NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES

Transcription

1 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES MATTHEW BOYLAN Abstract Let pn be the ordinary partition function We show, for all integers r and s with s 1 and 0 r < s, that #{n : n r mod s and pn 0 mod } r,s log We also prove a similar result on the non-vanishing of pn modulo l {5, 7, 11} 1 Introduction and statement of results A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n We denote by pn the number of partitions of n By convention, we set p0 := 1 Euler showed that the generating function for pn satisfies 11 pnq n = n=0 m=1 1 1 q m Among the most well-known contributions of Ramanujan to the study of partitions are the congruences which now bear his name These are, for all integers n 0, p5n mod 5 p7n mod 7 p11n mod 11 After Ramanujan, relatively few linear congruences for pn beyond 12, 1, 14, and their extensions modulo powers of 5, 7, and 11 had been discovered until recent work of Ahlgren and Ono [, 5, 11] They proved, for every modulus M coprime to 6, that there are infinitely many arithmetic progressions An + B, none contained in any other, such that pan + B 0 mod M for all n By contrast, less is known about pn modulo 2 or In this paper, we consider the following conjecture Conjecture S Let r and t be integers with 0 r < t 1 There are infinitely many integers n r mod t such that pn 0 mod 2 2 There are infinitely many integers m r mod t such that pm 0 mod Remark 1 Subbarao [15] conjectured 1 in 1966, while Ahlgren and Ono conjectured 2 in 2001 [4, Conjecture 52] For other conjectures and problems on the arithmetic of pn, many of which remain unsettled, the reader should consult [12, Section 54] Date: May 1, Mathematics Subject Classification 11P, 11F11 1

2 2 MATTHEW BOYLAN Remark 2 Conjecture S implies that there are no linear congruences for pn modulo 2 or Remark In earlier work, Ono [10] proved the even analogue of 1: For all integers r and t with 0 r < t, there are infinitely many n r mod t such that pn 0 mod 2 Combined with an estimate of Serre [9], this work gives #{n : n r mod t and pn 0 mod 2} r,t In recent unpublished work, Ono has also proved that there are infinitely many n for which pn 0 mod using a generalization of Borcherds theory of automorphic infinite products However, his result is not refined to the point where one knows that there are infinitely many such n in any fixed arithmetic progression We now summarize the main results on Conjecture S Prior to the mid 1990 s, part 1 was known see [10] for references for every arithmetic progression r mod t with t {1, 2,, 4, 5, 6,, 10, 12, 16, 20, 40} In [10], Ono showed that if there is at least one integer n r mod t with pn 0 mod 2, then there are infinitely many such n Hence, to verify part 1 of Conjecture S for a fixed arithmetic progression r mod t, it suffices to give a single integer n r mod t for which pn 0 mod 2 This was done by Ono in [10] for every arithmetic progression r mod t with t 10 5 and by Ono and the author in [6] for every arithmetic progression whose modulus is a power of 2, thus providing the first infinite family of arithmetic progressions for which part 1 of Conjecture S is true Theorem 11 Let r and s be integers with s 1 and 0 r < 2 s Then part 1 of Conjecture S holds for the arithmetic progression r mod 2 s Moreover, we have #{n : n r mod 2 s and pn 0 mod 2} r,s log The estimate in Theorem 11 follows from our work together with a theorem of Ahlgren [1, Theorem 14] In this paper, we furnish the first infinite family of arithmetic progressions for which part 2 of Conjecture S is true Our methods also apply to the study of the non-vanishing of pn modulo 5 and 7 We prove Theorem 12 Let s 1 be an integer 1 Suppose that r is an integer with 0 r < s Then part 2 of Conjecture S holds for the arithmetic progression r mod s Moreover, we have #{n : n r mod s and pn 0 mod } r,s log 2 There is an integer g s 4 mod 5 such that for all integers u with 0 5u + g s < 5 s, we have #{n : n 5u + g s mod 5 s and pn 0 mod 5} u,s log

3 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES There is an integer h s 5 mod 7 such that for all integers v with 0 7v +h s < 7 s, we have #{n : n 7v + h s mod 7 s and pn 0 mod 7} v,s log Remark 1 The methods we use to prove Theorem 12 are natural adaptations and generalizations of the methods used to prove Theorem 11 Most if not all of the results in Sections 2-4 are stated for l {, 5, 7}, but also hold for l = 2 Remark 2 Theorem 12 continues to hold for l = 11, and the proof in this case is a modification of the proof for l {5, 7} In particular, one can show that if s 1 is an integer, then there is an integer i s 6 mod 11 such that for all integers w with 0 11w + i s < 11 s, we have #{n : n 11w + i s mod 11 s and pn 0 mod 11} w,s log To see this, one observes that the proof of Theorem 12 for l {5, 7} depends on Lemma 22 In turn, Lemma 22 relies crucially on the existence of the Ramanujan congruences modulo l {5, 7} and the fact that, for l {5, 7}, a modular mod l Galois representation ramified only at l must be reducible Therefore, in view of the Ramanujan congruence modulo 11, to produce a generalization of Lemma 22 to l = 11, we need to study modular mod 11 Galois representations ramified only at 11 Unfortunately, there is an irreducible modular Galois representation of this type arising from the fact that z η 2 zη 2 11z mod 11, where ηz is Dedekind s eta-function This representation is induced by the Galois action on the 11-torsion points of 0 11 Using fundamental lemmas of Swinnerton-Dyer [16, Lemma 7] and Ono and Skinner [1], one can show that there exists a set of primes m 1 mod 11 which satisfy some other complicated conditions coming from the properties of 0 11 for which the conclusion of Lemma 22 holds for l = 11 We emphasize that for the purpose of proving Theorem 12 and its analogue for l = 11, we merely require the existence of such primes; a simple, explicit description is not necessary However, a simple, explicit description of the primes m in Lemma 22 is possible when l {5, 7} since the associated modular mod l Galois representations are reducible in these cases Before giving the proof of Theorem 12, we cite an important fact due to Ahlgren [2] Theorem 1 Let l be an odd prime, and let r and t be integers with 0 r < t Suppose that there is an integer n r mod t such that pn 0 mod l Then we have #{n : n r mod t and pn 0 mod l} r,t log Let l {, 5, 7}, and let s 1 In view of Theorem 1, to prove Theorem 12 for a fixed arithmetic progression r mod l s, it suffices to exhibit a single n r mod l s with pn 0 mod l We devote Sections 2-4 of this paper to this task In Section 2, we relate modular forms to partitions In particular, we state a precise lemma on the non-vanishing of modular form coefficients modulo l, and we show that its truth implies Theorem 12 In Section, we study the non-vanishing of modular form coefficients modulo l, and in Section 4, we use the results from Section to complete the proof of Theorem 12

4 4 MATTHEW BOYLAN Acknowledgments The author thanks K Ono for several helpful conversations in the preparation of this paper 2 Modular forms and partitions The proofs of our theorems require the theory of modular forms see, for example, [] for details We begin by fixing notation Let k 4 be an even integer We denote by M k the finite-dimensional complex vector space of holomorphic modular forms of weight k on SL 2 Z Setting q := e 2πiz, we identify a modular form fz M k with its q-expansion, fz = a f nq n n=0 The subspace of cusp forms in M k, denoted by S k, consists of forms fz with a f 0 = 0 The principal cusp form of interest to us is the Delta function of Ramanujan, given by 21 z := q 1 q n 24 S 12 Let s 1 be an integer, and let l {, 5, 7} For integers n 1, we define integers a s,l n by 22 a s,l nq n := { z l2s 1 if l = z l2s 1 24 if l {5, 7} We also define, for every integer j 0, integers r s,l j by 2 24 r s, jq 9sj := 1 q 9sn, r s,l jq l2sj := 1 q l2sn if l {5, 7} The following proposition establishes a recursive formula modulo l for the modular form coefficients a s,l n in terms of partition values in certain arithmetic progressions modulo powers of l 2 Proposition 21 Let n 1 be an integer, and let l {, 5, 7} 1 We have a s,l n r s,l jp r s,l jp l 2s j l 2s j n l2s 1 n l 2s If n 0 or 2 mod, then we have a s, n 0 mod mod l if l = mod l if l {5, 7} If l {5, 7} and n 0 mod l, then we have a s,l n 0 mod l

5 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES 5 Proof We prove the l = case of 1 The proofs of the l = 5 and 7 cases are similar Computing modulo using 11, 21, 22, and 2, we obtain a s, nq n = q 9s 1 1 q n 1 9s 1 q m q 9s 1 1 q 9sn m=1 m=1 r s, jq 9s j n=0 k=0 n 9 s 1 r s, jp 1 1 q m pkq k+ 9s 1 9 s j q n mod To prove 2, we note that 9s 1 1 mod for all s 1 Therefore, if n 0, 2 mod, part 1 of the proposition implies that a s, n 0 mod To prove for l = 5 respectively l = 7, we observe by 12 respectively 1 that for every integer n, p ln l2s 1 l 2s j 0 mod l 24 For l {5, 7}, it now follows by part 1 of the proposition that a s,l ln 0 mod l The following lemma gives, for our purposes, the crucial non-vanishing properties modulo l of the coefficients a s,l n Lemma 22 Let s 1 be an integer, and let l {, 5, 7} There is an integer n s,l 1 such that { 0, 2 mod l if l = 1 n s,l 0 mod l if l {5, 7} 2 a s,l n s,l 0 mod l For every prime m n s,l with m 1 mod l, we have a s,l n s,l m 2 0 mod l We defer the proof of Lemma 22 to Section 4 significance of Lemma 22 Proposition 2 Lemma 22 implies Theorem 12 The next proposition highlights the Proof We first prove the proposition in the case l = By Theorem 1, it suffices to show, for all integers r and s with s 1 and 0 r < 9 s, that there is an integer M s,r r mod 9 s for which pm s,r 0 mod By Lemma 22, there is a positive integer n s, 1 mod such that for every prime m n s, with m 1 mod, we have a s, n s, m 2 0 mod Using Proposition 21, we see that 0 a s, n s, m 2 ns, m 2 9s 1 r s, jp 9 s j Hence, it follows that there is an integer j 0 for which ns, m 2 9s 1 25 p 9 s j 0 mod mod

6 6 MATTHEW BOYLAN We emphasize that 25 holds for all primes m n s, with m 1 mod Using Hensel s Lemma, one can show that the quadratic congruence in the variable x, n s, x 2 9s 1 26 r mod 9 s, has a unique solution x mod 9 s with x 1 mod By Dirichlet s Theorem, there are infinitely many primes m x mod 9 s In particular, there must be a prime m s,r n s, with m s,r 1 mod satisfying 26 The result now follows with M s,r = n s,m 2 s,r 9s 1 The l = 5 case differs slightly from the l = case Let s 1 be an integer, and let n s,5 be the integer whose existence is guaranteed by the l = 5 case of Lemma 22 By Theorem 1, it suffices to show, for all integers r with r n s,5 1 mod 5 and 0 r < 25 s, that there is an integer N s,r r mod 25 s for which pn s,r 0 mod 5 We note that r 4 mod 5 since n s,5 0 mod 5 To conclude the proof, we argue as in the case of l =, using the properties of n s,5, Lemma 21, Hensel s Lemma, and Dirichlet s Theorem to produce an integer N s,r with the required properties The l = 7 case is similar Modular form coefficients modulo, 5, and 7 In view of Lemma 2, it suffices to prove Lemma 22 In this section, we use modular forms modulo primes l, Hecke operators, and Galois representations to develop general techniques for studying the non-vanishing modulo l {, 5, 7} of l-integral Fourier coefficients of modular forms on SL 2 Z To begin, let l be prime, and let fz = anq n and gz = bnq n M k Z l [[q]], where Z l is the localization of the integers at l We say that fz gz mod l if and only if for all integers n 0, an bn mod l For every prime p, there is a Hecke operator T p,k 1 anq n T p,k = n=0 We note that if p n, then a n p n=0 : S k S k whose action is given by n q n p apn + p k 1 a = 0 We say that fz S k Z l [[q]] is an eigenform for the Hecke operator T p,k modulo l with eigenvalue λ p Z/lZ if fz T p,k λ p fz mod l The following lemma gives the key fact underlying the proof of Lemma 22 Lemma 1 Fix l {2,, 5, 7} Suppose that for every prime p l, fz S k Z l [[q]] is an eigenform for T p,k modulo l If p 1 mod l, then we have fz T p,k 0 mod l Proof Suppose, for primes p l, that fz has eigenvalues λ p Z/lZ for the Hecke operators T p,k A well-known theorem of Deligne [7, Théorème 67] states that there is a continuous semisimple Galois representation ρ l,f : GalQ/Q GL 2 Z/lZ,

7 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES 7 unramified outside l, such that for all primes p l, Trρ l,f Frob p λ p mod l Now, let χ l : GalQ/Q Z/lZ be the mod l cyclotomic character By a result of Atkin, Serre, and Tate [14, Theorem ], there is an integer i 0 such that ρ l,f χ i l comes from an eigenform in M k with k l + 1 If l {2,, 5, 7} and k l + 1, then S k = {0}, so ρ l,f must be reducible For all primes p l, we have χ l Frob p p mod l Since ρ l,f is odd Detρ l,f c 1 mod l when c is complex conjugation and reducible, there are m, n Z/l 1Z with m + n odd for which ρ l,f = χ m l χ n l Hence, observing that n m must be odd, if p 1 mod l is prime, then we have which proves the lemma λ p Trρ l,f Frob p p m + p n p m 1 + p n m 0 mod l, Before giving the principal application of Lemma 1, we recall certain facts on modular forms modulo primes l Let k 4 be an even integer We require the normalized Eisenstein series of weight k on SL 2 Z defined by E k z := 1 2k B k d k 1 q n M k, where B k is the kth Bernoulli number We note the well-known fact that if l is prime and l 1 k, then 2 E k z 1 mod l If j 1 is an integer, it is also well-known that S 12j has bases given by, for example, 4 {E 4 z j z,, E 4 z j t z t,, z j }, {E 6 z 2j 2 z,, E 6 z 2j 2s z s,, z j } Now, fix l {, 5, 7}, and let fz S 12j Z l [[q]] Using 2,, and 4, it follows that there are integers a 1,l,, a j,l Z/lZ for which j 5 fz a i,l z i mod l i=1 In view of 5, for such primes l, we also require information on z j T l,12j d n mod l Proposition 2 Fix l {, 5, 7}, and let j 1 be an integer There are integers c j,l and d j,l with 0 d l,j < j for which z j T l,12j c j,l z d j,l mod l Remark To illustrate, when l =, we have { z j z j mod if j T,12j 0 mod if j This follows from formula 1, together with the easily verified facts that z T,12 0 mod and z 2 T,24 0 mod

8 MATTHEW BOYLAN We now turn to the principal application of Lemma 1 Lemma Fix l {, 5, 7}, let j 1 be an integer, and let fz S 12j Z l [[q]] have fz 0 mod l Then exactly one of the following must hold 1 For every prime p with p 1 mod l or p = l, we have fz T p,12j 0 mod l 2 There is an integer i f,l 2 for which the following are true: There are i f,l 1 distinct primes p 1,, p if,l 1 1 or 0 mod l such that fz T p1,12j T pif,l 1,12j 0 mod l For every collection of i f,l distinct primes m 1,, m if,l 1 or 0 mod l, we have fz T m1,12j T mif,l,12j 0 Remark If fz satisfies 1, we define i f,l := 1 mod l Proof We will prove the l = case The proofs of the other cases are similar In view of 5, it suffices to prove the lemma with fz = z j We proceed by induction on j To begin, we observe that for all primes p, z is an eigenform for T p,12 modulo Therefore, by Lemma 1, if p 2 mod, then z T p,12 0 mod As noted in the remark following Proposition 2, z T,12 0 mod Hence, z satisfies part 1 of the lemma Now, fix an integer j 2 We assume, for every integer k with 1 k j 1, that there is an integer i k := i k, 1 as in the proposition Set t := max{i 1,, i j 1 }, and let primes m 1,, m t+1 1 or 0 mod be given We propose to show that 6 z j T m1,12j T mt+1,12j 0 mod, thereby proving the proposition with 1 i j, t + 1 For every integer s 1, we define G s z := z s T m1,12s T mt,12s S 12s Z [[q]] It suffices to show that G j z T mt+1,12j 0 mod By the definition of t, we have 7 G s z 0 mod if 1 s j 1 If we also have G j z 0 mod, then 6 holds Thus, we assume that G j z 0 mod We claim, for all primes p, that G j z is an eigenform for T p,12j modulo Let p be an arbitrary prime By 5 it follows that there are integers a 1,p,, a j,p {0, 1, 2} for which j 1 z j T p,12j a j,p z j + a i,p z i mod Using the commutativity of the Hecke operators, 7, and, we verify that G j z 0 mod is an eigenform for T p,12j modulo : i=1 G j z T p,12j = z j T p,12j T m1,12j T mt,12j j 1 a j,p G j z + a i,p G i z a j,p G j z mod i=1

9 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES 9 Lemma 1 now implies that if m t+1 2 mod, then G j z T mt+1,12j 0 mod It remains to show that G j z T,12j 0 mod We compute as follows, using the commutativity of the Hecke operators and the remark after Proposition 2: { G G j z T,12j = z j j z mod if j T,12j T m1,12j T mt,12j 0 mod if j If j, then j < j 1, so G j z 0 mod by 7, proving the proposition If l is prime and fz is as in the hypothesis of Lemma, we define the nilpotency degree of fz modulo l by nil l f := i f,l 1 For example, one can show that nil 7 = 5 Moreover, for l {, 5, 7}, the proof of Lemma shows that if j 1 is an integer and fz S 12j Z l [[q]], then nil l f j We now describe how to use nil l f to deduce information about coefficients of fz which do not vanish modulo l Proposition 4 Fix l {, 5, 7}, let j 1 be an integer, and let fz = anq n S 12j Z l [[q]] have nil l f = i f,l 1 1 Suppose that i f,l = 1 Then there is an integer r f,l 1 such that ar f,l 0 mod l and r f,l is divisible neither by l nor by primes p 1 mod l 2 Suppose that i f,l 2 Then there is an integer t f,l 1 and distinct primes p 1,, p if,l 1 1 or 0 mod l such that at f,l p 1 p if,l 1 0 mod l and t f,l is divisible neither by l nor by primes p 1 mod l Proof We prove the proposition for l = The proofs of the other cases are similar Using 1, nil f = 1 implies that n fz T,12j = an + 12j 1 a q n anq n 0 mod, from which it follows that 9 an 0 mod Now, let p 1 mod be prime By the same reasoning, we have n fz T p,12j = anp + p 12j 1 a q n 0 mod, p which gives 10 anp a n p mod Suppose that k 1 is an integer with p k Then there are positive integers r and m with p m for which k = mp r If r is even, then repeated application of 10 with n replaced by mp b for suitable integers b yields 11 ak = amp r amp r 2 am mod

10 10 MATTHEW BOYLAN If r is odd, then by the same argument, we have 12 ak = amp r amp r 2 amp a m 0 mod p Hence, 9, 11, and 12 show that there must be an integer r f, with the properties stated in part 1 of the proposition If i f := i f, 2, then there are distinct primes p 1,, p if 1 1 or 0 mod such that fz T p1,12j T pif 1,12j 0 mod For convenience, we define gz S 12j Z [[q]] by gz := fz T p1,12j T pif 1,12j Therefore, the condition that nil f = i f implies that nil g = 1 Now, for all integers n and j with n 0 and 1 j i f 1, we define b j n Z by b1 nq n = fz T p1,12j b2 nq n = fz T p1,12j T p2,12j bif 1nq n = fz T p1,12j T = gz pif 1 Since nil g = 1, it follows by part 1 of the proposition that there is an integer t f 1 which is divisible neither by nor by primes p 1 mod and which has b if 1t f 0 mod Furthermore, since p 1,, p if 1 1 or 0 mod, we see that 1 gcdt f, p 1 p if 1 = 1 Repeatedly using formula 1 together with 1, we obtain 0 b if 1t f b if 2t f p if 1 b if 2 tf p if 1 tf p if 1 b if t f p if 2p if 1 b if p if 2 b if 2t f p if 1 b if t f p if 2p if tf p p if 1 b 1 t f p 1 p if 1 b 1 p 2 tf p 2 p if 1 at f p 1 p if 1 a p 1 b 1 t f p 2 p if 1 at f p 1 p if 1 mod The lemma follows with t f, = t f

11 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES 11 4 The proof of Lemma 22 In this section we prove Lemma 22, whose truth implies our theorems on the non-vanishing of partition values modulo l {, 5, 7} Recall that the lemma gives a uniform description of infinitely many integers n for which a s,l n 0 mod l, where the a s,l n are integers defined by { z l2s 1 if l = a s,l nq n := z l2s 1 24 if l {5, 7} Proof We prove the l = case of the lemma; the proofs of the other cases are similar We first note that for all integers s 1, z 9s 1 0 mod, so there is an integer i s := nil 9s 1 1 If i s = 1, then by part 1 of Proposition 4, there is an integer r s, 1 with a s, r s, 0 mod and such that r s, is divisible neither by nor by primes p 1 mod Now, let m r s, be prime with m 1 mod, and for convenience, denote by k s := 9s 1 2 the weight of the cusp form z 9s 1 Since nil 9s 1 = 1, we obtain z 9s 1 T m,ks = which implies, for all integers n 1, that 41 a s, nm a s, n m Replacing n by r s, m in 41 yields n a s, mn m ks 1 q n 0 mod, m mod a s, r s, m 2 a s, r s, 0 mod, which gives the lemma in this case with n s, = r s, Now, suppose that i s 2 By part 2 of Proposition 4, there is an integer t s, 1 which is divisible neither by nor by primes p 1 mod and a collection of i s 1 distinct primes p 1,, p is 1 1 or 0 mod such that a s, t s, p 1 p is 1 0 mod Part 2 of Proposition 21 implies that t s, p 1 p is 1 0 mod, so we must have p 1,, p is 1 1 mod Next, we let m t s, p 1 p is 1 be prime with m 1 mod For all integers n and j with n 1 and 0 j i s 1, we define integers c j n by c0 nq n = z 9s 1 T m,ks c1 nq n = z 9s 1 T m,ks T p1,k s cis 1nq n = z 9s 1 T m,ks T p1,k s T pis 1,k s

12 12 MATTHEW BOYLAN Since nil 9s 1 = i s, it follows that for all integers n 1, 42 c is 1n 0 mod In particular, setting n = t s, m in 42, we obtain c is 1t s, m 0 mod By the same argument as in the proof of Proposition 4, using formula 1 and the fact that gcdt s,, p 1 p is 1 = 1, we obtain 0 c is 1t s, m c is 2t s, p is 1m c is t s, p is 2p is 1m c 0 t s, p 1 p is 1m a s, t s, p 1 p is 1m 2 a s, t s, p 1 p is 1 mod Therefore, we have a s, t s, p 1 p is 1m 2 a s, t s, p 1 p is 1 mod, which proves the lemma when i s 2 with n s, = t s, p 1 p is 1 References [1] S Ahlgren, Distribution of the parity of the partition function in arithmetic progressions, Indag Math , no 2, [2] S Ahlgren, Non-vanishing of the partition function modulo odd primes l, Mathematika , no 1, [] S Ahlgren, Distribution of the partition function modulo positive integers M, Math Ann , no 4, [4] S Ahlgren and K Ono, Congruences and conjectures for the partition function, q-series with applications to combinatorics, number theory, and physics Urbana, IL, 2000, 1-10, Contemp Math, 291, Amer Math Soc, Providence, RI, 2001 [5] S Ahlgren and K Ono, Congruence properties of the partition function, Proc Natl Acad Sci USA , no 2, [6] M Boylan and K Ono, Parity of the partition function in arithmetic progressions, II, Bull London Math Soc 2001, no 5, [7] P Deligne and J-P Serre, Formes modulaires de poids 1, Ann Sci École Norm Sup , [] N Koblitz, Introduction to elliptic curves and modular forms, Second edition, Graduate Texts in Mathematics, 97, Springer-Verlag, New York, 199 [9] J-L Nicolas, I Z Rusza, and A Sárközy, On the parity of additive representation functions, with an appendix by J-P Serre, J Number Theory 7 199, no 2, [10] K Ono, Parity of the partition function in arithmetic progressions, J Reine Angew Math , 1-15 [11] K Ono, Distribution of the partition function modulo m, Ann of Math , no 1, [12] K Ono, The web of modularity: Arithmetic of the coefficients of modular forms and q-series, Conference Board of the Math Sciences, Regional Conference Series in Mathematics, 102, Amer Math Soc, Providence, RI, 2004 [1] K Ono and C Skinner, Fourier coefficients of half-integral weight modular forms modulo l, Ann of Math , no 2, [14] J-P Serre, Valeurs propres des opérateurs de Hecke modulo l, Astérisque ,

13 NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES 1 [15] M V Subbarao, Some remarks on the partition function, Amer Math Monthly , [16] H P F Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III Proc Internat Summer School, Univ Antwerp, 1972, 1-55 Lecture Notes in Math, Vol 50, Springer, Berlin, 197 Department of Mathematics, University of South Carolina, Columbia, SC address: boylan@mathscedu