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DISTRIBUTION OF THE PARTITION FUNCTION MODULO m
Ken Ono
Annals of Mathematics, 151, 2000, pages 293-
- Introduction and Statement of Results
A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n. Let p(n) denote the number of partitions of n (as usual, we adopt the convention that p(0) = 1 and p(α) = 0 if α 6 ∈ N). Ramanujan proved for every non-negative integer n that
p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11),
and he conjectured further such congruences modulo arbitrary powers of 5, 7, and 11. Although the work of A. O. L. Atkin and G. N. Watson settled these conjectures many years ago, the congruences have continued to attract much attention. For example, 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 spirit of F. Dyson, have gone a long way towards providing combinatorial and physical explanations for their existence. Ramanujan [Ra, p. xix] already observed that his congruences were quite special. For instance, he proclaimed that
“It appears that there are no equally simple properties for any moduli involving primes other than these three (i.e. m = 5, 7 , 11 ).”
1991 Mathematics Subject Classification. Primary 11P83; Secondary 05A17. Key words and phrases. partition function, The Erd¨os’ Conjecture, Newman’s Conjecture. The author is supported by NSF grants DMS-9508976 , DMS-9874947 and NSA grant MSPR- 97Y012.
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2 KEN ONO
Although there is no question that congruences of the form p(an + b) ≡ 0 (mod m) are rare (see recent works by the author [K-Ol, O1, O2]), the question of whether there are many such congruences has been the subject of debate. In the 1960’s, Atkin and O’Brien [At1, At2, At-Ob] uncovered further congruences such as
(1) p(11^3 · 13 n + 237) ≡ 0 (mod 13).
However, no further congruences have been found and proven since. In a related direction, P. Erd¨os and A. Ivi´c conjectured that there are infinitely many primes m which divide some value of the partition function [E-I], and Erd¨os made the following stronger conjecture [Go, I].
Conjecture (Erd¨os). If m is prime, then there is at least one non-negative integer nm for which p(nm) ≡ 0 (mod m).
A. Schinzel (see [E-I] for the proof) proved the Erd¨os-Ivi´c conjecture using the Hardy- Ramanujan-Rademacher asymptotic formula for p(n), and more recently Schinzel and E. Wirsing [Sc-W] have obtained a quantitative result in the direction of Erd¨os’ stronger con- jecture. They have shown that the number of primes m < X for which Erd¨os’ Conjecture is true is log log X. Here we present a uniform and systematic approach which settles the debate regarding the existence of further congruences, and yields Erd¨os’ Conjecture as an immediate corollary.
Theorem 1. Let m ≥ 5 be prime and let k be a positive integer. A positive proportion of the primes ` have the property that
p
mk`^3 n + 1 24
≡ 0 (mod m)
for every non-negative integer n coprime to `.
In view of work of S. Ahlgren [A], J. -L. Nicolas, I. Z. Ruzsa, A. S´ark¨ozy and J.-P. Serre [Ni-R-Sa] for m = 2, the fact that p(3) = 3, and Theorem 1, we obtain:
Corollary 2. Erd¨os’ Conjecture is true for every prime m. Moreover, if m 6 = 3 is prime, then
#{ 0 ≤ n ≤ X : p(n) ≡ 0 (mod m)} m
X if m = 2, X if m ≥ 5.
Surprisingly, it is not known whether there are infinitely many n for which p(n) ≡ 0 (mod 3).
4 KEN ONO
Theorem 5. If m ≥ 5 is prime, then there are integers 0 ≤ N (m) ≤ 48(m^3 − 2 m − 1) and 1 ≤ P (m) ≤ 48(m^3 − 2 m − 1) such that for every i > N (m) we have
p
min + 1 24
≡ p
mP^ (m)+i^ · n + 1 24
(mod m)
for every non-negative integer n.
For each class r (mod m) one obtains explicit sequences of integers nk such that p(nk) ≡ r (mod m) for all k. This is the subject of Corollaries 9 through 12 below. For example, taking n = 0 in Corollary 12 shows that for every non-negative integer k
(4) p
232 k+1^ + 1 24
≡ 5 k^ (mod 23) and p
232 k+3^ + 1 24
≡ 5 k+1^ (mod 23).
Similarly it is easy to show that
(5) p
1367 · 232 k+2^ + 1 24
≡ 0 (mod 23) and p
1297 · 232 k+1^ + 1 24
≡ 0 (mod 23).
Congruences of this sort mod 13 were previously discovered by Ramanujan and found by M. Newman [N2]. In fact, this paper was inspired by such entries in Ramanujan’s lost manuscript on p(n) and τ (n) (see [B-O]). A priori, one knows that the generating functions F (m, k; z) are the reductions mod m of weight − 1 /2 non-holomorphic modular forms, and as such lie in infinite dimensional Fm-vector spaces. This infinitude has been the main obstacle in obtaining results for the partition function mod m. In §3 we shall prove a theorem (see Theorem 8) which establishes that the F (m, k; z) are the reductions mod m of half-integral weight cusp forms lying in one of two spaces with Nebentypus. Hence, there are only finitely many possibilities for each F (m, k; z). This is the main observation which underlies all of the results in this paper. We then prove Theorems 1 and 3 by employing the Shimura correspondence and a theorem of Serre about Galois representations. In §4 we present detailed examples for 5 ≤ m ≤ 23.
- Preliminaries
We begin by defining operators U and V which act on formal power series. If M and j are positive integers, then
n≥ 0
a(n)qn
| U (M ) :=
n≥ 0
(6) a(M n)qn,
n≥ 0
a(n)qn
(^) | V (j) :=
n≥ 0
(7) a(n)qjn.
THE PARTITION FUNCTION MODULO m 5
We recall that Dedekind’s eta-function is defined by
(8) η(z) := q^1 /^24
∏^ ∞
n=
(1 − qn)
and that Ramanujan’s Delta-function is
(9) ∆(z) := η^24 (z),
the unique normalized weight 12 cusp form for SL 2 (Z). If m ≥ 5 is prime and k is a positive integer, then define a(m, k, n) by
∑^ ∞
n=
a(m, k, n)qn^ :=
∆δ(m,k)(z) | U (mk)
|V (24)
ηmk^ (24z)
(mod m),
where δ(m, k) := (m^2 k^ − 1)/24. Recall the definition (3) of F (m, k; z).
Theorem 6. If m ≥ 5 is prime and k is a positive integer, then
F (m, k; z) ≡
∑^ ∞
n=
a(m, k, n)qn^ (mod m).
Proof. We begin by recalling that Euler’s generating function for p(n) is given by the infinite product ∑∞
n=
p(n)qn^ :=
∏^ ∞
n=
(1 − qn)
Using this fact, one easily finds that
ηm
k (mkz) η(z)
| U (mk) =
n=
p(n)qn+δ(m,k)^ ·
∏^ ∞
n=
(1 − qm
k (^) n )m
k
| U (mk)
∑^ ∞
n=
p(mkn + β(m, k))qn+^
δ(m,k)+β(m,k) mk^ ·
∏^ ∞
n=
(1 − qn)m
k ,
where 1 ≤ β(m, k) ≤ mk^ − 1 satisfies 24β(m, k) ≡ 1 (mod mk).
Since (1 − Xm
k )m
k ≡ (1 − X)m
2 k (mod m), we find that
∑^ ∞
n=
p(mkn + β(m, k))qn+^
δ(m,k)+β(m,k) mk^ ≡ ∆δ(m,k)(z) | U (mk) ∏∞ n=1(1^ −^ q n)mk^ (mod^ m).
THE PARTITION FUNCTION MODULO m 7
Theorem 8. If m ≥ 5 is prime, then for every positive integer k we have
F (m, k; z) ∈ S (^) m (^2) −m− 1 2
(Γ 0 (576m), χχk m− 1 )m,
where χ is the non-trivial quadratic character with conductor 12, and χm is the usual Kro- necker character for Q(
m).
Proof. The U (m) operator defines a map (see [Lemma 1, S-St])
U (m) : Sλ+ 12 (Γ 0 (4N m), ν) −→ Sλ+ 12 (Γ 0 (4N m), νχm).
Therefore, in view of Proposition 7 it suffices to prove that
F (m, 1; z) ∈ S (^) m (^2) −m− 1 2
(Γ 0 (576m), χ)m.
If d ≡ 0 (mod 4), then it is well known that the space of cusp forms Sd(Γ 0 (1)) has a basis of the form (^) {
∆(z)j^ E 4 (z)
d 4 − 3 j : 1 ≤ j ≤
[
d 12
]}
Since the Hecke operator Tm is the same as the U (m) operator on S 12 δ(m,1)(Γ 0 (1))m, we know that ∆δ(m,1)(z) | U (m) ≡
j≥ 1
αj ∆(z)j^ E 4 (z)^3 δ(m,1)−^3 j^ (mod m),
where the αj ∈ Fm. However, since
∆δ(m,1)(z) = qδ(m,1)^ − · · · ,
it is easy to see that
∆δ(m,1)(z) | U (m) =
n≥n 0
t(n)qn
where n 0 ≥ δ(m, 1)/m. However, since δ(m, 1) ∈ Z, one can easily deduce that n 0 > m/24. The only basis forms in ∆δ(m,1)(z) | U (m) (mod m) are those ∆j^ (z)E 4 (z)^3 δ(m,1)−^3 j^ where j > m/24. This implies that ( ∆(z)δ(m,1)^ | U (m)
| V (24)
ηm(24z)
is a cusp form. Since
∆(z)δ(m,1)^ | U (m)
| V (24) is the reduction mod m of a weight m
(^2) − 1 2 cusp form with respect to Γ 0 (24), and η(24z) is a weight 1/2 cusp form with respect to Γ 0 (576) with character χ, the result follows.
Q.E.D.
Now we recall an important result due to Serre [6.4,S].
8 KEN ONO
Theorem (Serre). The set of primes ` ≡ −1 (mod N ) for which
f | T` ≡ 0 (mod m)
for every f (z) ∈ Sk(Γ 0 (N ), ν)m has positive density. Here Tdenotes the usual Hecke operator of index acting on Sk(Γ 0 (N ), ν).
Proof of Theorem 1. If F (m, k; z) ≡ 0 (mod m), then the conclusion of Theorem 1 holds for every prime `. Hence, we may assume that F (m, k; z) 6 ≡ 0 (mod m). By Theorem 8, we know that each F (m, k; z) belongs to S (^) m (^2) −m− 1 2
(Γ 0 (576m), χχk m− 1 )m. Therefore each
F (m, k; z) is the reduction mod m of a half-integral weight cusp form. Now we briefly recall essential facts about the “Shimura correspondence” [Sh], a family of maps which send modular of forms of half-integral weight to those of integer weight. Although Shimura’s original theorem was stated for half-integral weight eigenforms, the generalization we describe here follows from subsequent works by Cipra and Niwa [Ci, Ni]. Suppose that f (z) =
n=1 b(n)q
n (^) ∈ S λ+ 12 (Γ^0 (4N^ ), ψ) is a cusp form where^ λ^ ≥^ 2. If^ t^ is any square-free integer, then define At(n) by
∑^ ∞
n=
At(n) ns^
:= L(s − λ + 1, ψχλ − 1 χt) ·
∑^ ∞
n=
b(tn^2 ) ns^
Here χ− 1 (resp. χt) is the Kronecker character for Q(i) (resp. Q(
t)). These numbers At(n) define the Fourier expansion of St(f (z)), a cusp form
St(f (z)) :=
∑^ ∞
n=
At(n)qn
in S 2 λ(Γ 0 (4N ), ψ^2 ). Moreover, the Shimura correspondence St commutes with the Hecke algebra. In other words, if p - 4 N is prime, then
St(f |T (p^2 )) = St(f ) | Tp.
Here Tp (resp. T (p^2 )) denotes the usual Hecke operator acting on the space S 2 λ(Γ 0 (4N ), ψ^2 ) (resp. Sλ+ 12 (Γ 0 (4N ), ψ)).
Therefore, for every square-free integer t we have that the image St(F (m, k; z)) under the t-th Shimura correspondence is the reduction mod m of an integer weight form in Sm (^2) −m− 2 (Γ 0 (576m), χtriv ). Now let S(m) denote the set of primes ≡ −1 (mod 576m) for which G | T ≡ 0 (mod m)
for every G ∈ Sm (^2) −m− 2 (Γ 0 (576m), χtriv )m. By Serre’s theorem, the set S(m) contains a positive proportion of the primes.
10 KEN ONO
Although we have not conducted a thorough search, it is almost certain that many such congruences exist.
Proof of Theorem 3. By the proof of Theorem 6, recall that
F (m, 1; z) =
n≥ 0 , mn≡−1 (mod 24)
p
mn + 1 24
qn^ ∈ S (^) m (^2) −m− 1 2
(Γ 0 (576m), χ)m.
Since m is good, for each 0 ≤ r ≤ m − 1 let nr be a fixed non-negative integer for which mnr ≡ −1 (mod 24) and
p
mnr + 1 24
≡ r (mod m).
Let Mm be the set of primes p for which p | nr for some r, and define Sm by
Sm :=
p∈Mm
p.
Obviously, the form F (m, 1; z) also lies in S (^) m^2 −m− 1 2
(Γ 0 (576mSm, χ)m. Therefore, by Serre’s
theorem and the commutativity of the Shimura correspondence, a positive proportion of the primes ` ≡ −1 (mod 576mSm) have the property that
F (m, 1; z) | T (`^2 ) ≡ 0 (mod m).
By (11), for all but finitely many such ` we have for each r that
p
mnr `^2 + 1 24
m^2 −m− 2 (^2) nr `
`
m^2 − 2 m − 4 p
mnr + 1 24
≡ 0 (mod m).
However, since ` ≡ −1 (mod m) this implies that
(12) p
mnr `^2 + 1 24
≡ χ(`)
m^2 − 2 m− 2
`
m^2 −m− 2 2
nr `
r (mod m).
If nr =
i pi^ where the^ pi^ are prime, then ( nr `
i
pi `
THE PARTITION FUNCTION MODULO m 11
Since nr is odd, ≡ 3 (mod 4), and ≡ −1 (mod pi), we find by quadratic reciprocity that
( pi `
`
pi
pi
−`
pi
pi
Therefore, for all but finitely many such ` congruence (12) reduces to
(13) p
mnr `^2 + 1 24
≡ χ(`)
m^2 −m− 2 2 `
m^2 − 2 m − 2 r (mod m).
Hence for every sufficiently large such `, the m values p
mnr `^2 + 24
are distinct and represent
each residue class mod m. To complete the proof, it suffices to notice that the number of such primes ` < X, by Serre’s theorem again, is X/ log X. In view of (13), this immediately yields the
X/ log X estimate. The estimate when r = 0 follows easily from Theorem 1.
Q.E.D.
Proof of Theorem 5. Since F (m, k; z) is in S (^) m^2 −m− 1 2
(Γ 0 (576m), χχk m− 1 )m, it follows that
each F (m, k; z) lies in one of two finite dimensional Fm-vector spaces. The result now follows immediately from (3), Theorem 6, Proposition 7, and well known upper bounds for the dimensions of spaces of cusp forms (see [C-O]).
Q.E.D.
- Examples
In this section we list the Ramanujan cycles for the generating functions F (m, k; z) when 5 ≤ m ≤ 23. Although we have proven that each F (m, k; z) ∈ S (^) m^2 −m− 1 2
(Γ 0 (576m), χχk m− 1 )m,
in these examples it turns out that they all are congruent mod m to forms of smaller 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 positive 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).
THE PARTITION FUNCTION MODULO m 13
Corollary 9. Define integers a(n) and b(n) by
∑^ ∞
n=
a(n)qn^ :=
∏^ ∞
n=
(1 − qn)^11 ,
∑^ ∞
n=
b(n)qn^ :=
∏^ ∞
n=
(1 − qn)^23.
If k and n are non-negative integers, then
p
132 k+1(24n + 11) + 1 24
≡ 11 · 6 k^ · a(n) (mod 13),
p
132 k+2(24n + 23) + 1 24
≡ 10 · 6 k^ · b(n) (mod 13).
Case where m = 17.
By [Prop. 4, Gr-O], it is known that
∆^12 (z) | U (17) ≡ 7 E 4 (z)∆(z) (mod 17)
where E 4 (z) = 1 + 240
n=1 σ^3 (n)q
n (^) is the usual weight 4 Eisenstein series. Therefore by
(10) it turns out that
F (17, 1; z) ≡ 7 q^7 + 16q^31 + · · · ≡ 7 η^7 (24z)E 4 (24z) (mod 17).
Again using Sturm’s theorem one easily verifies that
η^7 (24z)E 4 (24z) | U (17) ≡ 7 η^23 (24z)E 4 (24z) (mod 17), η^23 (24z)E 4 (24z) | U (17) ≡ 13 η^7 (24z)E 4 (24z) (mod 17).
By Proposition 7 this implies that for every non-negative integer k
F (17, 2 k + 1; z) ≡ 7 · 6 kη^7 (24z)E 4 (24z) (mod 17), F (17, 2 k + 2; z) ≡ 15 · 6 kη^23 (24z)E 4 (24z) (mod 17).
As an immediate corollary we obtain:
14 KEN ONO
Corollary 10. Define integers c(n) and d(n) by
∑^ ∞
n=
c(n)qn^ := E 4 (z) ·
∏^ ∞
n=
(1 − qn)^7 ,
∑^ ∞
n=
d(n)qn^ := E 4 (z) ·
∏^ ∞
n=
(1 − qn)^23.
If k and n are non-negative integers, then
p
172 k+1(24n + 7) + 1 24
≡ 7 · 6 k^ · c(n) (mod 17),
p
172 k+2(24n + 23) + 1 24
≡ 15 · 6 k^ · d(n) (mod 17).
Case where m = 19.
Using [Prop. 4, Gr-O], and arguing as above it turns out that for every non-negative integer k
F (19, 2 k + 1; z) ≡ 5 · 10 kη^5 (24z)E 6 (24z) (mod 19), F (19, 2 k + 2; z) ≡ 11 · 10 kη^23 (24z)E 6 (24z) (mod 19).
Here E 6 (z) = 1 − 504
n=1 σ^5 (n)q
n (^) is the usual weight 6 Eisenstein series. As an immediate
corollary we obtain:
Corollary 11. Define integers e(n) and f (n) by
∑^ ∞
n=
e(n)qn^ := E 6 (z) ·
∏^ ∞
n=
(1 − qn)^5 ,
∑^ ∞
n=
f (n)qn^ := E 6 (z) ·
∏^ ∞
n=
(1 − qn)^23.
If k and n are non-negative integers, then
p
192 k+1(24n + 5) + 1 24
≡ 5 · 10 k^ · e(n) (mod 19),
p
192 k+2(24n + 23) + 1 24
≡ 11 · 10 k^ · f (n) (mod 19).
Case where m = 23.
Using [Prop. 4, Gr-O], and arguing as above we have for every non-negative integer k F (23, 2 k + 1; z) ≡ 5 kη(24z)E 4 (24z)E 6 (24z) (mod 23), F (23, 2 k + 2; z) ≡ 5 k+1η^23 (24z)E 4 (24z)E 6 (24z) (mod 23).
16 KEN ONO
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[Ra] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. (2) 19 (1919), 207-210. [Sc-W] A. Schinzel and E. Wirsing, Multiplicative properties of the partition function, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), 297-303. [S] J.-P. Serre, Divisibilit´e de certaines fonctions arithm´etiques, L’Ensein. Math. 22 (1976), 227-
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Dept. of Math., Penn State University, University Park, Pennsylvania 16802, USA. E-mail address: [email protected]
