THE ARITHMETIC OF THE COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO 1. Introduction and Statement of Results If H( n) denotes the Hurwitz-Kronecer class number of positive definite binary quadratic forms with discriminant n, then Zagier s weight 3/ Eisenstein series is given by (q = e πiz throughout) (1.1) H 3/ (z) = 1 1 + H(1, n)q n = 1 1 + 0<n 0,3 (mod 4) H( n)q n. This series fits into Cohen s general theory of weight + 1 Eisenstein series (1.) H + 1 (z) = H(, n)q n, n=0 where H(, n) = H 1 (, n) is defined in (??). If, then H + 1 (z) is a weight + 1 modular form on the congruence subgroup Γ 0 (4) (see Th. 3.1, [?]). If D 0, 1 (mod 4), and χ D ( ) = ( D ) denotes the usual Kronecer character, then it is a classical fact ([?], Ch. 5, 4) that H(1, D) = H(D) = D 1 χ D (a)a, when D < 4 is a fundamental discriminant. We generalize this classical result in a variety of ways. See, for example, Corollary??. Moreover, we obtain an elegant uniform description of the H(, n) in terms of values of the Bernoulli polynomials B (x), weighted against the mod n representation numbers of sums of squares. Date: January 1, 004. 1991 Mathematics Subject Classification. 11F11. The authors than the National Science Foundation for their generous support. The second and third authors than the Number Theory Foundation for their generous support, and the third author is grateful for the support of the Alfred P. Sloan, David and Lucile Pacard, and H. I. Romnes Fellowships. 1
ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO To state our result, if χ is a Dirichlet character, we let L(s, χ) = χ(n)n s its L-series. For positive integers r, n and with r odd, let D := ( 1) n and set ( 1) [/] χ 8 (r)( 1)!n r L(, χ D ) r+1 if ( 1) n 0 (mod 4), π (1.3) h r (, n) = ( 1) [/] ( 1)!n r L(, χ D ) if ( 1) n 1 (mod 4), 1 π 0 if ( 1) n, 3 (mod 4). Define also d n h r(, n/d ) if ( 1) n 0, 1 (mod 4), (1.4) H r (, n) = ζ(1 ) if n = 0, 0 otherwise. Note that h r (, n) n r log(n) (where the log(n) is necessary only for = 1). In particular, (1.5) H r (, n) n max( r,0)+ε. Next, we define, for positive integers r and n, (1.6) R r (a, n) := #{(ν 1, ν,..., ν r ) (Z/nZ) r : ν 1 + ν + + ν r a (mod n)}. If is a positive integer, then let B be the th Bernoulli number and let B (x) denote the usual th Bernoulli function with the convention that { x 1 if x 0 (1.7) B 1 (x) = 0 if x = 0. Finally, define the two formal Dirichlet series H r, (s) and F r, (s) by H r (, n) (1.8) H r, (s) = n s and (1.9) F r, (s) = χ ( 1) (r) n 1 R r(a, n)b (a/n) n s. By (??), H r, (s) converges for Re(s) > max(1, + 1 r ). For F r,(s), we note that B (x) is bounded in the interval [0, 1] and R r (a, n) = n r. This implies that F r, (s) converges for Re(s) > r + 1. As a consequence of the following theorem, we can extend the region of convergence for F r, (s) to Re(s) > max( r + 1, r + 1). By modifying an argument of J. Bruinier (see Th. 11 [?]) in the case where = 1, for positive odd integers r, we obtain the following description of H r, (s) as a formal
COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES 3 product of F r, (s) and a simple quotient of the Riemann zeta function ζ(s). In particular, if r = 1, we obtain a convenient description of the Mellin transform of Cohen s half integral weight Eisenstein series. Theorem 1.1. If r is a positive odd integer and 1, then As is well-nown, ζ(s) ζ(s) H r, (s) = ζ(s) ζ(s) F r,(s + r). = λ(n)n s, where λ is the Liouville function, the totally multiplicative function on positive integers defined on primes p by λ(p) = 1. Then Theorem?? gives an immediate formula for H r (, n). Corollary 1.. For r, n, Z with r odd, H r (, n) = χ ( 1) (r) d 1 λ(n/d)d r R r (a, d)b (a/d). d n We cannot resist interpreting the r = 1 case of Theorem?? in terms of the logarithmic derivatives of the infinite products (1.10) F (q) = (1 q n ) H(,n)/n. Corollary 1.3. If is a positive integer, then q d dq (F (q)) F (q) + a,b 1 c b square-free H(, a)q abc = 1 n 1 R 1 (a, n)b (a/n)n 1 q n. Acnowledgments: The authors than the referee, whose contributions improved this paper.. Proofs We begin with the following elementary lemmas. The first lemma is standard and can be found in ([?], p. 5). Lemma.1. The Fourier expansion of B (x) is given by: (.1) B (x) =! (πi) where n Z e(nx) n. indicates that we are summing over all non-zero integers, and e(x) = e πix. The proof of the following lemma is a straightforward analogue of the proof of Möbius inversion and will be left to the reader.
4 ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO Lemma.. Let f : Z + C. Define F, G : Z + C by F (N) = ( ) N f and d d N (.) G(N) = ( ) N f. d N d Then (.3) (i) F (N) = (ii) G(N) = a N a N N/a square-free λ ( N a G(a) and ) F (a). Fixing notation, we define, for integers m and a, (.4) G(m, a) = e(mν ). ν Z/aZ In the following lemma, we recall some basic facts of the Gauss sums in (??). For their proofs, we refer the reader to ([?], IV.3). Lemma.3. Suppose that a, b and c are integers with b, c 1. (i) If c (a, b), then G(a, b) = c G ( a, ) b c c (ii) If (b, c) = 1 and (a, bc)=1, then G(a, bc) = G(ab, c)g(ac, b) (iii) If b is odd and (a, b) = 1, the G(a, b) = ( a b) G(1, b) (iv) If a is odd, then G(a, s ) = ( ) s a ε((a 1)/)G(1, s ) (1 + i) b if b 0 (mod 4) b if b 1 (mod 4) (v) G(1, b) = 0 if b (mod 4) i b if b 3 (mod 4) Here, ε(b) = 1 if b is even and ε(b) = i if b is odd. For positive integers and r, let { Re(G(m, a) r ) if Z (.5) g,r (m, a) = Im(G(m, a) r ) if Z + 1. Then we see immediately from the definition and (i) above that, for b (m, a), ( m (.6) g,r (m, a) = b r g,r b, a ). b Then we have the following proposition.
COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES 5 Proposition.4. For integers m and a such that (m, a) = 1, ( ) (.7) g,r (m, a) = a r ( 1) a r 1 χ ( 1) 8(r) if ( 1) a 0 (mod 4) χ m ( 1) (r) if ( 1) a 1 (mod 4) 0 if ( 1) a, 3 (mod 4). Proof. If ( 1) a 1 (mod 4), then G(m, a) r = ( ) m rε() r a r/. Therefore, a ( ) ( ) (.8) g,r (m, a) = a r m ε() r 1 = a r m χ a a ( 1) (r). Clearly, by Lemma??, if ( 1) a (mod 4), g,r (m, a) = 0. If ( 1) a 3 (mod 4), then G(m, a) r = ( ) m rε( a 1) r a r/ and g,r (m, a) = 0. Therefore, the only case left is ( 1) a 0 (mod 4). Suppose a = s a 0 with a 0 odd; then, by Lemma??, we see that ( )( ) ( ) m s ma0 1 (.9) G(m, a) = ε G(1, s )G(1, a 0 ). m a 0 Therefore, if a 0 1 (mod 4), then ( ) ( s a 0 m 1 (.10) G(m, a) = ε m ) (1 + i) s/ a 0 = ( ) a ε m and if a 0 3 (mod 4), then ( )( ) ( ) a0 s m + 1 G(m, a) = ε i a 0 (1 + i) s/ m m ( ) ( ) a m + 1 (.11) = ε i a(1 + i) m ( ) ( ) a m 1 a(1 = ε + i). m Then we see that (.1) G(m, a) r = (r 1)/ a r/ ( a m After a lengthy verification, we obtain ( ) ( 1) a (.13) g,r (m, a) = r 1 r a m as desired. ) ( ) r m 1 ε i (r 1)/. χ( 1) 8(r), The following proposition provides the foundation for Theorem??. Proposition.5. For r,, n Z + with r odd, n 1 R r (a, n)b (a/n) = χ ( 1) (r) n r h r (, d). d n ( ) m 1 a(1 + i),
6 ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO Proof. Let N be a positive integer; then by Lemma??, (.14) N 1 ( ) ν R r (a, N)B (a/n) = B 1 N + + ν r N ν 1,...,ν r (mod N) ( e n =! (πi) n Z n ν 1,...,ν r (mod N) =! n G(n, N) r (πi) n Z ( )) ν 1 N + + ν r N =! n (Re(G(n, N) r ) + iim(g(n, N) r )) (πi) n Z = ε()! (πi) n g,r (n, N). n 1 Reindexing this sum on the divisors of N, we find (.15) N 1 R r (a, N)B (a/n) = ε()! (πi) a N n 1 (n,n)= N a = ε()! N r a r (πi) a N n g,r (n, N) (m,a)=1 m g,r (m, a). Since χ a (m) = 0 if (m, a) 1, Proposition?? yields a r m g,r (m, a) = r 1 χ( 1) 8(r)a r/ m χ ( 1) a(m) (.16) (m,a)=1 = r 1 χ( 1) 8(r)a r/ L(, χ ( 1) a) if ( 1) a 0 (mod 4), and m g,r (m, a) = χ ( 1) (r)a r/ m χ ( 1) a(m) (.17) a r (m,a)=1 = χ ( 1) (r)a r/ L(, χ ( 1) a) if ( 1) a 1 (mod 4). If ( 1) a, 3 (mod 4), the sum is zero. (??) yields (.18) as desired. N 1 R r (a, N)B (a/n) = χ ( 1) (r) N r We are now prepared to prove the main theorem. h r (, d) d N
COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES 7 Proof of Theorem??. Combining Proposition?? with Lemma??(ii) yields (.19) H r (, n) = χ ( 1) (r) d 1 λ(n/d)d r R r (a, d)b (a/d). d n A straightforward calculation involving formal Dirichlet series verifies that, for any series n 1 b(n)n s, ζ(s) b(n)n s = λ(n/d)b(d) n s. ζ(s) n 1 n 1 d n Applying this to (??), we obtain the theorem. Proof of Corollary??. This is (??) in the proof of Theorem?? Proof of Corollary??. If we define a formal product (.0) F (q) = (1 q n ) c(n), then one obtains formally that (.1) q d (F (q)) dq F (q) = c(d)d q n. d n Combining Proposition?? with Lemma??(i), we get (.) χ ( 1) (r) n r n 1 R r (a, n)b (a/n) = The result follows from this and (??) when r = 1. d n n/d square-free H r (, d). 3. Remars We remar, here, that these calculations can be carried out in similar settings with similar results. We state here some interesting formulas that are obtained when r is even in the above calculations. We obtain the following Theorem. Theorem 3.1. With notation as above, (i) If r (mod 4) and is even, 1 B n 1 R r (a, n)b (a/n) = n r d n d ( ) 1 d (1 r/ 1p ). d p d
8 ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO (ii) If r (mod 4) and is odd, ( 1) (r )/4 n 1 r/ B ( 1) 4 R r (a, n)b (a/n) = n r d n 4 d d r/ p d p ( 1 ( 1 p p )) (iii) If r 0 (mod 4) and is even, 1 n 1 R r (a, n)b (a/n) = n r d (1 r/ 1p ) η(d), where B d n p d 1 if d η(d) = 0 if d (mod 4) ( 1) r/4 r/ if 4 d. (iv) If r 0 (mod 4) and is odd, n 1 R r (a, n)b (a/n) = 0.. References [1] Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New Yor, 1966. [] J. Bruinier, On the ran of Picard groups of modular varieties attached to orthogonal groups, Compositio Mathematica, in press. [3] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann., 17, (1975), pages 71-85. [4] K. Knopp, Theory and Application of Infinite Series, nd edition, Hafner Publishing Company, New Yor, 1947. [5] S. Lang, Algebraic Number Theory, Springer-Verlag, Graduate Texts in Mathematics 110, 1986. Mathematical Institute of the Hungarian Academy of Sciences, P. O. Box 17, Budapest 1364, Hungary E-mail address: balog@hexagon.math-inst.hu Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 E-mail address: mcgraw@math.wisc.edu Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 E-mail address: ono@math.wisc.edu