Distribution of the partition function modulo m

Transcription

1 Annas of Mathematics, 151 (000), Distribution of the artition function moduo m By Ken Ono* 1. Introduction and statement of resuts A artition of a ositive integer n is any nonincreasing sequence of ositive integers whose sum is n. Let (n) denote the number of artitions of n (as usua, we adot the convention that (0) = 1 and (α) =0ifα N). Ramanujan roved for every nonnegative integer n that (5n +4) 0 (mod 5), (7n +5) 0 (mod 7), (11n +6) 0 (mod 11), and he conjectured further such congruences moduo arbitrary owers of 5, 7, and 11. Athough the work of A. O. L. Atkin and G. N. Watson setted these conjectures many years ago, the congruences have continued to attract much attention. For exame, subsequent works by G. Andrews, A. O. L. Atkin, F. Garvan, D. Kim, D. Stanton, and H. P. F. Swinnerton-Dyer ([An-G], [G], [G-K-S], [At-Sw]), in the sirit of F. Dyson, have gone a ong way towards roviding combinatoria and hysica exanations for their existence. Ramanujan [Ra,. xix] aready observed that his congruences were quite secia. For instance, he rocaimed that It aears that there are no equay sime roerties for any modui invoving rimes other than these three (i.e. m =5, 7, 11). Athough there is no question that congruences of the form (an + b) 0 (mod m) are rare (see recent works by the author ([K-O], [O1], [O])), the question of whether there are many such congruences has been the subject of debate. In the 1960 s, Atkin and O Brien ([At], [At-Sw1], [At-Ob]) uncovered *The author is suorted by NSF grants DMS , DMS and NSA grant MSPR- 97Y Mathematics Subject Cassification. Primary 11P83; Secondary 05A17. Key words and hrases. artition function, The Erdös conjecture, Newman s conjecture.

2 94 KEN ONO further congruences such as (1) ( n + 37) 0 (mod 13). However, no further congruences have been found and roven since. In a reated direction, P. Erdös and A. Ivić ([E-I]) conjectured that there are infinitey many rimes m which divide some vaue of the artition function, and Erdös made the foowing stronger conjecture [Go], [I]. If m is rime, then there is at east one nonneg- Conjecture (Erdös). ative integer n m for which (n m ) 0 (mod m). A. Schinze (see [E-I] for the roof) roved the Erdös-Ivić conjecture using the Hardy-Ramanujan-Rademacher asymtotic formua for (n), and more recenty Schinze and E. Wirsing [Sc-W] have obtained a quantitative resut in the direction of Erdös stronger conjecture. They have shown that the number of rimes m<x for which Erdös conjecture is true is og og X. Here we resent a uniform and systematic aroach which settes the debate regarding the existence of further congruences, and yieds Erdös conjecture as an immediate coroary. Theorem 1. Let m 5 be rime and et k be a ositive integer. ositive roortion of the rimes have the roerty that ( m k 3 ) n +1 0 (mod m) for every nonnegative integer n corime to. A In view of work of S. Ahgren [A], J.-L. Nicoas, I. Z. Ruzsa, A. Sárközy [Ni-R-Sa] and J-P. Serre [S] for m =, the fact that (3) = 3, and Theorem 1, we obtain: Coroary. Erdös conjecture is true for every rime m. Moreover, if m 3is rime, then { X if m =, #{0 n X : (n) 0 (mod m)} m X if m 5. Surrisingy, it is not known whether there are infinitey many n for which (n) 0 (mod 3). As an exame, we sha see that = 59 satisfies the concusion of Theorem 1 when m = 13 and k = 1. In this case, by considering integers in the

3 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 95 arithmetic rogression r 1 (mod 59), we find for every nonnegative integer n that () ( n ) 0 (mod 13). Our resuts are aso usefu in attacking a famous conjecture of M. Newman [N1]. Conjecture (M. Newman). If m is an integer, then for every residue cass r (mod m) there are infinitey many nonnegative integers n for which (n) r (mod m). Works by Atkin, Newman, and O. Koberg ([At], [N1], [K]) have verified the conjecture for m =, 5, 7, 11 and 13 (in fact, the case where m =11isnot roved in these aers, but one may easiy modify the arguments to obtain this case). Here we resent a resut which, in rincie, may be used to verify Newman s conjecture for every remaining rime m 3. We sha ca a rime m 5 good if for every r (mod m) there is a nonnegative integer n r for which mn r 1 (mod ) and ( ) mnr +1 r (mod m). Theorem 3. If m 5 is a good rime, then Newman s conjecture is true for m. Moreover, for each residue cass r (mod m) we have { X/og X if 1 r m 1, #{0 n X : (n) r (mod m)} r,m X if r =0. Athough it aears ikey that every rime m 13 is good, roving that a rime m is good invoves a substantia comutation, and this comutation becomes raidy infeasibe as the size of m grows. The author is indebted to J. Hagund and C. Hayna who wrote efficient comuter code to attack this robem. As a resut, we have the foowing. Coroary 4. Newman s conjecture is true for every rime m<1000 with the ossibe excetion of m =3. We aso uncover surrising eriodic reations for certain vaues of the artition function mod m. In articuar, we rove that if m 5 is rime, then the sequence of generating functions ( m k ) n +1 (3) F (m, k; z) := q n (mod m) n 0 m k n 1 (mod )

4 96 KEN ONO (q := e πiz throughout) is eventuay eriodic in k. We ca these eriods Ramanujan cyces. Their existence imies the next resut. Theorem 5. If m 5 is rime, then there are integers 0 N(m) 48(m 3 m 1) and 1 P (m) 48(m 3 m 1) such that for every i>n(m) we have ( m i ) ( ) n +1 m P (m)+i n +1 (mod m) for every nonnegative integer n. For each cass r (mod m) one obtains exicit sequences of integers n k such that (n k ) r (mod m) for a k. This is the subject of Coroaries 9 through 1 beow. For exame, taking n = 0 in Coroary 1 shows that for every nonnegative integer k (4) ( 3 k+1 ) ( k k+3 ) +1 (mod 3) and 5 k+1 (mod 3). Simiary it is easy to show that ( k+ ) +1 (5) 0 (mod 3) and ( k+1 ) +1 0 (mod 3). Congruences of this sort mod 13 were reviousy discovered by Ramanujan and found by M. Newman [N]. In fact, this aer was insired by such entries in Ramanujan s ost manuscrit on (n) and τ(n) (see [B-O]). A riori, one knows that the generating functions F (m, k; z) are the reductions mod m of weight 1/ nonhoomorhic moduar forms, and as such ie in infinite dimensiona F m -vector saces. This infinitude has been the main obstace in obtaining resuts for the artition function mod m. In Section 3 we sha rove a theorem (see Theorem 8) which estabishes that the F (m, k; z) are the reductions mod m of haf-integra weight cus forms ying in one of two saces with Nebentyus. Hence, there are ony finitey many ossibiities for each F (m, k; z). This is the main observation which underies a of the resuts in this aer. We then rove Theorems 1 and 3 by emoying the Shimura corresondence and a theorem of Serre about Gaois reresentations. In Section 4 we resent detaied exames for 5 m 3.

5 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 97. Preiminaries We begin by defining oerators U and V which act on forma ower series. If M and j are ositive integers, then (6) a(n)q n U(M) := a(mn)q n, n 0 n 0 (7) a(n)q n V (j) := a(n)q jn. n 0 n 0 We reca that Dedekind s eta-function is defined by (8) η(z) :=q 1/ (1 q n ) and that Ramanujan s Deta-function is (9) (z) :=η (z), the unique normaized weight 1 cus form for SL (Z). If m 5 is rime and k is a ositive integer, then define a(m, k, n) by ( (10) a(m, k, n)q n δ(m,k) (z) U(m k ) ) V () := (mod m), η mk (z) where δ(m, k) :=(m k 1)/. Reca the definition (3) of F (m, k; z). Theorem 6. If m 5 is rime and k is a ositive integer, then F (m, k; z) a(m, k, n)q n (mod m). Proof. We begin by recaing that Euer s generating function for (n) is given by the infinite roduct (n)q n 1 := (1 q n ). Using this fact, one easiy finds that { η mk (m k } z) U(m k )= (n)q n+δ(m,k) (1 q mkn ) mk U(m k ) η(z) = (m k δ(m,k)+β(m,k) n+ n + β(m, k))q m k where 1 β(m, k) m k 1 satisfies β(m, k) 1 (mod m k ). (1 q n ) mk,

6 98 KEN ONO Since (1 X mk ) mk (1 X) mk (mod m), we find that (m k δ(m,k)+β(m,k) n+ n + β(m, k))q m k δ(m,k) (z) U(m k ) (1 qn ) mk (mod m). Reacing q by q and mutiying through by q mk one obtains (m k β(m,k) 1 n+ n + β(m, k))q m k a(m, k, n)q n (mod m). It is easy to see that (m k β(m,k) 1 n+ n + β(m, k))q m k = n 0 m k n 1 (mod ) ( m k n +1 ) q n. We concude this section with the foowing eementary resut which estabishes that the F (m, k; z) form an inductive sequence generated by the action of the U(m) oerator. Proosition 7. If m 5 is rime and k is a ositive integer, then F (m, k +1;z) F (m, k; z) U(m) (mod m). Proof. Using definition of the F (m, k; z) and the convention that (α) =0 for α Z, one finds that ( m k ) n +1 F (m, k; z) U(m) q n U(m) = n 0 m k n 1 (mod ) n 0 m k+1 n 1 (mod ) F (m, k +1;z) (mod m). ( m k+1 n +1 ) q n 3. Proof of the resuts First we reca some notation. Suose that w 1 Z, and that N is a ositive integer (with 4 N if w Z). Let S w (Γ 0 (N),χ) denote the sace of weight w cus forms with resect to the congruence subgrou Γ 0 (N) and with Nebentyus character χ. Moreover, if is rime, then et S w (Γ 0 (N),χ) denote the F -vector sace of the reductions mod of the q-exansions of forms in S w (Γ 0 (N),χ) with rationa integer coefficients.

7 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 99 Theorem 8. If m 5 is rime, then for every ositive integer k we have F (m, k; z) S m m 1 (Γ 0 (576m),χχ k 1 m ) m, where χ is the nontrivia quadratic character with conductor 1, and χ m is the usua Kronecker character for Q( m). Proof. The U(m) oerator defines a ma (see [S-St, Lemma 1]) U(m) : S λ+ 1 (Γ 0 (4Nm),ν) S λ+ 1 (Γ 0 (4Nm),νχ m ). Therefore, in view of Proosition 7 it suffices to rove that F (m, 1; z) S m m 1 (Γ 0 (576m),χ) m. If d 0 (mod 4), then it is we known that the sace of cus forms S d (Γ 0 (1)) has a basis of the form { (z) j E 4 (z) d 4 3j : 1 j [ ]} d. 1 Since the Hecke oerator T m is the same as the U(m) oerator on S 1δ(m,1) (Γ 0 (1)) m, we know that δ(m,1) (z) U(m) j 1 α j (z) j E 4 (z) 3δ(m,1) 3j (mod m), where the α j F m. However, since it is easy to see that δ(m,1) (z) =q δ(m,1), δ(m,1) (z) U(m) = n n 0 t(n)q n where n 0 δ(m, 1)/m. However, since δ(m, 1) Z, one can easiy deduce that n 0 >m/. The ony basis forms in δ(m,1) (z) U(m) (mod m) are those j (z)e 4 (z) 3δ(m,1) 3j where j>m/. This imies that ( (z) δ(m,1) U(m) ) V () η m (z) is a cus form. Since ( (z) δ(m,1) U(m) ) V () is the reduction mod m of a weight m 1 cus form with resect to Γ 0 (), and η(z) is a weight 1/ cus form with resect to Γ 0 (576) with character χ, the resut foows. Now we reca an imortant resut due to Serre [S, 6.4].

8 300 KEN ONO Theorem (Serre). The set of rimes 1 (mod N) for which f T 0 (mod m) for every f(z) S k (Γ 0 (N),ν) m has ositive density. Here T denotes the usua Hecke oerator of index acting on S k (Γ 0 (N),ν). Proof of Theorem 1. If F (m, k; z) 0 (mod m), then the concusion of Theorem 1 hods for every rime. Hence, we may assume that F (m, k; z) 0 (mod m). By Theorem 8, we know that each F (m, k; z) beongs to S m m 1 (Γ 0 (576m),χχ k 1 m ) m. Therefore each F (m, k; z) is the reduction mod m of a haf-integra weight cus form. Now we briefy reca essentia facts about the Shimura corresondence ([Sh]), a famiy of mas which send moduar of forms of haf-integra weight to those of integer weight. Athough Shimura s origina theorem was stated for haf-integra weight eigenforms, the generaization we describe here foows from subsequent works by Cira and Niwa [Ci], [Ni]. Suose that f(z) = b(n)qn S λ+ 1 (Γ 0 (4N),ψ) is a cus form where λ. If t is any square-free integer, then define A t (n) by A t (n) n s := L(s λ +1,ψχ λ 1χ t ) b(tn ) n s. Here χ 1 (res. χ t ) is the Kronecker character for Q(i) (res. Q( t)). These numbers A t (n) define the Fourier exansion of S t (f(z)), a cus form S t (f(z)) := A t (n)q n in S λ (Γ 0 (4N),ψ ). Moreover, the Shimura corresondence S t commutes with the Hecke agebra. In other words, if - 4N is rime, then S t (f T ( )) = S t (f) T. Here T (res. T ( )) denotes the usua Hecke oerator acting on the sace S λ (Γ 0 (4N),ψ ) (res. S λ+ 1 (Γ 0 (4N),ψ)). Therefore, for every square-free integer t we have that the image S t (F (m, k; z)) under the t th Shimura corresondence is the reduction mod m of an integer weight form in S m m (Γ 0 (576m),χ triv ). Now et S(m) denote the set of rimes 1 (mod 576m) for which G T 0 (mod m) for every G S m m (Γ 0 (576m),χ triv ) m. By Serre s theorem, the set S(m) contains a ositive roortion of the rimes.

9 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 301 By the commutativity of the corresondence, if S(m), then we find that F (m, k; z) T ( ) 0 (mod m), where T ( ) is the Hecke oerator of index on S m m 1 (Γ 0 (576m),χχ k 1 m ). In articuar (see [Sh]), if f = a f (n)q n S λ+ 1 (N,χ f ) is a haf-integra weight form, then (11) f T ( ):= ( a f ( n)+χ f () ) λ 1 a f (n) ( ( 1) λ n ) + χ f ( ) λ 1 a f (n/ ) q n. Therefore, if S(m) and n is a ositive integer which is corime to, then, by reacing n by n, wehave ( m ( 1) m a(m, k, n 3 )+χχ k 1 n m () Since ( ) n = 0, by Theorem 6 we find that ( m k 3 ) n +1 ) m m 4 a(m, k, n) 0 (mod m). a(m, k, 3 n) 0 (mod m). Remark. Athough Theorem 1 is a genera resut guaranteeing the existence of congruences, there are other congruences which foow from other simiar arguments based on (11). For exame, suose that is a rime for which F (m, k; z) T ( ) λ()f (m, k; z) (mod m) for some λ() F m. If n is a nonnegative integer for which - n, then (11) becomes ( m ( 1) m ) n a(m, k, n) λ() χχk 1 m () m m 4 Hence, if it turns out that λ() ± m m 4 (mod m), then there are arithmetic rogressions of integers n for which ( m a(m, k, n k ) n +1 ) 0 (mod m). a(m, k, n ) (mod m).

10 30 KEN ONO Athough we have not conducted a thorough search, it is amost certain that many such congruences exist. Proof of Theorem 3. By the roof of Theorem 6, reca that ( ) mn +1 F (m, 1; z) = n 0, mn 1 (mod ) q n S m m 1 (Γ 0 (576m),χ) m. Since m is good, for each 0 r m 1 et n r be a fixed nonnegative integer for which mn r 1 (mod ) and ( ) mnr +1 r (mod m). Let M m be the set of rimes for which n r for some r, and define S m by S m := M m. Obviousy, the form F (m, 1; z) aso ies in S m m 1 (Γ 0 (576mS m,χ) m. Therefore, by Serre s theorem and the commutativity of the Shimura corresondence, a ositive roortion of the rimes 1 (mod 576mS m ) have the roerty that F (m, 1; z) T ( ) 0 (mod m). By (11), for a but finitey many such we have for each r that ( mnr ) ( m +1 ( 1) m ) ( ) n r +χ() m m 4 mnr +1 0 (mod m). However, since 1 (mod m) this imies that ( mnr ) ( m +1 ( 1) m ) ( ) (1) χ() ( 1) m m nr r (mod m). If n r = i i where the i are rime, then ( ) nr := ( ) i. i Since n r is odd, 3 (mod 4), and 1 (mod i ), we find by quadratic recirocity that ( ) ( )( ) i 1 = i i ( ) ( ) 1 = = =1. i i

11 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 303 Therefore, for a but finitey many such congruence (1) reduces to ( mnr ) ( m +1 ( 1) m ) (13) χ() ( 1) m m r (mod m). ( ) Hence for every sufficienty arge such, the m vaues mnr +1 are distinct and reresent each residue cass mod m. To comete the roof, it suffices to notice that the number of such rimes <X, by Serre s theorem again, is X/ og X. In view of (13), this immediatey yieds the X/og X estimate. The estimate when r = 0 foows easiy from Theorem 1. Proof of Theorem 5. Since F (m, k; z) isins m m 1 (Γ 0 (576m),χχ k 1 m ) m, it foows that each F (m, k; z) ies in one of two finite-dimensiona F m -vector saces. The resut now foows immediatey from (3), Theorem 6, Proosition 7, and we-known uer bounds for the dimensions of saces of cus forms (see [C-O]). 4. Exames In this section we ist the Ramanujan cyces for the generating functions F (m, k; z) when 5 m 3. Athough we have roven that each F (m, k; z) S m m 1 (Γ 0 (576m),χχ k 1 m ) m, in these exames it turns out that they a are congruent mod m to forms of smaer weight. Cases where m =5, 7, and 11. In view of the Ramanujan congruences mod 5, 7, and 11, it is immediate that for every ositive integer k we have F (5,k; z) 0 (mod 5), F (7,k; z) 0 (mod 7), F (11,k; z) 0 (mod 11). Therefore, these Ramanujan cyces are degenerate. Case where m = 13. By [Gr-O, Pro. 4] it is known that 7 (z) U(13) 11 (z) (mod 13). Therefore by (10) and Theorem 6 it turns out that F (13, 1; z) 11q 11 +9q η 11 (z) (mod 13). Using a theorem of Sturm [St, Th. 1], one easiy verifies with a finite comutation that F (13, 1; z) T (59 ) 0 (mod 13).

12 304 KEN ONO By the roof of Theorem 1, we find that every nonnegative integer n 1 (mod ) that is corime to 59 has the roerty that ( ) n +1 0 (mod 13). Congruence () foows immediatey. Using Sturm s theorem again, one readiy verifies that By Proosition 7 this imies that η 11 (z) U(13) 8η 3 (z) (mod 13), η 3 (z) U(13) 4η 11 (z) (mod 13). F (13, ; z) 10η 3 (z) (mod 13), and more generay it imies that for every nonnegative integer k (14) (15) F (13, k +1;z) 11 6 k η 11 (z) (mod 13), F (13, k +;z) 10 6 k η 3 (z) (mod 13). These two congruences aear in Ramanujan s unubished manuscrit on τ(n) and (n), and their resence in arge art insired this entire work. From (14) and (15) we obtain the foowing easy coroary. Coroary 9. Define integers a(n) and b(n) by a(n)q n := (1 q n ) 11, b(n)q n := (1 q n ) 3. If k and n are nonnegative integers, then ( 13 k+1 ) (n +11) k a(n) (mod 13), ( 13 k+ ) (n +3) k b(n) (mod 13). Case where m =17. By [Gr-O, Pro. 4], it is known that 1 (z) U(17) 7E 4 (z) (z) (mod 17) where E 4 (z) = σ 3(n)q n is the usua weight 4 Eisenstein series. Therefore by (10) it turns out that F (17, 1; z) 7q 7 +16q η 7 (z)e 4 (z) (mod 17).

13 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 305 Again using Sturm s theorem one easiy verifies that η 7 (z)e 4 (z) U(17) 7η 3 (z)e 4 (z) (mod 17), η 3 (z)e 4 (z) U(17) 13η 7 (z)e 4 (z) (mod 17). By Proosition 7 this imies that for every nonnegative integer k F (17, k +1;z) 7 6 k η 7 (z)e 4 (z) (mod 17), F (17, k +;z) 15 6 k η 3 (z)e 4 (z) (mod 17). As an immediate coroary we obtain: Coroary 10. Define integers c(n) and d(n) by c(n)q n := E 4 (z) (1 q n ) 7, d(n)q n := E 4 (z) (1 q n ) 3. If k and n are nonnegative integers, then ( 17 k+1 ) (n +7) k c(n) (mod 17), ( 17 k+ ) (n +3) k d(n) (mod 17). Case where m =19. Using [Gr-O, Pro. 4], and arguing as above it turns out that for every nonnegative integer k F (19, k +1;z) 5 10 k η 5 (z)e 6 (z) (mod 19), F (19, k +;z) k η 3 (z)e 6 (z) (mod 19). Here E 6 (z) =1 504 σ 5(n)q n is the usua weight 6 Eisenstein series. As an immediate coroary we obtain: Coroary 11. Define integers e(n) and f(n) by e(n)q n := E 6 (z) (1 q n ) 5, f(n)q n := E 6 (z) (1 q n ) 3. If k and n are nonnegative integers, then ( 19 k+1 ) (n +5) k e(n) (mod 19), ( 19 k+ ) (n +3) k f(n) (mod 19).

14 306 KEN ONO Case where m =3. Using [Gr-O, Pro. 4], and arguing as above we have for every nonnegative integer k F (3, k +1;z) 5 k η(z)e 4 (z)e 6 (z) (mod 3), F (3, k +;z) 5 k+1 η 3 (z)e 4 (z)e 6 (z) (mod 3). Coroary 1. Define integers g(n) and h(n) by g(n)q n := E 4 (z)e 6 (z) (1 q n ), h(n)q n := E 4 (z)e 6 (z) (1 q n ) 3. If k and n are nonnegative integers, then ( 3 k+1 ) (n +1)+1 5 k g(n) (mod 3), ( 3 k+ ) (n +3)+1 5 k+1 h(n) (mod 3). Pennsyvania State University, University Park, Pennsyvania E-mai address: ono@math.su.edu References [A] S. Ahgren, Distribution of arity of the artition function in arithmetic rogressions, Indagationes Math., to aear. [An-G] G. E. Andrews and F. Garvan, Dyson s crank of a artition, Bu. Amer. Math. Soc. 18 (1988), [At] A. O. L. Atkin, Mutiicative congruence roerties and density robems for (n), Proc. London Math. Soc. 18 (1968), [At-Ob] A. O. L. Atkin and J. N. O Brien, Some roerties of (n) and c(n) moduo owers of 13, Trans. Amer. Math. Soc. 16 (1967), [At-Sw1] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Moduar forms on noncongruence subgrous, Combinatorics (Proc. Symos. Pure Math., Univ. of Caifornia, Los Angees, 1968), XIX (1971), 1 5. [At-Sw], Some roerties of artitions, Proc. Lond. Math. Soc. 4 (1954), [B-O] B. C. Berndt and K. Ono, Ramanujan s unubished manuscrit on the artition and tau functions with roof and commentary, Sém. Lothar. Combin. 4 (1999). [Ci] B. Cira, On the Niwa-Shintani theta kerne ifting of moduar forms, Nagoya Math. J. 91 (1983), [C-O] H. Cohen and J. Oesteré, Dimensions des esaces de formes moduaires, Moduar functions of one variabe, VI (Proc. Second Int. Conf. Univ. Bonn, 1976), 69 78, Lecture Notes in Math. 67 (1977), Sringer-Verag, New York. [E-I] P. Erdös and A. Ivić, The distribution of certain arithmetica functions at consecutive integers, in Proc. Budaest Conf. Number Th., Co. Math. Soc. J. Boyai, North-Hoand, Amsterdam 51 (1989),

15 DISTRIBUTION OF THE PARTITION FUNCTION MODULO m 307 [G] F. Garvan, New combinatoria interretations of Ramanujan s artition congruences mod 5, 7, and 11, Trans. Amer. Math. Soc. 305 (1988), [G-K-S] F. Garvan, D. Kim, and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), [Go] B. Gordon, rivate communication. [Gr-O] A. Granvie and K. Ono, Defect zero -bocks for finite sime grous, Trans. Amer. Math. Soc. 348 (1996), [I] A. Ivić, rivate communication. [K-O] I. Kiming and J. Osson, Congruences ike Ramanujan s for owers of the artition function, Arch. Math. (Base) 59 (199), [K] O. Koberg, Note on the arity of the artition function, Math. Scand. 7 (1959), [N1] M. Newman, Periodicity moduo m and divisibiity roerties of the artition function, Trans. Amer. Math. Soc. 97 (1960), [N], Congruences for the coefficients of moduar forms and some new congruences for the artition function, Canad. J. Math. 9 (1957), [Ni-R-Sa] J.-L. Nicoas, I. Z. Ruzsa, and A. Sárközy, On the arity of additive reresentation functions (with an aendix by J-P. Serre), J. Number Theory 73 (1998), [Ni] S. Niwa, Moduar forms of haf integra weight and the integra of certain theta functions, Nagoya Math. J. 56 (1975), [O1] K. Ono, Parity of the artition function in arithmetic rogressions, J. Reine Angew. Math. 47 (1996), [O], The artition function in arithmetic rogressions, Math. Ann. 31 (1998), [Ra ] S. Ramanujan, Congruence roerties of artitions, Proc. London Math. Soc. 19 (1919), [Sc-W] A. Schinze and E. Wirsing, Mutiicative roerties of the artition function, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), [S] J-P. Serre, Divisibiité de certaines fonctions arithmétiques, L Ensein. Math. (1976), [S-St] J-P. Serre and H. Stark, Moduar forms of weight 1/, Lecture Notes in Math. 67 (1971), [Sh] G. Shimura, On moduar forms of haf integra weight, Ann. of Math. 97 (1973), [Stu] J. Sturm, On the congruence of moduar forms, in Number Theory, 75 80, Lecture Notes in Math. 10 (1987), Sringer-Verag, New York. (Received December 14, 1998)