JACKSON–MELLIN's TRANSFORM of MODULAR FORMS and Q
Kyushu J. Math. Vol . 61, 2007, pp. 551–563
JACKSON–MELLIN’S TRANSFORM OF MODULAR FORMS AND q-ZETA FUNCTIONS
Shai HARAN, Nobushige KUROKAWA and Masato WAKAYAMA (Received 24 July 2006)
Dedicated to Leonhard Euler on the 300th anniversary of his birthday
Abstract. This note studies a q-analogue of Mellin’s transform of a modular form via Jackson’s integral.
1. Introduction
The Mellin transform sends modular forms to L-functions and it is basic for the study of L-functions. The main purpose of the present note is to develop an elementary study of a q-analogue of this situation. In fact, a q-Mellin transform which we give here via the Jackson integral (see, e.g., [KC]) can indeed be applied to various modular forms for obtaining a natural approximation when q → 1 to the classical (completed) L-functions having functional equation and yet are periodic mod 2πi/log q. Let 0 It is immediate to see that Mq (f )(s) becomes the usual Mellin transform when q → m (f )(s) := 1. To simplify the description we sometimes use the notation q ns n n∈Z q f(q ),thatis,Mq (f ) = (1 − q)mq(f ). 2000 Mathematics Subject Classification: Primary 11M36. Keywords and Phrases: Mellin’s transform; Jackson’s integral; modular forms; Riemann’s zeta function; Ramanujan’s delta function; q-series. This article is an invited contribution to a special issue of the Kyushu Journal of Mathematics commemorating the sixtieth volume. 552 S. Haran et al The first aim is to show that mq (f )(s) has a functional equation when f is a modular form. For motivation and guidance, we give here a very simple example as a −a follows. Consider the function fa(x) = 1/(x + x ) for a>0. Then it is clear that − fa(x 1) = fa(x).Putφa(s) = mq (fa)(s). Then we can calculate φa(s) as ns n ns n −ns −n φa(s) = q fa(q ) = fa(1) + q fa(q ) + q fa(q ) n∈Z n≥1 n≥1 ns −ns n = fa(1) + (q + q )fa(q ). n≥1 The series obviously converges when |Re s| Hence, φa(s) is meromorphic in C and has a simple pole at s =±{a(1 + 2m) + (2πi/log q)} (m, ∈ Z,m ≥ 0) with residue (−1)m−1/log q. The location of the zeros φa(s), however, seems difficult to detect. Throughout the paper, we assume that 0 2. Jackson–Mellin’s transform of modular forms We prove the following basic theorem. THEOREM 2.1. Let F be a holomorphic modular form of weight k (k must be a non- negative even integer) for SL(2, Z); in particular, F(−z−1) = zkF(z) for z ∈ H, H being the upper half plane. Put f(y) = F(iy) for y>0. Define a q-analogue of Jackson–Mellin’s transform of modular forms and q-zeta functions 553 the completed L-function q (s, F ) by q (s, F ) :=mq (f − F(i∞))(s) − F(i) ns n = q (f (q ) − a0) − f(1). (2.1) n∈Z Here a0 is the constant term of the Fourier expansion of F(z): 2πinz F(z)= ane . n≥0 Then q (s, F ) can be analytically continued to a meromorphic function on C and 1 i−k q (s, F ) + a + (2.2) 0 1 − qs 1 − qk−s is entire. In particular, if F isacuspform,thenq (s, F ) is an entire function. Furthermore, q (s, F ) has a functional equation: −k q (s, F ) = i q (k − s, F). (2.3) Proof. Since f(q−n) = (iqn)kf(qn) = ikqnkf(qn),wehave ns n −ns −n q (s, F ) = f(1) − a0 + q (f (q ) − a0) + q (f (q ) − a0) − f(1) n≥1 n≥1 ns −k −nk −n −ns −n =−a0 + q (i q f(q ) − a0) + q (f (q ) − a0) n≥1 n≥1 ns −k −nk −n −k −nk =−a0 + q {i q (f (q ) − a0) + a0(i q − 1)} n≥1 −ns −n + q (f (q ) − a0) n≥1 −ns −k −n(k−s) −n =−a0 + (q + i q )(f (q ) − a0) n≥ 1 i−kqs−k qs + a − 0 1 − qs−k 1 − qs for Re s>k. It follows that −k −ns −k −n(k−s) −n 1 i q (s, F ) = (q + i q )(f (q ) − a ) − a + . 0 0 1 − qs 1 − qk−s n≥1 (2.4) 554 S. Haran et al f(q−n) − a = ∞ a e−2πmq−n Remark 1. Similarly, we have the following expression of q (s, F ): k ns k n(k−s) n 1 i q (s, F ) = (q + i q )(f (q ) − a ) − a + . 0 0 1 − q−s 1 − q−k+s n≥1 (2.5) We notice that, however, the series in the first term may also involve (actually involves) the poles located on the lines Re s = 0andRes = k because 0 (i) = (1 )24/ 24π18 q (s, ) ( 4 2 .) It follows that is entire and satisfies the functional equation q (s, ) = q (12 − s, ). Note also that the same procedure used in Remark 1 yields 6n n q (6 + it,) = q cos(nt log q)(iq ), (2.7) n=0 because q6n(iqn) = q−6n(iq−n). The proof of the previous theorem provides the following variant. − k COROLLARY 2.2. Let f be a function satisfying f(x 1) = x f(x) for x>0. Suppose f has the following series expansion: −2πnx f(x)= ane . n≥0 Define Lq (s, f ) by ns n/k Lq (s, f ) := q (f (q ) − a0). (2.8) n∈Z Then Lq (s, f ) can be analytically continued to a meromorphic function on C. Furthermore, we have Lq (s, f ) = Lq (1 − s, f ). (2.9) Proof. Since f(q−n/k) = (qn/k)kf(qn/k) = qnf(qn/k), in a similar way as in the preceding proof, we obtain ns n/k −ns −n/k Lq (s, f ) = f(1) − a0 + q (f (q ) − a0) + q (f (q ) − a0) n≥1 n≥1 ns −n −n/k −ns −n/k = f(1) − a0 + q (q f(q ) − a0) + q (f (q ) − a0) n≥1 n≥1 556 S. Haran et al ns −n −n/k −n = f(1) − a0 + q {q (f (q ) − a0) + a0(q − 1)} n≥1 −ns −n/k + q (f (q ) − a0) n≥1 −ns −n(1−s) −n/k = f(1) − a0 + (q + q )(f (q ) − a0) n≥ 1 qs−1 qs + a0 − 1 − qs−1 1 − qs for Re s>1. Hence, we get −ns −n(1−s) −n/k Lq (s, f ) = f(1) + (q + q )(f (q ) − a0) n≥ 1 1 1 − a0 + . (2.10) 1 − qs 1 − q1−s The result follows easily from this expression. 2 Example 2. A q-analogue of the Riemann zeta function (which is called a ‘baby zeta’ in [H]) ζ (q)(s) defined by (q) ns −πm2q2n ζ (s) := q e (Re s>1) (2.11) n∈Z m=0 has the expression ζ (q)(s) = e−πm2 + (q−ns + q−n(1−s)) e−πm2q−2n − 1 − 1 . 1 − qs 1 − q1−s m∈Z n≥1 m=0 (2.12) ϑ(z) := eiπm2z This is because√ the theta function m∈Z is a modular form of weight 1 θ(t−1) = tθ(t) (t > ) θ(t) := ϑ(it) = e−πm2t 2 ; 0 ,where m∈Z . Expression (2.12) above immediately shows that the (2πi/log q)Z-periodic function ζ (q)(s) has meromorphic continuation to the entire plane C, has the unique (mod (2πi/log q)Z) simple poles at s = 0, 1, and satisfies the functional equation ζ (q)(1 − s) = ζ (q)(s). Remark 3. The q-analogue ζq (s) of the Riemann zeta function introduced in [KKW] (q) is obviously different from the present q-analogue ζ (s). The function ζq (s) enables us to justify easily the evaluation of ζ(−m)(m ∈ Z≥0) done by Euler around 1740 but does not have a functional equation as ζ (q)(s) possesses. Jackson–Mellin’s transform of modular forms and q-zeta functions 557 Remark 4. We may also establish similar results as in the theorem above for Maass wave forms. In general, it seems very difficult to determine the location of the zeros of q (s, F ). We close this section by giving a quite elementary example whose zeros can be easily seen. Example 3. Let N be a positive integer. Let χ[qN ,q−N ](x) be the characteristic function of the interval [qN ,q−N ]⊂R. Then, we have N ns n ns −ns mq (χ[qN ,q−N ])(s) = q χ[qN ,q−N ](q ) = 1 + (q + q ) n∈Z n=1 qs(1 − qNs) q−s(1 − q−Ns) = 1 + + 1 − qs 1 − q−s − q(2N+1)s = q−Ns 1 . 1 − qs Hence, one sees clearly that the functional equation mq (χ[qN ,q−N ])(s) := mq (χ[qN ,q−N ])(−s) holds and the zeros of the entire function mq (χ[qN ,q−N ])(s) are givenby2πim/(2N + 1) log q(m ∈ (2N + 1)Z,m∈ Z). Question. Can one obtain any relation between Example 2 and Example 3 (where the zeros are known completely) via the central limit theorem relative to a multiplicative convolution? 3. One-parameter deformation of ζ (q)(s) Define a one-parameter deformation of ζ (q)(s) by (q) (n+t)s −πm2q2(n+t) ζ (s, t) := q e (Re s>1). (3.1) n∈Z m=0 The series converges absolutely when Re q2t is positive, that is, when the imaginary part of t satisfies |Im t| <π/4|log q|. Also, one sees obviously that ζ (q)(s, t + 1) = ζ (q)(s, t) and ζ (q)(s, 0) = ζ (q)(s). The following formula shows that the Fourier coefficients of ζ (q)(s, t) (for the Fourier expansion with respect to the variable t)are given by the 2πi/log q-translations of the complete Riemann zeta function ζ(s)ˆ := π−s/2(s/2)ζ(s). 558 S. Haran et al PROPOSITION 3.1. Put δq = 2πi/log q.Then ζ (q)(s, t) =− 1 ζ(sˆ − δ n)e2πint. q q (3.2) log n∈Z In particular, 1 (q) ζ(s)ˆ =−(log q) ζ (s, t) dt. (3.3) 0 t | t| < 1 |δq|=π/ | q| s Moreover, for a fixed satisfying Im 8 4 log , as a function of , ζ (q)(s, t) has meromorphic continuation to C, has the unique (mod (2πi/log q)Z) simple poles at s = 0, 1, and satisfies a functional equation: (q) (q) ζ (1 − s, t) = ζ (s, −t). (3.4) Proof. Formula (3.2) is deduced from the Poisson summation formula. In fact, the 2 2(x+t) Fourier transform of the function q(x+t)se−πm q (m = 0) is calculated as ∞ q(x+t)se−πm2q2(x+t)e−2πixξ dx −∞ ∞ =− 1 e−swe−πm2e−2w e2πi(t+w/logq)ξ dw log q −∞ πitξ ∞ e2 − w =− e−w(s−δq ξ)e−πm2e 2 dw log q −∞ 2πitξ ∞ e 1 (s−δ ξ)− −πm2α =− α 2 q 1e dα 2logq 0 2πitξ e − 1 (s−δ ξ) s − δq ξ =− (πm2) 2 q , 2logq 2 whence the Poisson summation formula yields q(n+t)s e−πm2q2(n+t) n∈Z m=0 1 πitn − 1 (s−δ n) −(s−δ n) s − δq n =− e2 π 2 q |m| q 2logq 2 n∈Z m=0 1 − 1 (s−δ n) s − δq n πint =− π 2 q ζ(s − δ n)e2 . q q log n∈Z 2 Hence, (3.2) follows. The integral representation (3.3) follows immediately from (3.2) as the constant term of the Fourier expansion with respect to the variable t. Jackson–Mellin’s transform of modular forms and q-zeta functions 559 The functional equation can be obtained by the functional equation ζ(ˆ 1 − s) = ζ(s)ˆ . The remaining√ assertions are also clear from (3.2) together with the Stirling formula x− 1 −π|y|/ |(x + iy)|∼ 2π|y| 2 e 2 when |y|→∞for a fixed real x. 2 From the definition of the Jackson–Mellin transform, we note that (q) 1 s ζ (s) = (Mq2 θ) . 1 − q2 2 ∞ −πm2t Hence, if we put θ+(t) = m= e for t>0, we have 1 − q s ∞ (q) log 1 s/2−1 −( q)ζ (s) = (M θ) −→ · t θ+(t) dt = ζ(s).ˆ log 2 q2 2 1 − q 2 2 0 (3.5) (q) Therefore, we now discuss briefly some estimate of |ζ (s0,t)| for t ∈ R when s0 is a non-trivial zero ζ(s).Lett ∈ R. Then, by (3.2) one gets (q) 1 ˆ 1 ˆ ζ (s, t) + ζ(s) ≤ |ζ(s − δqn)| log q |log q| n=0 − 1 Re s π 2 s − δq n = |ζ(s − δq n)| . |log q| 2 n=0 Suppose now 0 < Re s<1. Then, it is known (see, e.g., [T]) that there is a positive 1 constant C such that |ζ(s)|≤C|Im s| 2 log |Im s| holds. Hence, 1 2πn 2 2πn |ζ(s − δq n)|≤C Im s − log Im s − . log q log q It follows again from the Stirling formula that (q) 1 ˆ ζ (s, t) + ζ(s) log q − 1 s 1 s ( π) Re πn 2 Re πn 2 2 2 2 −(π/4)|Im s−2πn/log q| ≤ C Im s − log Im s − e |log q| log q log q n=0 for some constant C > 0. Hence, if 0 < Re s<1and|Im s| < 2π/|log q|,wehave (q) 1 ˆ (π/8)|Im s| −(π2/4)|n/logq| ζ (s, t) + ζ(s) ≤ De e log q n=0 e(π/8)|Ims| ≤ 2D eπ2/4|log q| − 1 560 S. Haran et al for some positive number D when q is sufficiently close to 1. Thus, we easily have the following. COROLLARY 3.2. Let s0 = ρ + iγ (0 <ρ<1,γ ∈ R) be a non-trivial zero of ζ(s). Suppose |γ | < 2π/|log q|. Then we have (q) −(π2/4)|1/logq| |ζ (s0,t)| e (3.6) when |q − 1| is sufficiently small. Remark 5. Note that the q-series ζ (q)(s, t) converges absolutely if |Im t| < π/4|log q|. Therefore, when s(>1) is a positive integer, one finds from (3.1) that the q-series ζ (q)(s, t) defines essentially an elliptic function in the t-plane. Actually, s = ∈ Z φ (t) := ζ (q)( , t − 1 δ ) when 4 4 >0, the period lattice for 4 8 q is given by Z · + Z · 1 δ 1 4 q . It is hence interesting to find any (algebraic) differential equation for φ(t) as the q-series studied in [KW]. 4. Remarks on some generalizations We give here generalizations of the previous q-analogue on two different directions. 4.1. Two-variable q-analogue of L-functions One may introduce a two-variable q-analogue of zeta and L-functions. A typical example is given by a q-analogue of the following Arakelov two-variable zeta function ZQ(w, s) attached to Q (see [LR]): ∞ w−s dt 2 s 1 ZQ(w, s) := θ(t ) θ ( w< s< ). 2 Re Re 0 (4.1) 0 t t It can be easily rewritten in the form ∞ 1 w s/2−1 ZQ(w, s) = θ(t) t dt. (4.2) 2 0 The function ZQ(w, s) shares many properties of the Riemann zeta function. For instance, it satisfies the functional equation ZQ(w, s) = ZQ(w, w − s) and the function (s(s − w)/2w)ZQ(w, s) isshowntobeentirelyonC2 (see [LR, Section 2]). Since the second expression of (4.1) shows that ZQ(w, s) is the Mellin Jackson–Mellin’s transform of modular forms and q-zeta functions 561 transform of 2θ(t)w, which is a modular form of weight w/2 for a congruence subgroup of SL2(Z). Thus, what we arrive at is the following definition: (q) 1 ns n w ζ (w, s) := q 2 (θ(q ) − 1). (4.3) n∈Z Then, similarly to the above, we obtain (q) w −(n/ )s −(n/ )(w−s) −n w ζ (w, s) = θ(1) + (q 2 + q 2 )(θ(q ) − 1) n≥1 1 1 − − , (4.4) 1 − qs/2 1 − q(w−s)/2 whence it follows that ζ (q)(w, s) = ζ (q)(w, w − s). 4.2. Hilbert modular forms Let F/Q be a real quadratic field. Let F(z1,z2) be a Hilbert modular form for := SL2(OF ) of weight (k1,k2).Put f(t1,t2) := F(it1,it2)(t1,t2 > 0). γ = 0 −1 ∈ From the modular invariance for 10 ,wehave k k f(t−1,t−1) = t 1 t 2 f(t ,t ), 1 2 1 2 1 2 (4.5) where for simplicity, we assume that 4|k1,k2. If we define the two-variable Mellin transform by ∞ ∞ s − s − M(s ,s ; f):= f(t ,t )t 1 1t 2 1 dt dt , 1 2 1 2 1 2 1 2 0 0 the following functional equation follows immediately from (4.5): M(s1,s2; f)= M(k1 − s1,k2 − s2; f). Let q = (q1,q2) for 0 Then, similarly to a single q-analogue case, we have the following functional equation M (s ,s ; f) of q1,q2 1 2 : M (s ,s ; f)= M (k − s ,k − s ; f). q1,q2 1 2 q1,q2 1 1 2 2 (4.7) Actually, direct calculation shows that −n −n −s n −s n Mq ,q (s ,s ; f)= ( − q )( − q ) f(q 1 ,q 2 )q 1 1 q 2 2 1 2 1 2 1 1 1 2 1 2 1 2 n ,n ∈Z 12 n n k n k n −s n −s n = ( − q )( − q ) f(q 1 ,q 2 )q 1 1 q 2 2 q 1 1 q 2 2 1 1 1 2 1 2 1 2 1 2 n ,n ∈Z 12 n n (k −s )n (k −s )n = ( − q )( − q ) f(q 1 ,q 2 )q 1 1 1 q 2 2 2 1 1 1 2 1 2 1 2 n1,n2∈Z = M (k − s ,k − s ; f). q1,q2 1 1 2 2 Hence, the functional equation (4.7) follows. Acknowledgement. This work was partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012. REFERENCES [H] S. Haran. The Mysteries of The Real Prime (London Mathematical Society Monographs, New Series, 25). Clarendon Press, Oxford, 2001. [KC] V. Kac and P. Cheung. Quantum Calculus. Springer, New York, 2002. [KKW] M. Kaneko, N. Kurokawa and M. Wakayama. A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 75–192. [KW] N. Kurokawa and M. Wakayama. Certain families of elliptic functions defined by q-series. Ramanujan J. 10 (2005), 23–41. [LR] J. C. Lagarias and E. Rains. On a two-variable zeta function for number fields. Ann. Inst. Fourier (Grenoble) 53 (2003), 1–68. [T] E. C. Titchmarsh. The Theory of the Riemann Zeta-function. Oxford University Press, Oxford, 1951. Shai Haran Department of Mathematics Technion — Israel Institute of Technology Haifa 32000 Israel (E-mail: [email protected]) Jackson–Mellin’s transform of modular forms and q-zeta functions 563 Nobushige Kurokawa Department of Mathematics Tokyo Institute of Technology Oh-okayama, Meguro Tokyo 152-0033 Japan (E-mail: [email protected]) Masato Wakayama Faculty of Mathematics Kyushu University Hakozaki Fukuoka 812-8581 Japan (E-mail: [email protected])