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Math. Ann. 312, 251–260 (1998)
Mathematische
Annalen
©^ c Springer-Verlag 1998
The partition function in arithmetic progressions
Ken Ono
Department of Mathematics, Penn State University, University Park, PA 16802, USA (e-mail: [email protected])
Received 3 March 1998 / Revised version 30 March 1998
In celebration of G.E. Andrews’ 60 th^ birthday.
Mathematics Subject Classification (1991): 05A17; 11P
1. Introduction and statement of results
A partition of a non-negative integer n is a non-increasing sequence of positive integers whose sum is n. Euler gave the following generating function for p ( n ), the number of partitions of an integer n :
∑^ ∞
n =
p ( n ) q n^ =
∏^ ∞
n =
1 − q n^
(1) = 1 + q + 2 q^2 + 3 q^3 + 5 q^4 + 7 q^5 + 11 q^6 + · · ·.
Ramanujan observed various surprising congruences for p ( n ) when n is in certain, very special, arithmetic progressions. For instance, in [Ra, p. xix] Ramanujan proclaims:
“ I have proved a number of arithmetic properties of p(n)...in particular that
p (5 n + 4) ≡ 0 (mod 5), and p (7 n + 5) ≡ 0 (mod 7).
...I have since found another method which enables me to prove all of these properties and a variety of others, of which the most striking is
p (11 n + 6) ≡ 0 (mod 11).
There are corresponding properties in which the moduli are powers of 5 , 7 , or 11 ... It appears that there are no equally simple properties for any moduli involving primes other than these three. ”
The author is supported by NSF grant DMS-9508976 and NSA grant MSPR-97Y012.
252 K. Ono
There are now many proofs of these congruences (and their generalizations) in the literature (for instance, see [An, An-G, At, G, G-Ki-St, H-Hu, W]), often involving modular equations and various combinatorial constructions. Although subsequent works show that there are indeed congruences where the modulus contains prime divisors other than 5, 7 , and 11, it is still widely believed, as Ramanujan suggested, that “simple” congruence properties are “rare”. The quantification of this expectation has remained as one of the open problems in the area. For instance, there do not seem to be any such congruences modulo 2 or
- The parity of p ( n ) seems to be quite random. A widely believed “Folklore Conjecture” asserts that p ( n ) is ‘equally often’ even and odd; that is that p ( n ) is even for ∼ 12 X positive integers n ≤ X. Parkin and Shanks [P-S] conducted the first extensive numerical study, and their evidence strongly supports the “Folklore Conjecture.” In the direction of this conjecture, Nicolas, Ruzsa and S´ark¨ozy [N- R-S-Se] have proved that
#{ N ≤ X | p ( N ) is even}
X ,
#{ M ≤ X | p ( M ) is odd}
X · exp
−(log 2 + ) log X log log X
Subbarao [Su] made the following conjecture on the parity of p ( n ), for those integers n belonging to any given arithmetic progression:
Conjecture 1. In every progression r (mod t ) there are infinitely many M ≡ r (mod t ) for which p ( M ) is odd, and infinitely many N ≡ r (mod t ) for which p ( N ) is even.
This conjecture had been proved for every arithmetic progression with mod- ulus t (see [O] for precise references) where
t ∈ { 1 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 12 , 16 , 20 , 40 },
using a variety of elegant combinatorial methods, from the works of Garvan, Kolberg, Hirschhorn, Stanton and Subbarao (Note: This corrects the list of such t that appears in [O]. The author carelessly omitted t = 6, 8 , and 20.) In [O] the author went a step further by proving that in any progression r (mod t ) there are infinitely many N ≡ r (mod t ) for which p ( N ) is even, and that there are infinitely many M ≡ r (mod t ) for which p ( M ) is odd, provided there is one such M. Furthermore, if there is such an M , then the first one is less than an explicit constant C (^) r , t < 1010 t^7. Hence, the “even” case of Conjecture 1 has now been verified for every pro- gression, but the “odd” case remains open. However, we have a simple algorithm to determine the truth of the “odd” case for any given progression r (mod t ): Compute p ( M ) (mod 2) for M = r , r + t , r + 2 t ,... for all such M up to Cr , t. As soon as we find one odd number we have verified the conjecture. If all these numbers are even, then we have proved that the conjecture is false. Using an efficient version of this algorithm, K. Burrell (Universal Analytics, Inc.) verified the “odd” case of Conjecture 1 for every progression r (mod t ) with t ≤ 10 5.
254 K. Ono
2. Preliminaries
In this section we develop the essential preliminaries regarding modular forms (see [K] for background). As usual, if k is a positive integer, then let Sk (Γ 1 ( N )) denote the space of weight k cusp forms with respect to the congruence subgroup Γ 1 ( N ). Similarly, if ψ is a Dirichlet character modulo N , then let Sk ( N , ψ) denote the space of weight k cusp forms with respect to the congruence subgroup Γ 0 ( N ) with Nebentypus character ψ. As usual, we shall identify all such modular forms f ( z ) by their Fourier expansions
f ( z ) =
∑^ ∞
n =
a ( n ) q n^.
Here q := e^2 π iz^ is the uniformizing variable for the point at infinity. We begin with the following well known Lemma:
Lemma 1 [III, Sect. 3, Prop. 17 (b), K]. If f ( z ) =
n =1 a ( n ) q^ n (^) ∈ Sk ( N , ψ) and
χ is a Dirichlet character modulo t, then
f χ( z ) :=
∑^ ∞
n =
χ( n ) a ( n ) q n^ ∈ S (^) k ( Nt^2 , ψ χ^2 ).
Now we recall the Hecke operators. If f ( z ) =
n =1 a ( n ) q^ n (^) ∈ Sk ( N , ψ) and
p - N is prime, then the Hecke operator T ( p , k , ψ) acts on f ( z ) and returns the cusp form
T ( p , k , ψ)| f ( z ) =
∑^ ∞
n =
a ( np ) + p k^ −^1 ψ( p ) a ( n / p )
(4) q n^ ∈ S (^) k ( N , ψ).
Here a ( n / p ) = 0 if p - n.
Lemma 2. Let f ( z ) =
n =1 a ( n ) q^
n (^) ∈ Sk ( N , ψ) , and χ a Dirichlet character
modulo t. If p - Nt^2 is prime, then T ( p , k , ψ χ^2 )| f χ( z ) ∈ Sk ( Nt^2 , ψ χ^2 ) and is given by
T ( p , k , ψ χ^2 )| f χ( z ) =
∑^ ∞
n =
χ( np )
a ( np ) + p k^ −^1 ψ( p ) a ( n / p )
q n^.
Proof. That T ( p , k , ψ χ^2 )| f χ( z ) ∈ Sk ( Nt^2 , ψ χ^2 ) is immediate from Lemma 1 and (4). The claim follows immediately by (4) and the definition of f χ( z ) since
T ( p , k , ψ χ^2 )| f χ( z ) :=
∑^ ∞
n =
χ( np ) a ( np ) + p k^ −^1 ψ( p )χ^2 ( p ) · χ( n / p ) a ( n / p )
q n^.
ut
Using Dirichlet orthogonality we shall obtain the following result.
The partition function in arithmetic progressions 255
Lemma 3. Suppose that f ( z ) =
n =1 a ( n ) q^
n (^) ∈ Sk ( N , ψ) , and let 1 ≤ r < t be
integers for which gcd( r , t ) = 1_. If p_ - Nt^2 is prime, then
F ( r , t , p ; z ) :=
n ≡ rp (mod t )
a ( np ) q n^ + p k^ −^1 ψ( p )
n ≡ r (mod t )
a ( n ) q np^ ∈ S (^) k (Γ 1 ( Nt^2 )).
Proof. Recall Dirichlet’s theorem that for every integer n
∑
χmod t
χ( r )χ( n ) =
φ( t ) if n ≡ r (mod t ), 0 otherwise.
If p - Nt^2 , then define F ( r , t , p ; z ) by
F ( r , t , p ; z ) :=
φ( t )
χmod t
χ( rp^2 )
T ( p , k , ψ χ^2 )| f χ( z )
It is easy to see that F ( r , t , p ; z ) ∈ S (^) k (Γ 1 ( Nt^2 )) since each f χ ∈ Sk (Γ 1 ( Nt^2 )). By Lemma 2 and (5) we find that
F ( r , t , p ; z ) :=
φ( t )
χmod t
χ( rp^2 )
∑^ ∞
n =
χ( np )
a ( np ) + p k^ −^1 ψ( p ) a ( n / p )
q n
φ( t )
∑^ ∞
n =
χmod t
χ( rp^2 )χ( np )
a ( np ) + p k^ −^1 ψ( p ) a ( n / p )
q n
np ≡ rp^2 (mod t )
a ( np ) + p k^ −^1 ψ( p ) a ( n / p )
q n
n ≡ rp (mod t )
a ( np ) q n^ + p k^ −^1 ψ( p )
n ≡ r (mod t )
a ( n ) q np^.
ut
Using a theorem of Sturm and Lemma 3, we obtain the following general theorem guaranteeing the existence of non-zero coefficients in arithmetic pro- gressions.
Theorem 3. Let ` be prime. If f ( z ) =
n =1 a ( n ) q^
n (^) ∈ Sk ( N , ψ) has integer
coefficients, and 1 ≤ r < t are coprime integers for which there is an
n 0 ≡ r (mod t ) with a ( n 0 ) 6 ≡ 0 (mod `),
then for every sufficiently large prime p - ` Nt^2 there is an integer n for which
n ≡ rp (mod t ) and a ( np ) 6 ≡ 0 (mod `).
The partition function in arithmetic progressions 257
Here χ 2 denotes the usual Kronecker character for Q(
In view of this Lemma we obtain the following easy Lemma.
Lemma 5. If t is a positive integer, then let j be any non-negative integer for which 2 j^ > t / 24_. There is an integer n_ ≡ r (mod t ) for which p ( n ) is odd if and only if there exists an integer n ′^ ≡ 24 r − 1 (mod 24 t ) for which a (^) t , j ( n ′) is odd.
Proof. By (7) it is easy to see that
a (^) t , j ( n ′) ≡
∑^ ∞
k =
p
n ′^ − 2 j^ · 24 t (2 k + 1) 2 + 1 24
(mod 2).
The result now follows easily. ut
In view of these preliminary lemmas and Theorem 3 we obtain the following general theorem.
Theorem 4. Let 0 ≤ r < t be integers for which gcd(24 r − 1 , t ) = 1_. If there exists an integer n_ ≡ r (mod t ) for which p ( n ) is odd, then for every integer s coprime to 24 t there are infinitely many integers M ≡ s^2 ( r − 24 −^1 ) + 24−^1 (mod t ) for which p ( M ) is odd.
Proof. Let j be a positive integer for which 2 j^ > t /24 and let ft , j ( z ) be as defined in Lemma 4. By Lemma 5, there is an integer n 0 ≡ 24 r − 1 (mod 24 t ) for which a (^) t , j ( n 0 ) is odd. By Theorem 3, for every sufficiently large prime p there is an integer m ≡ (24 r − 1) p (mod 24 t ) for which at , j ( mp ) is odd. In particular, for every residue class s (mod 24 t ) with gcd( s , 24 t ) = 1 we find an integer m ≡ (24 r − 1) s^2 (mod 24 t ) for which at , j ( m ) is odd. Therefore, by Lemma 5 again, for each such s there are integers M ≡ s^2 ( r − 24 −^1 ) + 24−^1 (mod t ) for which p ( M ) is odd. Hence by [Main Theorem 2, O], there are infinitely many such M for which p ( M ) is odd. ut
Corollary 1. If t > 3 is prime, and there are two integers n 0 and n 1 for which
( i ) p ( n 0 ) ≡ p ( n 1 ) ≡ 1 (mod 2),
( ii )
24 n 0 − 1 t
24 n 1 − 1 t
then in every progression 24 −^1 6 ≡ r (mod t ) there are infinitely many M ≡ r (mod t ) for which p ( M ) is odd.
Proof. Since t is prime, the only r (mod t ) for which gcd(24 r − 1 , t ) = 1 is the/ residue class r ≡ 24 −^1 (mod t ). Since t is prime, it is easy to see that Theorem 4 will cover all the residue classes 24−^1 6 ≡ r (mod t ). ut
We now have an algorithm for proving Conjecture 2 for almost every prime: Given any finite set of integers n 1 , n 2 ,... ns for which p ( ni ) are odd, simply find
258 K. Ono
all the arithmetic progressions of primes for which
( (^24) ni − 1 t
) (^24 n (^) j − 1 t
= −1 for
any 1 ≤ i , j ≤ s. A MAPLE calculation that computed p ( n ) (mod 2) for every n ≤ 15000 yields Theorem 1.
4. The partition function modulo `
In this section we consider the reduction of p ( n ) modulo odd primes ` for those n belonging to an arithmetic progression. We begin with the following important lemma.
Lemma 6. Let _be an odd prime, and t a positive integer. If j is a positive integer for which_ j^ > 24 t, then
f ( t , `, j ; z ) =
∑^ ∞
n =
a ( t , `, j ; n ) q n^ :=
η` j (576 tz ) η(24 z )
∈ S (^) ` j (^) − 1 2
243 t , χ t ,`, j
where χ t ,`, j is the usual Kronecker character for Q
` j^ − 1 (^2) · 6 t
. Moreover,
the Fourier expansion of f ( t , , _j_ ; _z_ ) _factors modulo_ as
f ( t , `, j ; z ) =
∑^ ∞
n =
a ( t , `, j ; n ) q n^ ≡
n =
p ( n ) q^24 n −^1
k =−∞
(−1) k^ q^24 t `
j (^) (6 k +1) 2
(8) (mod `).
Proof. Gordon, Hughes, Ligozat and Newman proved the following well known fact about eta-products. If N is a positive integer, and f ( z ) :=
δ| N η
r δ (^) (δ z ) is an
eta-product for which
∑
δ| N
δ r δ ≡ 0 (mod 24),
δ| N
N
δ
r δ ≡ 0 (mod 24), and
δ| N
N · gcd( d , δ)^2 r δ gcd( d , Nd ) d δ
for every integer d | N , then f ( z ) ∈ S (^) k ( N , χ) where k := 12 ·
∏^ δ| N^ r δ^ ,^ s^ := δ| N δ r δ (^) , and χ is the Kronecker character for Q(
(−1) k^ s ). That f ( t , `, j ; z ) ∈
S (^) ` j (^) − 1 2
(24^3 t , χ t ,`, j ) is now immediate. By combining Euler’s Pentagonal Number
Theorem [Corollary 1.7, An], that
∏^ ∞
n =
(1 − q n^ ) =
∑^ ∞
k =−∞
(−1) k^ q
3 k^22 + k ,
with the fact that (1 − X )_j_ ≡ 1 − _X_ j (mod `), we obtain (8) from (1). ut
260 K. Ono
Corollary 2. If ` and t > 3 are odd primes, and there are two integers n 0 and n 1 for which
( i ) p ( n 0 ) p ( n 1 ) 6 ≡ 0 (mod `),
( ii )
24 n 0 − 1 t
24 n 1 − 1 t
then in every arithmetic progression 24 −^1 6 ≡ r (mod t ) there are infinitely many integers M ≡ r (mod t ) for which p ( M ) 6 ≡ 0 (mod `).
Remark 1. Let be prime. By Corollaries 1 and 2, since _p_ (0) = _p_ (1) = 1 6 ≡ 0 (mod), for any prime t > 3 with
t
t
= −1 every arithmetic progression 24 −^1 6 ≡ r (mod t ) has the property that there are infinitely many M ≡ r (mod t ) with p ( M ) 6 ≡ 0 (mod `). This holds for every prime t for which
( (^) t 23
A MAPLE calculation that computed p ( n ) for every n ≤ 750 yields Theorem
- Almost certainly one can obtain a better density by computing many more values of p ( n ) (mod `).
Acknowledgements. The author thanks the referee and K. Ribet for making many suggestions that improved and clarified the exposition of this paper.
References
[A] S. Ahlgren, The Distribution of parity of the partition function in arithmetic progressions, preprint. [An] G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976. [An-G] G. E. Andrews, F. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167–171. [At] A.O.L. Atkin, Proof of a conjecture of Ramanujan, Glasgow J. Math. 8 (1967), 14–32. [G] F. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7, and 11, Trans. Amer. Math. Soc. 305 (1988), 47–77. [G-Ki-St] F. Garvan, D. Kim, D. Stanton, Cranks and t -cores, Invent. Math. 101 (1990), 1–17. [H-Hu] M. Hirschhorn, D. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Ange. Math. 336 (1981), 1–17. [K] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag, 1984. [N-R-S-Se] J.-L. Nicolas, I. Z. Ruzsa, A. S´ark¨ozy [Appendix by J.-P. Serre], On the parity of the additive representation functions, preprint. [O] K. Ono, Parity of the partition function in arithmetic progressions, J. reine angew. Math. 472 (1996), 1– [P-S] T. R. Parkin, D. Shanks, On the distribution of parity in the partition function, Math. Comp. 21 (1967), 466–480. [Ra ] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. (2) 19 (1919), 207–210. [Stu] J. Sturm, On the congruence of modular forms, Springer Lect. Notes 1240 , Springer,
[Su] M. Subbarao, Some remarks on the partition function, Amer. Math. Monthly 73 (1966), 851–854. [W] G.N. Watson, Ramanujan’s vermutung ¨uber zerf¨allungsanzahlen, J. reine angew. Math. 179 (1938), 97–128.
