EXTENSION OF RAMANUJAN S CONGRUENCES FOR THE PARTITION FUNCTION MODULO POWERS OF 5. Jeremy Lovejoy and Ken Ono. Appearing in Crelle s Journal

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1 EXTENSION OF RAMANUJAN S CONGRUENCES FOR THE PARTITION FUNCTION MODULO POWERS OF 5 Jeremy Lovejoy ad Ke Oo Appearig i Cree s Joura 1. Itroductio ad Statemet of Resuts A partitio of a positive iteger is ay o-icreasig sequece of positive itegers whose sum is. Let p( deote the umber of partitios of. (As usua, we adopt the covetio that p(0 = 1 ad p(α = 0 if α N. Ramauja s famous cogrueces, which were proved by Atki, Ramauja ad Watso [2, 3, 13], assert that if j is a positive iteger, the p(5 j N + β 5 (j 0 (mod 5 j, (1 p(7 j N + β 7 (j 0 (mod 7 [j/2]+1, (2 p(11 j N + β 11 (j 0 (mod 11 j (3 for every o-egative iteger N where β m (j := 1/24 (mod m j. These cogrueces are quite strikig sice a cursory examiatio of vaues of the partitio fuctio fais to revea further cogrueces. The mere questio as to whether there are ifiitey may other cogrueces of the form p(an + B 0 (mod M had remaied ope for some time. Athough works by the secod author [7, 8] have goe some way toward quatifyig the rarity of these cogrueces, it is ow kow that there 1991 Mathematics Subject Cassificatio. Primary 11P83; Secodary 05A17. Key words ad phrases. partitio fuctio, Ramauja s Cogrueces. The secod author is supported by a NSF Eary Career grat, a Afred P. Soa Foudatio Research Feowship, ad a David ad Lucie Packard Research Feowship. 1 Typeset by AMS-TEX

2 2 JEREMY LOVEJOY AND KEN ONO are ideed ifiitey may such cogrueces. I fact, Ahgre ad the secod author [1, 9] have show that there are such cogrueces for every moduus M which is coprime to 6. Ufortuatey, these resuts are ot costructive. I fact, to our kowedge o expicit exampes of such cogrueces are kow with prime moduus M > 31. I a simiar directio, it is atura to ask whether the modui i Ramauja s cogrueces (1-3 are optima. I particuar, are there subprogressios, besides those foud by Ramauja, where the kow cogruece moduo m j is a cogruece moduo m j+1? I this paper we revisit (1 ad aswer this questio i the affirmative by expicity exhibitig ifiitey may such progressios for each j. Usig the ideas foud i [9], oe ca obtai simiar extesios of (2 ad (3. Ufortuatey, these extesios are ot paatabe. For coveiece, defie ratioa umbers β(j, by { 19 5j if j is odd, β(j, := 23 5 j if j is eve. Notice that β(j, 1 = β 5 (j. I this otatio, we obtai the foowig systematic famiies of mutipicative cogrueces. Theorem 1. Let 7 be prime. 1 If j 1 is odd, the for every o-egative iteger we have p(5 j 2 + β(j, (4 ( ( ( ( p(5 j j + β 5 (j p 2 + β(j, 1 (mod 5 j+1. 2 If j 2 is eve, the for every o-egative iteger we have p(5 j 2 + β(j, ( ( ( ( 5 p(5 j j + β 5 (j p 2 + β(j, 1 (mod 5 j+1. Usig Theorem 1, we obtai two coroaries which revea extesios of a of Ramauja s cogrueces moduo powers of 5. I both cases, for every positive iteger j we costruct ifiitey may distict o-trivia subprogressios of the arithmetic progressio 5 j N + β 5 (j for which Ramauja s cogruece moduo 5 j is a cogruece moduo 5 j+1.

3 RAMANUJAN S CONGRUENCES MODULO POWERS OF 5 3 Coroary 2. Let 4 (mod 5 be prime. 1 If j 1 is odd, et 0 r, s 1 be itegers such that (i 24r (mod, (ii 24s + 24r (mod 2. 2 If j 2 is eve, et 0 r, s 1 be itegers such that (i 24r (mod, (ii 24s + 24r (mod 2. The for every o-egative iteger N we have p(5 j 4 N + 5 j 3 s + 5 j 2 r + β(j, 0 (mod 5 j+1. Coroary 3. Let 7 3 (mod 5 be prime. If j 1 is odd (resp. eve ad 0 r 1 is a iteger for which ( ( 24r 19 = 1 (resp. 24r 23 = 1, the for every o-egative iteger N we have p(5 j 3 N + 5 j 2 r + β(j, 0 (mod 5 j+1. It is easy to see that a of the arithmetic progressios i Coroaries 2 ad 3 are ot subprogressios of 5 j+1 N + β 5 (j + 1. For exampe, if j is odd i Coroary 3 the 5 j 3 N + 5 j 2 r + β(j, β(j + 1, 1 0 (mod 5 j+1 5 j 3 N + 5 j 2 r j N r 1 0 (mod 5 which is obviousy ot true for a N. 23 5j (mod 5 j+1 Exampes. Here we iustrate the utiity of Coroaries 2 ad 3. If j = 1, = 19, ad r = s = 0 i Coroary 2, the we have p(651605n (mod 25. Simiary, if j = 1, = 13 ad r = 1 i Coroary 3, the we have p(10985n (mod 25. If we et = 13 ad r = 1 i Coroary 3, the for every j 1 we have p(5 j 13 3 N + 5 j β(j, 13 0 (mod 5 j+1.

4 4 JEREMY LOVEJOY AND KEN ONO 2. The importat observatios As usua, et η(z deote Dedekid s eta-fuctio give by the ifiite product η(z := q 1 24 (1 q where q := e 2πiz. If χ is the quadratic character 1 if ±1 (mod 12, χ( := 1 if ±5, (mod 12, 0 otherwise, the Euer s cassica Petagoa Number Theorem asserts that η(24z = χ(q 2. This fact is very usefu for computig the coefficiets of a the moduar forms i this paper. We sha study the two moduar cusp forms F (z := η 19 (24z = a(q = q 19 19q q 67..., (6 G(z := η 23 (24z := (5 b(q = q 23 23q q (7 It is easy to deduce (see [5] that F (z is a cusp form i S 19/2 (Γ 0 (576, χ ad that G(z S 23/2 (Γ 0 (576, χ. The foowig theorem was proved by Newma. Lemma 2.1. (Newma [Th. 1, 6] If 5 is prime, the defie λ a ( ad λ b ( by ( 57 λ a ( := a( , ( 69 λ b ( := b( For every positive iteger we have λ a (a( = a( ( 3 λ b (b( = b( ( 3 a( + 17 a(/ 2, b( + 21 b(/ 2.

5 RAMANUJAN S CONGRUENCES MODULO POWERS OF 5 5 Lemma 2.1 states that F (z ad G(z are eigeforms of the haf iteger weight Hecke operators. Reca that if g(z = c(q M λ+ 1 (Γ 0(4N, ψ is a haf iteger weight 2 moduar form ad p 4N is prime, the the Hecke operator T λ (p 2 is give by g T λ (p 2 := ( ( ( 1 c(p 2 λ + ψ(p p λ 1 c( + ψ(p 2 p 2λ 1 c(/p 2 q. p Moreover, g is a eigeform if for every prime p 4N there is a compex umber λ g (p for which g T λ (p 2 = λ g (pg. It turs out that the eigevaues λ a ( ad λ b ( satisfy the foowig coveiet cogrueces. Without such cogrueces, it seems imposssibe to obtai cea extesios of (1. Theorem 2.2. If 5 is prime, the λ a ( λ b ( ( 15 (1 + (mod 5. To prove this theorem we sha empoy some we kow facts about moduar forms moduo ad the Shimura correspodece [10]. This correspodece is a famiy of maps which sed moduar of forms of haf-itegra weight to those of iteger weight. Suppose that f(z = b(q S λ+ 1 (Γ 0(4N, ψ is a eigeform with λ 2. If t is ay square-free 2 iteger, the defie A t ( by A t ( s := L(s λ + 1, ψχ λ 1χ t b(t 2 s. Here χ 1 (resp. χ t = ( t is the Kroecker character for Q(i (resp. Q( t. These umbers A t ( defie the Fourier expasio of S t,λ (f(z, a cusp form S t,λ (f(z := A t (q i S 2λ (Γ 0 (4N, ψ 2. For us, the importat feature of the Shimura correspodece S t,λ is the fact that it commutes with the Hecke agebra. I other words, if p 4N is prime, the S t,λ (f T λ (p 2 = S t,λ (f T λ p. (8

6 6 JEREMY LOVEJOY AND KEN ONO Here T λ p (resp. T λ (p 2 deotes the usua Hecke operator o the space S 2λ (Γ 0 (4N, ψ 2 (resp. S λ+ 1 2 (Γ 0(4N, ψ. Proof of Theorem 2.2. Lemma 2.1 impies that F (z = η 19 (24z S 19/2 (Γ 0 (576, χ ad G(z = η 23 (24z i S 23/2 (Γ 0 (576, χ are eigeforms of the haf-iteger weight Hecke operators o M 19/2 (Γ 0 (576, χ ad M 23/2 (Γ 0 (576, χ respectivey. Now et F(z be the eigeform which is the image of F (z uder S 19,9, ad et G(z be the image of G(z uder S 23,11. The first few terms of F(z ad G(z are F(z = G(z = A(q = q q q 7..., B(q = q q q 7... Sice A(1 = B(1 = 1, for every prime 5 we have that A( (resp. B( is the eigevaue of F(z (resp. G(z with respect to T 9 11 (resp. T. Therefore, by the commutativity of the Shimura correspodece (8 we have that A( = λ a ( ad B( = λ b ( for every prime 5. Athough Shimura s correspodece guaratees that F(z S 18 (Γ 0 (288, χ triv (resp. G(z S 22 (Γ 0 (288, χ triv, it turs out that F(z is i S 18 (Γ 0 (144, χ triv ad that G(z is i S 22 (Γ 0 (144, χ triv (see the Appedix. If σ k ( deotes the sum of the kth powers of the positive divisors of a iteger, the it suffices to prove that ( 60 F(z G(z σ 1 (q (mod 5. (9 This foows from the fact that A( = B( = 0 if gcd(, 6 1. This fact is easiy deduced from the defiitio of S 19,9 ad S 23,9 ad the fact that a( = 0 (resp. b( = 0 uess 19 (mod 24 (resp. 23 (mod 24. Reca that the cassica weight 6 Eisestei series E 6 (z o SL 2 (Z is give by E 6 (z = σ 5 (q 1 + σ 1 (q (mod 5. O the basis of Ramauja s study of differetia operators o moduar forms, Swierto- Dyer [Lemma 5, 12] proved that if 5 is prime ad c(q is a weight k moduar form with iteger coefficiets, the there is a weight k moduar form α(q o SL 2 (Z with iteger coefficiets whose Fourier expasio satisfies α(q c(q (mod.

7 RAMANUJAN S CONGRUENCES MODULO POWERS OF 5 7 By appyig this procedure twice to E 6 (z with = 5, we fid that there is a weight 18 moduar form H 0 (z = C(q with respect to SL 2 (Z such that H 0 (z = C(q 2 σ 1 (q If H 1 (z is the χ quadratic twist of H 0 (z, the we have H 1 (z = χ(c(q ( 5 σ 1 (q (mod 5. (10 ( 60 σ 1 (q (mod 5. By [III 3 Prop. 17, 5], H 1 (z is i the space M 18 (Γ 0 (144, χ triv. Therefore, cogruece (9 for F(z is equivaet to the assertio that H 1 (z F(z (mod 5. By a theorem of Sturm [Th. 1, 11], it suffices to show that A( ( 60 σ 1 ( (mod 5 for every 433. A simpe computatio verifies the cogruece (see Appedix for detais o computig F(z. Cogruece (9 for G(z ca be haded simiary. Usig the fact that the cassica Eisestei series E 4 (z = σ 3 (q 1 (mod 5, it suffices to check that the weight 22 moduar form H 1 (ze 4 (z i M 22 (Γ 0 (144, χ triv satisfies the cogruece H 1 (ze 4 (z G(z (mod 5. Usig Sturm s theorem agai, this cogruece is easiy verified by checkig that B( ( 60 σ 1 ( (mod 5 for every 529 (see Appedix for detais o computig G(z.

8 8 JEREMY LOVEJOY AND KEN ONO 3. Proof of Theorem 1 ad Coroaries 2 ad 3 We begi by recaig the foowig cassica fact (see [p. 111, 13]. Theorem 3.1. If j 1, the the geeratig fuctio for the umbers p(5 j + β 5 (j is of the form p(5 j + β 5 (jq = ( i 1 i 1 x j,i q i 1 ( x j,i q i 1 (1 q 5 6i 1 (1 q 6i, if j is odd, (1 q 5 6i (1 q, if j is eve, 6i+1 (11 where { 3 j 1 5 j (mod 5 j+1 if i = 1, x j,i = 0 (mod 5 j+1 if i 2. The foowig coroary carifies the importace of Lemma 2.1. Coroary 3.2. If j 1, the for every o-egative iteger we have { 3 j 1 p(5 j 5 j a( (mod 5 j+1 if j is odd, + β 5 (j 3 j 1 5 j b( (mod 5 j+1 if j is eve. Proof. From Theorem 3.1, if j 1 is odd, the 1 3 j 1 5 j p(5 j + β 5 (jq q 19 (1 q (1 q 24 6 a(q (mod 5. Simiary, if j 2 is eve, the 1 3 j 1 5 j p(5 j + β(5, jq q 23 (1 q (1 q 24 7 b(q (mod 5. The resut foows immediatey. Proof of Theorem 1. Lemma 2.1 impies that a( 2 {( ( ( } a( a(/ 2 (mod 5 (12

9 ad RAMANUJAN S CONGRUENCES MODULO POWERS OF 5 9 b( 2 {( ( ( } b( b(/ 2 (mod 5. (13 Cogruece (12 foows from the simpe observatio that ( 5 2 (mod 5. Repacig by ad i (12 ad (13, respectivey, ad appyig Coroary 3.2 immediatey estabishes the resut. We cocude with the proofs of Coroaries 2 ad 3. Proof of Coroary 2. Repace by N 2 + s + r i Theorem 1 ad ote that 5 j ( 2 N + s + r 2 + β(j, 1 is ot a iteger. Therefore, sice ( a = 0 if a ad p(5 j + β 5 (j 0 (mod 5 j, we have that p(5 j 4 N + 5 j 3 s + 5 j 2 r + β(j, 0 (mod 5 j+1. Proof of Coroary 3. Repace by N + r ad it is easy to see that 5 j (N + r 2 + β(j, 1 caot be a iteger. Therefore, sice p(5 j + β 5 (j 0 (mod 5 j we have that p(5 j 3 N + 5 j 2 r + β(j, 0 (mod 5 j+1. Appedix Here we obtai a cosed formua for the coefficiets of the ewform F(z. Computig G(z is haded simiary. Let F 0 (z = A 0(q (resp. G 0 (z = B 0(q be the uique ewform i the space S 18 (Γ 0 (6, χ triv (resp. S 22 (Γ 0 (6, χ triv whose Fourier expasio are F 0 (z = q 256q q q q q G 0 (z = q q q q q q 6....

10 10 JEREMY LOVEJOY AND KEN ONO The ewform F(z (resp. G(z is the χ quadratic twist of F 0 (z (resp. G 0 (z. I particuar, we have that F(z = G(z = A(q = B(q = gcd(,6=1 gcd(,6=1 ( 12 A 0 (q, (14 ( 12 B 0 (q. (15 The proof of Theorem 2.2 requires the first 865 (resp terms of F(z. Defie ratioa umbers D( by D(q = η 29 (zη 5 (2zη(3zη(6z η 25 (zη(2zη 5 (3zη 5 (6z 4734 ( η 24 (zη 6 (2zη 6 (6z η 20 (zη 2 (2zη 4 (3zη 10 (6z η18 (zη 18 (3z + ( η 29 (zη 5 (2zη(3zη(6z U(2 + ( η24 (zη 6 (2zη 6 (6z η25 (zη(2zη 5 (3zη 5 (6z U(2. As usua, the U-operator is defied by ( b(q U(M := b(mq. It turs out that if is coprime to 6, the D( = A 0 ( = χ(a(. (17 Oe may use (14 ad (17 to compute the first 865 coefficiets of the ewform F(z. Simiary, it is staightforward to obtai G(z i terms of eta-products ad their images uder certai Hecke operators. For brevity we omit the detais.

11 RAMANUJAN S CONGRUENCES MODULO POWERS OF 5 11 Ackowedgemets The authors thak the referee for severa importat poits which improved ad corrected a earier versio of this paper. Refereces [1] S. Ahgre, Distributio of the partitio fuctio moduo composite itegers M, accepted for pubicatio, Math. Aae. [2] A. O. L. Atki, Proof of a cojecture of Ramauja, Gasgow Math. J. 8 (1967, [3] B. C. Berdt ad K. Oo, Ramauja s upubished mauscipt o the partitio ad tau fuctios with proofs ad commetary, Sém. Lothar. Combi., The Adrews Festschrift 42 (1999, Art. B42c. [4] M. Hirschhor ad D. C. Hut, A simpe proof of the Ramauja cojecture for powers of 5, J. reie agew. math. 336 (1981, [5] N. Kobitz, Itroductio to eiptic curves ad moduar forms, Spriger Verag, New York, [6] M. Newma, Further idetities ad cogrueces for the coefficiets of moduar forms, Caadia J. Math. 10 (1958, [7] K. Oo, Parity of the partitio fuctio i arithmetic progressios, J. reie age. math. 472 (1996, [8] K. Oo, The partitio fuctio i arithmetic progressios, Math. A. 312 (1998, [9] K. Oo, The distributio of the partitio fuctio moduo m, Aas of Mathematics 151 (2000, [10] G. Shimura, O moduar forms of haf-itegra weight, Aas of Mathematics 97 (1973, [11] J. Sturm, O the cogruece of moduar forms, Spriger Lect. Notes i Math (1984, [12] H. P. F. Swierto-Dyer, O -adic represetatios ad cogrueces for coefficiets of moduar forms, Moduar fuctios of oe variabe III, Spriger Lect. Notes i Math. 350 (1973, [13] G. N. Watso, Ramauja s vermutug über zerfäugsazahe, J. reie agew. math. 179 (1938, Dept. of Mathematics, Uiversity of Wiscosi, Madiso, Wiscosi E-mai address: ovejoy@math.wisc.edu Dept. of Math., Uiversity of Wiscosi, Madiso, Wiscosi E-mai address: oo@math.wisc.edu