationsofthecotangentsums
(1.2), which involve trigonometric sums (i.e., degenerate elliptic functions). It is
still an open problem to (cid:12)nd true analogues in the elliptic world of the classical
Apostol sums which involve Bernoulli polynomials.
2. The main results
First we recall the three elliptic functions which play central roles in our dis-
cussion. Fix a complex number (cid:28) with positive imaginary part. The Weierstrass
}-function (introduced by Weierstrass) is a meromorphic function of a complex
variable z given as follows:
(cid:18) (cid:19)
X
1 1 1
}((cid:28);z):= + (cid:0) :
z2 (z(cid:0)(cid:13))2 (cid:13)2
(cid:13)22(cid:25)i(Z(cid:28)+Z)
(cid:13)6=0
Associated to }((cid:28);z), the function e ((cid:28)) is de(cid:12)ned by putting z =(cid:25)i in }((cid:28);z):
1
e ((cid:28)):=}((cid:28);(cid:25)i):
1
It is known that the }-function has the following expansion at z =0:
X1
1
}((cid:28);z)= + (2n(cid:0)1)G ((cid:28))z2n(cid:0)2;
z2 2n
n=2
where G ((cid:28)) is the Eisenstein series of weight 2n, namely,
2n
X
1
G ((cid:28)):= :
2n (cid:13)2n
(cid:13)22(cid:25)i(Z(cid:28)+Z)
(cid:13)6=0
Let(cid:0)(n)and(cid:0) (n)bethe congruencesubgroupsofSL (Z)de(cid:12)nedrespectivelyby
0 2
(cid:8)(cid:0) (cid:1) (cid:9)
(cid:0)(n):= a b 2SL (Z) j a(cid:17)d(cid:17)1 (n); b(cid:17)c(cid:17)0 (n) ;
(cid:8)(cid:0)c d(cid:1) 2 (cid:9)
(cid:0) (n):= a b 2SL (Z) j c(cid:17)0 (n) :
0 c d 2
It is well known that G ((cid:28)) (n>1) and e ((cid:28)) are modular forms for SL (Z) and
2n 1 2
(cid:0) (2) of weight 2n and 2, respectively.
0
4240 SHINJIFUKUHARAANDNORIKOYUI
The Weierstrass (cid:16)-function (also introduced by Weierstrass) is a meromorphic
function of a complex variable z given as follows:
(cid:18) (cid:19)
X
1 1 1 z
(cid:16)((cid:28);z):= + + + :
z z(cid:0)(cid:13) (cid:13) (cid:13)2
(cid:13)22(cid:25)i(Z(cid:28)+Z)
(cid:13)6=0
Then it is easy to see that the two functions }((cid:28);z) and (cid:16)((cid:28);z) have the following
relation:
@(cid:16)((cid:28);z)=@z =(cid:0)}((cid:28);z):
We see that the }-function }((cid:28);z) is periodic with respect to 2(cid:25)i(Z(cid:28) +Z). The
(cid:16)-function (cid:16)((cid:28);z) is subject to the following identities:
(cid:16)((cid:28);z+2(cid:25)i)=(cid:16)((cid:28);z)+(cid:17) ((cid:28)); (cid:16)((cid:28);z+2(cid:25)i(cid:28))=(cid:16)((cid:28);z)+(cid:17) ((cid:28));
1 2
where (cid:17) ((cid:28)) and (cid:17) ((cid:28)) are independent of z. This implies that the function (cid:16)((cid:28);z)
1 2
is not periodic but quasi-periodic with respect to 2(cid:25)i(Z(cid:28) +Z).
Now we de(cid:12)ne the function ’((cid:28);z) in a complex variable z as follows:
p
’((cid:28);z)= }((cid:28);z)(cid:0)e ((cid:28));
1
(2.1) 1 (cid:0) (cid:1)
’((cid:28);z)= +O 1 (z !0):
z
It is therefore easy to see that the function ’((cid:28);z) is periodic with respect to
2(cid:25)i(2Z(cid:28) +Z) and that it has the following Laurent expansion at z =0:
X1
1
(2.2) ’((cid:28);z)= H ((cid:28))z2n;
2n
z
n=0
where H ((cid:28)) is determined recursively using (2.1):
2n
1
H ((cid:28))=1; H ((cid:28))=(cid:0) e ((cid:28));
0 2 2 1
1nX(cid:0)1 2n(cid:0)1
H2n((cid:28))=(cid:0) H2j((cid:28))H2n(cid:0)2j((cid:28))+ G2n((cid:28)) (n>1):
2 2
j=1
Clearly H ((cid:28)) is a polynomial in e ((cid:28)), G ((cid:28)), G ((cid:28)), ..., G ((cid:28)) with rational
2n 1 4 6 2n
coe(cid:14)cients. Furthermore, we see that H ((cid:28)) is a modular form of weight 2n for
2n
(cid:0) (2). Also ’((cid:28);z) satis(cid:12)es the following transformation formulae:
0
’((cid:28);(cid:0)z)=(cid:0)’((cid:28);z)
and
’((cid:28);z+2(cid:25)i(cid:28))=(cid:0)’((cid:28);z):
For a (cid:12)xed (cid:28) with positive imaginary part, to ease the notations, we will simply
write}(z),(cid:16)(z), ’(z), e , (cid:17) , (cid:17) , G andH dropping the variable(cid:28). We denote
1 1 2 2n 2n
byf0(z)the(cid:12)rstderivativeofafunctionf(z)withrespecttoz,andmoregenerally,
by f(k)(z) the k-th derivative of a function f(z).
ELLIPTIC APOSTOL SUMS AND THEIR RECIPROCITY LAWS 4241
Retaining these notations we can now formulate our results:
Theorem 2.1. Let p and q be relatively prime positive integers, and z 2C.
(1) Suppose that p+q (cid:17)1 (2). Then we have
pX(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
((cid:0)1)(cid:21)’( )’(z+ )
p p p
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
Xq(cid:0)1
(2.3) + 1 ((cid:0)1)(cid:21)’(2(cid:25)ip((cid:21)(cid:28) +(cid:22)))’(z+ 2(cid:25)i((cid:21)(cid:28) +(cid:22)))
q q q
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
1
=(cid:0)’(pz)’(qz)(cid:0) ’0(z):
pq
(2) Suppose that p+q (cid:17)0 (2). Then we have
Xp(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
((cid:0)1)(cid:21)’( )(cid:16)(z+ )
p p p
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
Xq(cid:0)1
(2.4) + 1 ((cid:0)1)(cid:21)’(2(cid:25)ip((cid:21)(cid:28) +(cid:22)))(cid:16)(z+ 2(cid:25)i((cid:21)(cid:28) +(cid:22)))
q q q
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
1
=(cid:0)’(pz)’(qz)(cid:0) (cid:16)0(z)+C(p;q;(cid:28));
pq
where C(p;q;(cid:28)) is a constant with respect to z.
Remark 2.1. InTheorem2.1, the parityconditionsonthe relativelyprime positive
integers p and q turn out to be rather essential. As we see, the only di(cid:11)erence in
two identities lies in the second factor on the right-hand side. Indeed, in (2.3), the
secondfactor is the derivativeofthe ’-function, while in(2.4), the secondfactor is
the derivative ofthe(cid:16)-function. If we dropthe parityconditions, the identity (2.3)
(resp. (2.4)) no longer holds true.
In fact, if the opposite parity for p+q is taken in (2.3) (resp. (2.4)), the left-
hand side and the right-hand side of the equation end up having di(cid:11)erent periods,
namely,2(cid:25)i(Z(cid:28)+Z) and2(cid:25)i(2Z(cid:28)+Z), respectively. This implies that the identity
does not hold under this parity. For instance, (2.4) is not valid if we take p = 2
and q =1, while (2.3) does not hold for p=q =1. Therefore, the parity condition
on p+q is rather essential for the validity of the identity.
Next we de(cid:12)ne the \elliptic Apostol sums" which we may regard as elliptic
analogues of the Apostol sums.
We need two di(cid:11)erent types of elliptic Apostol sums depending on the parity
conditions of relatively prime positive integers p and q.
De(cid:12)nition 2.1. Let p and q be relatively prime positive integers, and let k be a
positive integer.
4242 SHINJIFUKUHARAANDNORIKOYUI
Depending on the parity of p+q, we de(cid:12)ne the sum s~ (q;p;(cid:28)) as follows:
k
s~ (q;p;(cid:28))
k
8 P
><(cid:0) k p(cid:0)1 ((cid:0)1)(cid:21)’(2(cid:25)iq((cid:21)(cid:28)+(cid:22)))’(k(cid:0)1)(2(cid:25)i((cid:21)(cid:28)+(cid:22))) (if p+q (cid:17)1 (2));
pk (cid:21);(cid:22)=0 p p
:= P((cid:21);(cid:22))6=(0;0)
>:(cid:0) k p(cid:0)1 ((cid:0)1)(cid:21)’(2(cid:25)iq((cid:21)(cid:28)+(cid:22)))(cid:16)(k(cid:0)1)(2(cid:25)i((cid:21)(cid:28)+(cid:22))) (if p+q (cid:17)0 (2)):
pk (cid:21);(cid:22)=0 p p
((cid:21);(cid:22))6=(0;0)
We call s~ (q;p;(cid:28)) the elliptic Apostol sum.
k
Remark 2.2. Weadoptthenotations~ (q;p;(cid:28))fortheellipticApostolsum,although
k
this notation really stands for two di(cid:11)erent sums. We are sticking to this notation
mainly because of the following two reasons: (1) we wish to emphasize the fact
that our sum is indeed an elliptic analogue of the classical Apostol sums (which
was denoted by s (p;q)), and (2) even though there is no single de(cid:12)ning formula
k
for elliptic Apostol sums which represent both odd and even parities for p+q , we
believe there is no danger of confusing the two cases. As we see, the di(cid:11)erence is
the second factor in each sum, namely, the (k(cid:0)1)-th derivative of the ’-function
in the (cid:12)rst case, and the (k(cid:0)1)-th derivative of the (cid:16)-function in the second case.
Remark 2.3. As we will show in Section 6 below, the functions ’(2(cid:25)iq((cid:21)(cid:28) +(cid:22))=p),
’(k(cid:0)1)(2(cid:25)i((cid:21)(cid:28)+(cid:22))=p)and(cid:16)(k(cid:0)1)(2(cid:25)i((cid:21)(cid:28)+(cid:22))=p)aremodularformsfortheprincipal
congruencesubgroup(cid:0)(p) (or (cid:0)(2p)) of weight1,k andk, respectively. In fact, we
see that
2(cid:25)i((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
(cid:16)(k(cid:0)1)( )=(cid:0)}(k(cid:0)2)( )
p p
is a \level p Eisenstein series" (cf. [15, pp. 131-135]). Hence s~ (q;p;(cid:28)) is also a
k
modular form for the same congruence subgroup (cid:0)(p) (or (cid:0)(2p)) of weight k+1.
Moreover we can show that the limit of s~ (q;p;(cid:28)) is the classical Apostol sum
k
(2.5) lim s~2n(cid:0)1(q;p;(cid:28))=s2n(cid:0)1(q;p)
(cid:28)!i1
for any n>1. This fact may justify our terminologyof \elliptic Apostol sums" for
the sums s~ (q;p:(cid:28)).
k
Another fundamental property of Apostol sums is that they satisfy the equality
s2n(cid:0)1(q+p;p)=s2n(cid:0)1(q;p):
Thisimplies thatanApostolsumisa(generalized)Dedekindsymbol(cf. [10]). As
for elliptic Apostol sums, we see that they also satisfy a similar equality:
s~2n(cid:0)1(q+2p;p;(cid:28))=s~2n(cid:0)1(q;p;(cid:28)):
Indeed, this follows easily from the identity that ’(z+2(cid:1)2(cid:25)i(m(cid:28) +n))=’(z) for
m;n 2 Z. An implication of this formula is that an elliptic Apostol sum is not a
Dedekind symbol.
Recall that if k is even, then the classical Apostol sums are trivial [1, p. 156].
As for the elliptic Apostol sums: if k is even, we have that s~ (q;p;(cid:28))=0. When k
k
is odd, there was a reciprocity law for the Apostol sums. This property should be
generalized to the elliptic Apostol sums. Indeed, this is the case, and establishing
a reciprocity law for the elliptic Apostol sums is one of our main results and is
formulated in the following theorem.
ELLIPTIC APOSTOL SUMS AND THEIR RECIPROCITY LAWS 4243
Theorem 2.2. Let p and q be relatively prime positive integers.
(1) Suppose that n is an integer with n>0 and p+q (cid:17)1 (2). Then we have
p2n(cid:0)2s~2n(cid:0)1(q;p;(cid:28))8+q2n(cid:0)2s~2n(cid:0)1(p;q;(cid:28)) 9
(2.6) (2n(cid:0)1)!<Xn =
= pq : H2jH2n(cid:0)2jp2jq2n(cid:0)2j +(2n(cid:0)1)H2n;:
j=0
(2) Suppose that n is an integer with n>1 and p+q (cid:17)0 (2). Then we have
p2n(cid:0)2s~2n(cid:0)1(q;p;(cid:28))8+q2n(cid:0)2s~2n(cid:0)1(p;q;(cid:28)) 9
(2.7) (2n(cid:0)1)!<Xn =
= pq : H2jH2n(cid:0)2jp2jq2n(cid:0)2j (cid:0)(2n(cid:0)1)G2n;:
j=0
Remark 2.4. Anotherinterpretationof (2.6)and(2.7)isintermsofmodularforms.
The left-hand sides of (2.6) and (2.7) representmodular forms for (cid:0)(p) if p is even
(resp. (cid:0)(2p) if p is odd) and for (cid:0)(q) if q is even (resp. (cid:0)(2q) if q is odd) of weight
2n, while the right-handsides represent modular forms for (cid:0) (2) of weight 2n. We
0
mayview the equations asthe identities betweenthese speci(cid:12)c modular forms. We
are not able to (cid:12)nd any literature dealing with these kinds of relations between
these speci(cid:12)c modular forms,andthese relationsappear to be new. One intriguing
question arises: Is there any geometric interpretation of these formulae?
3. Proof of Theorem 2.1
In this section we will give a proof for Theorem 2.1.
Proof of Theorem 2.1(1). Wewillestablishtheidentity(2.3). Weletf(z)andg(z)
stand for the left-hand side, and respectively, the right-hand side of (2.3). Namely
pX(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
f(z)= ((cid:0)1)(cid:21)’( )’(z+ )
p p p
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
Xq(cid:0)1
1 2(cid:25)ip((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
+ ((cid:0)1)(cid:21)’( )’(z+ )
q q q
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
and
1
g(z)=(cid:0)’(pz)’(qz)(cid:0) ’0(z):
pq
We claim that
f(z)=g(z):
First note that both f(z) and g(z) are meromorphic functions, and both have
simple poles at the points z =(cid:0)2(cid:25)i((cid:21)(cid:28) +(cid:22))=p+2(cid:25)i(m(cid:28) +n) with (cid:21);(cid:22)=0;1;:::;
p(cid:0)1; ((cid:21);(cid:22)) 6= (0;0); m;n 2 Z and at z = (cid:0)2(cid:25)i((cid:21)(cid:28) +(cid:22))=q+2(cid:25)i(m(cid:28) +n) with
(cid:21);(cid:22)=0;1;:::;q(cid:0)1; ((cid:21);(cid:22))6=(0;0); m;n2Z. Furthermore, at these points, both
functions have the same residues, that is,
((cid:0)1)(cid:21) 2(cid:25)iq((cid:21)(cid:28) +(cid:22))
Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=p+2(cid:25)i(m(cid:28)+n)(f)= p ’( p )
=Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=p+2(cid:25)i(m(cid:28)+n)(g)
4244 SHINJIFUKUHARAANDNORIKOYUI
and
((cid:0)1)(cid:21) 2(cid:25)ip((cid:21)(cid:28) +(cid:22))
Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=q+2(cid:25)i(m(cid:28)+n)(f)= q ’( q )
=Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=q+2(cid:25)i(m(cid:28)+n)(g):
Next we investigate other poles. We may assume without loss of generality that
g(z)hasnopoleofordergreaterthan1. Indeed,theprincipalpartsof(cid:0)’(pz)’(qz)
and(cid:0)(1=pq)’0(z)atz =0are(cid:0)1=(pqz2)and1=(pqz2),respectively,andtherefore,
they cancel out in g(z). Further notice that the principal part of (cid:0)’(pz)’(qz) at
z = 2(cid:25)i(cid:28) is 1=pq(z(cid:0)2(cid:25)i(cid:28))2 (note that p+q (cid:17) 1 (2)), while the principal part of
(cid:0)(1=pq)’0(z) at z =2(cid:25)i(cid:28) is (cid:0)1=pq(z(cid:0)2(cid:25)i(cid:28))2. Hence, again these principal parts
atz =2(cid:25)i(cid:28) cancelouting(z). Sinceg(z)isperiodicwithperiod2(cid:25)i(2Z(cid:28)+Z),any
poles at z = 2(cid:25)i(m(cid:28) +n)(m;n 2 Z) also cancel out in g(z). On the other hand,
clearly f(z) has no pole of order greater than 1. Thus it follows that the principal
parts of f(z) and g(z) coincide at all of their poles. Therefore f(z)(cid:0)g(z) is an
entire function.
Summarizing the abovediscussion,we canconclude that f(z)(cid:0)g(z)is a doubly
periodic entire function on C. Then by the well-known Liouville theorem it must
be a constant. Hence
f(z)(cid:0)g(z)=A(p;q;(cid:28));
where A(p;q;(cid:28)) is a constant with respect to z.
Next we will determine the constant function A(p;q;(cid:28)). First note that ’((cid:28);z)
has the expansion at z = 0 given by the formula (2.2). This implies that the
function g(z)=(cid:0)’(pz)’(qz)(cid:0)(1=pq)’0(z) has the expansion of the form
1 1
g(z)=(cid:0) (1+H p2z2+(cid:1)(cid:1)(cid:1))(1+H q2z2+(cid:1)(cid:1)(cid:1))+ (1(cid:0)H z2(cid:0)(cid:1)(cid:1)(cid:1)):
pqz2 2 2 pqz2 2
Tending z !0, we obtain
1 1
limg(z)=(cid:0) (p2+q2)H (cid:0) H :
2 2
z!0 pq pq
Now look at the limit of f(z) as z ! 0. We have that limz!0f(z) = f(0). Since
A(p;q;(cid:28)) is independent of z, we then obtain
A(p;q;(cid:28))=limff(z)(cid:0)g(z)g
z!0
pX(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
= ((cid:0)1)(cid:21)’( )’( )
p p p
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
1 Xq(cid:0)1 2(cid:25)ip((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22)) p2+q2+1
+ ((cid:0)1)(cid:21)’( )’( )+ H :
2
q q q pq
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
Proof of Theorem 2.1(1) will be complete if we show that A(p;q;(cid:28)) is indeed
equal to 0, and this will be done in Lemma 3.1 below. (cid:3)
ELLIPTIC APOSTOL SUMS AND THEIR RECIPROCITY LAWS 4245
Lemma3.1. Letpandqberelativelyprimepositiveintegerssuchthatp+q(cid:17)1(2).
Then we have
pX(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
((cid:0)1)(cid:21)’( )’( )
p p p
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
(3.1) 1 Xq(cid:0)1 2(cid:25)ip((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22)) p2+q2+1
+ ((cid:0)1)(cid:21)’( )’( )+ H
2
q q q pq
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
=0:
Proof. Let h(z) be an elliptic function de(cid:12)ned by
h(z)=’(pz)’(qz)’(z):
Since p+q (cid:17)1 (2), h(z) is periodic with period 2(cid:25)i(Z(cid:28) +Z). Then it follows that
the sum of the residues of h(z) in any fundamental domain is zero. For instance,
we may take the following fundamental domain:
1 1
f2(cid:25)i(s(cid:28) +t) j (cid:0)1(cid:20)s;t(cid:20) g:
2pq 2pq
In this fundamental domain, h(z) has simple poles at z = (cid:0)2(cid:25)i((cid:21)(cid:28) +(cid:22))=p with
(cid:21);(cid:22) = 0;1;:::;p (cid:0) 1; ((cid:21);(cid:22)) 6= (0;0) and at z = (cid:0)2(cid:25)i((cid:21)(cid:28) + (cid:22))=q with (cid:21);(cid:22) =
0;1;:::;q(cid:0)1; ((cid:21);(cid:22))6=(0;0). The residues of h(z)at these poles canbe computed
as follows:
((cid:0)1)(cid:21) 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=p(h)= p ’( p )’( p )
and
((cid:0)1)(cid:21) 2(cid:25)ip((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=q(h)= q ’( q )’( q ):
Next we ought to calculate the residue of h(z) at z = 0. For this we note that
h(z)hasthefollowingexpansionatz =0(whichcanbederivedfromtheexpansion
for ’(z) given in (2.2)):
1
h(z)=’(pz)’(qz)’(z)= (1+H p2z2+(cid:1)(cid:1)(cid:1))(1+H q2z2+(cid:1)(cid:1)(cid:1))(1+H z2+(cid:1)(cid:1)(cid:1)):
pqz3 2 2 2
This enables us to compute the residue of h(z) at z =0. We have
p2+q2+1
Res (h)= H :
z=0 2
pq
Finally summing overthe residuesfor h(z)atallpoles, we obtain(3.1)that weare
after. (cid:3)
Remark 3.1. The formula in Lemma 3.1 is a special case of Egami’s result [8,
Theorem 1]. Unfortunately Egami’s original formula as well as its proof contained
errors (e.g., the right-hand side of his formula [8, Eq. 5] should be divided by
a (cid:1)(cid:1)(cid:1)a ). In the above lemma and its proof, we reproduced Egami’s result by a
0 r
di(cid:11)erent method from that of Egami, correcting all the errors in Egami’s paper.
We will need another lemma corresponding to Lemma 3.1 to prove the case
p+q (cid:17)0 (2) in Theorem 2.1(2).
4246 SHINJIFUKUHARAANDNORIKOYUI
Lemma3.2. Letpandqberelativelyprimepositiveintegerssuchthatp+q(cid:17)0(2).
Then we have
Xp(cid:0)1 Xq(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 1 2(cid:25)ip((cid:21)(cid:28) +(cid:22))
(3.2) ((cid:0)1)(cid:21)’( )+ ((cid:0)1)(cid:21)’( )=0:
p p q q
(cid:21);(cid:22)=0 (cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0) ((cid:21);(cid:22))6=(0;0)
Proof. Let h(z) be an elliptic function de(cid:12)ned by
h(z)=’(pz)’(qz):
Since p+q (cid:17) 0 (2), h(z) is periodic with period 2(cid:25)i(Z(cid:28) +Z). Hence the sum of
the residues of h(z) in any fundamental domain is zero. We may take the same
fundamental domain as in the proof of Lemma 3.1, namely,
1 1
f2(cid:25)i(s(cid:28) +t) j (cid:0)1(cid:20)s;t(cid:20) g:
2pq 2pq
In this fundamental domain, h(z) has simple poles at z = (cid:0)2(cid:25)i((cid:21)(cid:28) +(cid:22))=p with
(cid:21);(cid:22) = 0;1;:::;p (cid:0) 1; ((cid:21);(cid:22)) 6= (0;0) and at z = (cid:0)2(cid:25)i((cid:21)(cid:28) + (cid:22))=q with (cid:21);(cid:22) =
0;1;:::;q(cid:0)1; ((cid:21);(cid:22))6=(0;0). The residues of h(z)at these poles canbe computed
as follows:
((cid:0)1)(cid:21) 2(cid:25)iq((cid:21)(cid:28) +(cid:22))
Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=p(h)= p ’( p )
and
((cid:0)1)(cid:21) 2(cid:25)ip((cid:21)(cid:28) +(cid:22))
Resz=(cid:0)2(cid:25)i((cid:21)(cid:28)+(cid:22))=q(h)= q ’( q ):
We need to calculate the residue of h(z) at z = 0. For this we expand h(z) at
z =0:
1
h(z)=’(pz)’(qz)= (1+H p2z2+(cid:1)(cid:1)(cid:1))(1+H q2z2+(cid:1)(cid:1)(cid:1)):
pqz2 2 2
Thenwe obtainthat Res (h)=0. Finally summing overthe residues for h(z) at
z=0
all poles, we arrive at the equation (3.2). (cid:3)
In the rest of this section, we will give a proof for Theorem 2.1(2).
Proof of Theorem 2.1(2). We will establish the identity (2.4) in the case p+q (cid:17)
0(2). OurproofisalongthesamelineasforTheorem2.1(1)inthecasep+q(cid:17)1(2)
(although we need to use the (cid:12)rst derivatives f0(z) and g0(z) in place of f(z) and
g(z)to establishthe assertionthat f(z)(cid:0)g(z)is a constant). Letf(z)andg(z)be
the functions de(cid:12)ned, respectively, as follows:
pX(cid:0)1
1 2(cid:25)iq((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
f(z)= ((cid:0)1)(cid:21)’( )(cid:16)(z+ )
p p p
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
Xq(cid:0)1
1 2(cid:25)ip((cid:21)(cid:28) +(cid:22)) 2(cid:25)i((cid:21)(cid:28) +(cid:22))
+ ((cid:0)1)(cid:21)’( )(cid:16)(z+ )
q q q
(cid:21);(cid:22)=0
((cid:21);(cid:22))6=(0;0)
and
1
g(z)=(cid:0)’(pz)’(qz)(cid:0) (cid:16)0(z):
pq