TRANSACTIONSOFTHE AMERICANMATHEMATICALSOCIETY Volume356,Number10,Pages4237{4254 S0002-9947(04)03481-6 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