ARITHMETIC OF THE PARTITION FUNCTION Department of ...

ARITHMETIC OF THE PARTITION FUNCTION

KEN ONO

Department of Mathematics University of Wisconsin at Madison Madison, Wisconsin 53706 USA

1. Introduction Here we describe some recent advances that have been made regarding the arithmetic of the unrestricted partition function p(n). A partition of a nonnegative integer n is any nonincreasing sequence of positive integers whose sum is n. As usual, we let p(n) denote the number of partitions of n. For example, it is easy to see that p(4) = 5 since the partitions of 4 are: 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. Partitions have played an important role in many aspects of combinatorics, Lie theory, physics, and representation theory. Here we describe some of the recent discoveries regarding the arithmetic of the partition function, including a relationship between partitions and Tate-Shafarevich groups of modular motives in arithmetic algebraic geometry. Euler [4] showed that the generating function for p(n) is given by the convenient infinite product ∞ X

p(n)q n =

n=0

∞ Y

1 = 1 + q + 2q 2 + 3q 3 + 5q 4 + · · · , n 1 − q n=1

(1)

and his Pentagonal Number Theorem asserts that ∞ Y

(1 − q n ) =

n=1

∞ X

(−1)n q (3n

2 +n)/2

.

(2)

n=−∞

It is easy to see that (1) and (2) together imply, for every positive n, that p(n) =

∞ X

(−1)k+1 p(n − 3(k2 + k)/2) + (−1)k+1 p(n − (3k2 − k)/2) .

k=1

(3)

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KEN ONO

Although (3) is an efficient recursive device for computing p(n), it is not a formula. Fortunately, an improvement, by Rademacher, of the HardyRamanujan asymptotic formula p(n) ∼

p 2n 1 √ · eπ 3 4n 3

leads to an ‘exact formula’ for p(n). Although one would hope that these formulas would be powerful tools for proving theorems about p(n), it has turned out that many of the most basic questions remain open. In this section we review some of the remaining classical problems, and in the next section we describe the recent progress that has been made on these questions. In the last section we describe the relationship between p(n) and certain families of modular L-functions in the context of the Bloch-Kato Conjecture. 1.1. PARITY

One of the simplest questions concerns the parity of p(n). Using (1) and (2), Kolberg [21] proved that there are infinitely many even (resp. odd) values of p(n). However, much more is conjectured to be true. The following widely believed conjecture is one of the most notorious problems in the subject. Conjecture 1 (Parkin and Shanks [30]). As n → +∞ we have #{n ≤ X : p(n) ≡ 0 (mod 2)} 1 = . X→+∞ X 2 lim

In 1983 Mirsky [23] proved that #{n ≤ X : p(n) is even (resp. odd)} log log X, and in 1995 Nicolas and S´ arközy [25] improved this estimate and proved that there is a constant c > 0 for which #{n ≤ X : p(n) is even (resp. odd)} logc X.

(4)

Subbarao made the following more accessible conjecture [33]. Conjecture 2 (Subbarao). In an arithmetic progression r (mod t), there are infinitely many integers M ≡ r (mod t) for which p(M ) is odd, and there are infinitely many integers N ≡ r (mod t) for which p(N ) is even.


3

Garvan, Kolberg, Hirschhorn, Stanton and Subbarao ([15], [18], [19], [20], [21]) have verified this conjecture for every arithmetic progression with modulus t ∈ {1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 40}. (5)

1.2. ARBITRARY MODULI

One is also naturally interested in the reduction of p(n) modulo arbitrary integers M . In this direction Newman [24] made the following similar conjecture regarding the behavior of p(n) (mod M ), as one varies n, for arbitrary M . Conjecture 3 (Newman) If M is a positive integer, then in every residue class r (mod M ) there are infinitely many integers N for which p(N ) ≡ r

(mod M ).

Works by Atkin, Kolberg and Newman ([7], [21], [24]) have verified Newman’s conjecture for every M ∈ {2, 5, 7, 13}.

(6)

To clarify the nature of these problems, consider the following conjecture due to Erdös [16]: Conjecture 4 (Erdös) If M is prime, then there is at least one nonnegative integer NM for which p(NM ) ≡ 0 (mod M ). Obviously, Erd¨ os’ conjecture is implied by Newman’s Conjecture. Erd¨ os’ Conjecture has also been very difficult to handle. Using the asympotic formula for p(n), Schinzel (see [12]) proved that there are infinitely many prime divisors among the values of p(n), and later Schinzel and Wirsing [32] proved that the number of primes M ≤ X which divide at least one value of p(n) is log log X. In view of the Prime Number Theorem (i.e. that the number of primes p ≤ X is asymptotically X/ log X), these results fall far short of Conjecture 4. 1.3. RAMANUJAN-TYPE CONGRUENCES

The most striking results concerning the congruence properties of p(n) are due to Ramanujan, the legendary Indian mathematician. Ramanujan’s

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KEN ONO

findings are particularly shocking in view of the conjectures and problems above. For instance, Ramanujan [31] proved that p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11)

(7) (8) (9)

for every non-negative integer n. These three congruences are the simplest cases of three infinite families of congruences which were conjectured by Ramanujan. By the works of Atkin, Ramanujan, and Watson (see [6], [10], [34]) it is now known that for every integer k we have p(5k n + δ5,k ) ≡ 0 (mod 5k ),

(10)

p(7k n + δ7,k ) ≡ 0 (mod 7[k/2] ),

(11)

p(11 n + δ11,k ) ≡ 0 (mod 11 )

(12)

k

k

for every non-negative integer n where 24δ`,k ≡ 1 (mod `k ). Such congruences are particularly suprising since a cursory examination of values of p(n) fails to reveal any further congruences. In fact, a study of the values of p(n) suggests that p(n) (mod M ) is random apart from progressions where there are Ramanujan-type congruences. Therefore, it is natural to ask the following two questions: Question 1 How rare are congruences of the form p(an + b) ≡ 0

(mod M )?

Question 2 If M > 1 is an integer, is there a progression b the property that p(an + b) ≡ 0 (mod M )

(mod a) with

for every non-negative integer n? There is some evidence that supports the view that there might be many congruences. In the 1960s, some further congruences for p(n), such as p(113 · 13n + 237) ≡ 0

(mod 13)

were discovered by Atkin, O’Brien and Swinnerton-Dyer (see [7], [8], [9]). Since partitions are combinatorial objects, it is natural to ask whether there are combinatorial explanations for Ramanujan’s congruences. In 1944, Dyson [11] conjectured that the ‘rank’ provides such an explanation for (7)


5

and (8). The rank of a partition is the difference between the number of its parts and its largest part. If ` = 5 or 7 and 0 ≤ i ≤ ` − 1, then Dyson conjectured, for every non-negative integer n, that p(`n + δ`,1 )/` equals the number of partitions of `n + δ`,1 with rank congruent to i (mod `). In 1954, Atkin and Swinnerton-Dyer [7] proved Dyson’s conjecture. More recent works by Andrews, Garvan, Kim, and Stanton ([5], [13], [14]) have produced a number of further statistics (a.k.a. ‘cranks’) which explain some of the other Ramanujan congruences. Question 3 Are there systematic statistics which uniformly describe (10), (11) and (12)? We conclude this sections with the following question. Question 4 Do the numbers p(n) (mod M ) play a fundamental role in other areas of mathematics? 2. Recent Results In this section we highlight the recent advances that have been made on the problems described in the previous section. 2.1. PARITY

Conjecture 1 remains wide open. However, recent work by Nicolas, Ruzsa, and Sárközy [26] have made a substantial improvement on (4) using a careful analysis of (3). Theorem 2.1 (Nicolas-Ruzsa-Sárközy) √ a) For large X, we have #{n ≤ X : p(n) is even} X. b) If > 0, then √ log X #{n ≤ X : p(n) is odd} X · exp (− log 2 + ) · . log log X Regarding Subbarao’s Conjecture, much more is now known. Using the theory of modular forms and Galois representations, as developed by Deligne and Serre, the author was able to prove the following theorem [27]. Theorem 2.2 (Ono) a) In any arithmetic progression r (mod t), there are infinitely many integers n such that p(n) is even. b) In any arithmetic progression r (mod t), there are infinitely many integers n such that p(n) is odd, provided that there is at least one such n. Furthermore, if such an n exists, then the smallest such n is < 1010 t7 .

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KEN ONO

Therefore, the ‘even’ case of Subbarao’s Conjecture is always true, and there is now a simple algorithm which determines the truth of the ‘odd part’ of the conjecture for any given arithmetic progression. Using this algorithm, the odd case has now been verified for every arithmetic progression with modulus t ≤ 105 . It is natural to ask for quantitative forms of Theorem 2.2. Ahlgren and Serre (see [1], [26]) have obtained such results. Theorem 2.3 (Ahlgren and Serre) In any arithmetic progression r (mod t) we have #{n ≤ X : n ≡ r

(mod t) and p(n) even} r,t

Theorem 2.4 (Ahlgren) If there is an integer n ≡ r p(n) is odd, then #{n ≤ X : n ≡ r

(mod t) and p(n) odd} r,t

√

X.

(mod t) for which √ X/ log X.

(Note. Ahlgren [2] has obtained a generalization of Theorem 2.4 which holds for arbitrary prime modulus M , not just M = 2.) 2.2. ARBITRARY MODULI AND RAMANUJAN CONGRUENCES

We begin by considering the rarity of Ramanujan-type congruences. The first author was able to quantify [27] their rarity using the fact that the Hecke operators commute with the action of twisting a modular form. Here we describe a typical result in this direction. Let ` be prime, and let S` denote the set of primes t with the property that in every arithmetic progression 24−1 6≡ r (mod t) there are infinitely many integers n ≡ r

(mod t) for which

p(n) 6≡ 0 (mod `). Theorem 2.5 (Ono) If ` is prime, then the set of primes S` has density exceeding 1 − 10−100 within the set of prime numbers. Such results clarify the rarity of Ramanujan-type congruences and provides a reasonable resolution to Question 1. Recently, we have learned a lot about Question 2. Using the theory of modular Galois representations and the commutativity of Hecke algebras across Shimura’s correspondence, the author was able to prove the following theorem [29].


7

Theorem 2.6 (Ono) Let M ≥ 5 be prime and let k be any positive integer. A positive proportion of the primes ` have the property that p

m k `3 n + 1 24

!

≡ 0 (mod M )

for every non-negative integer n coprime to `. It is easy to see that this theorem immediately implies that if M ≥ 5 is prime, then there are infinitely many distinct arithmetic progressions b (mod a) for which p(an + b) ≡ 0 (mod M ) for every non-negative integer n. Weaver [35] has computed over 70,000 explicit examples of such congruences. For instance, she has found that p(48037937 · N + 1122838) ≡ 0 (mod 17), p(1977147619 · N + 815655) ≡ 0 (mod 19), p(14375 · N + 3474) ≡ 0 (mod 23), p(348104768909 · N + 43819835) ≡ 0 (mod 29), p(4063467631 · N + 30064597) ≡ 0 (mod 31).

(13) (14) (15) (16) (17)

Furthermore, Theorem 2.6 implies the truth of Erdös’ Conjecture (Conjecture 4). Unfortunately, the following problem remains open. Question 5 Show that there are infinitely many integers n for which p(n) ≡ 0

(mod 3).

The methods which proved Theorem 2.6 are also useful in attacking Newman’s Conjecture 3. In particular, the author [29] proved Conjecture 3 for every prime modulus M < 1000 with the possible exception of M = 3. Recently, Ahlgren [3] has extended Theorem 2.6 to include composite moduli using an elegant p-adic completion of the forms employed by the author in [29]. The most elegant consequence of Ahlgren’s theorem is the following result. Theorem 2.7 (Ahlgren) If M is a positive integer coprime to 6, then there are infinitely many distinct arithmetic progressions b (mod a) for which p(an + b) ≡ 0 (mod M ) for every non-negative integer n.

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KEN ONO

Therefore, by Theorems 2.6 and 2.7, it is now apparent that Ramanujantype congruences are plentiful. However, it is typical that such congruences are monstrous like those appearing in (13-17). We conclude this section by noting that there are some new congruences which are elegant and systematic. In a recent preprint [22], the author and Lovejoy have extended (10) in infinitely many ways. The following result is one special case of these general results. Theorem 2.8 (Lovejoy and Ono) If j is a positive integer and β(j) = (3887 · 52j + 1)/24, then for every non-negative integer N we have p(25j · 133 N + 25j · 132 + β(j)) ≡ 0 (mod 52j+1 ). Although many questions such as Conjecture 1 remain open, it is refreshing to see that progress is being made on some of the questions described in Section 1. 3. Modular L-functions and p(n) In this last section we address Question 4 regarding the role that p(n) plays in other areas of mathematics. To the author’s surprise, it turns out that the residues of p(n) (mod M ) play a fundamental role in the arithmetic of certain modular motives [17]. Here we briefly describe the results proved P in [17]. If 13 ≤ ` ≤ 31 is prime, then let G` (z) = ∞ a (n)q n (q := e2πiz ) ` n=1 be the unique newform in S`−3 (Γ0 (6)) whose Fourier expansion begins with the terms

G` (z) = q +

2 `

·2

(`−5)/2 2

q +

3 `

· 3(`−5)/2 q 3 + · · · .

(18)

Here x` denotes the Legendre symbol modulo `. If D is a fundamental discriminant of a quadratic field, then √ let χD denote the usual Kronecker character for the quadratic field Q( D) and let L(G` ⊗ χD , s) denote the L-function of the twisted form given by L(G` ⊗ χD , s) =

∞ X χD (n)a` (n) n=1

ns

.

(19)

Define integers 1 ≤ δ` ≤ ` − 1 and 1 ≤ r` ≤ 23 by 24δ` ≡ 1 (mod `), r` ≡ −` (mod 24).

(20) (21)


9

For every non-negative integer n let D(`, n) be the integer given by D(`, n) := (−1)(`−3)/2 · (24n + r` ).

(22)

Using Shimura’s correspondence and a deep theorem of Waldspurger, the author and Guo proved the following theorem which relates the values of the partition function to these L-functions. Theorem 3.1 If 13 ≤ ` ≤ 31 is prime and n ≥ 0 is an integer for which D(`, n) is square-free, then

L G` ⊗ χD(`,n) , `−3 (24n + r` )(`−4)/2 2

(`−4)/2

L G` ⊗ χD(`,0) , `−3 r` 2

≡

p(`n + δ` )2 p(δ` )2

(mod `).

This theorem has deep implications regarding the arithmetic of certain motives. If 13 ≤ ` ≤ 31 is prime, then let M (`) be the (` − 3)/2-th Tate twist of the motive associated to G` (z) by the work of Scholl. Similarly, let M (`,n) denote the twisted motive obtained by twisting M (`) by χD(`,n) . For each D(`, n), let X(M (`,n) ) denote the Tate-Shafarevich group of M (`,n) . The celebrated conjectures of Bloch and Kato asserts that if `−3 L G` ⊗ χD(`,n) , 2 6= 0, then

`−3 L G` ⊗ χD(`,n) , 2

= Γ`,n × #X(M (`,n) ),

(23)

where Γ`,n is an explicit non-zero number depending on n and `. By a careful analysis of the factor Γ`,n in (23), the author and Guo have been able to prove [17] the following theorem which gives further importance to the partition function. Theorem 3.2 (Guo and Ono) Suppose that 13 ≤ ` ≤ 31 is prime and n ≥ 0 is an integer for which (i) n 6≡ −[(` + 1)/12] (mod `), (ii) D(`, n) is square-free,

(iii) L G` ⊗ χD(`,n) , `−3 6= 0. 2 Assuming the truth of the Bloch-Kato Conjecture, we have that ord`

#X(M (`,n) ) #X(M (`,0) )

!

⇐⇒ p(`n + δ` ) ≡ 0 (mod `).

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KEN ONO

Theorem 3.2 is a new role for the partition function in mathematics. The divisibility of p(n) now dictates, subject to the truth of the Bloch-Kato Conjecture, the presence of elements of order ` in certain Galois cohomology groups which are central in arithmetic geometry. 4. Acknowledgements The author thanks the Alfred P. Sloan Foundation, the David and Lucile Packard Foundation, and the National Science Foundation for their generous support. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Ahlgren, S. (1999) Distribution of parity of the partition function in arithmetic progressions, Indagationes Math., 10, 173-181. Ahlgren, S., The partition function modulo odd primes `, Mathematika, accepted for publication. Ahlgren, S., Distribution of the partition function modulo composite integers M , preprint. Andrews, G. E., (1998) The Theory of Partitions, Cambridge University Press, Cambridge. Andrews, G. E. and F. Garvan (1988) Dyson’s crank of a partition, Bull. Amer. Math. Soc., 18, 167-171. Atkin, A. O. L., (1967) Proof of a conjecture of Ramanujan, 8, Glasgow J. Math., 7, 14-32. Atkin, A. O. L., (1968) Multiplicative congruence properties and density problems for p(n), Proc. London Math. Soc., 18, 563-576. Atkin, A. O. L. and J. N. O’Brien, (1967) Some properties of p(n) and c(n) modulo powers of 13, Trans. Amer. Math. Soc., 126, 442-459. Atkin, A. O. L. and H. P. F. Swinnerton Dyer, (1971) Modular forms on noncongruence subgroups, Combinatorics, Proc. Sympos. Pure Math., UCLA, 1968, 19, 1-25. Berndt, B. C. and K. Ono, (1999) Ramanujan’s unpublished manuscript on the partition and tau functions with commentary, Seminaire Lotharingien de Combinatoire, 42. Dyson, F. (1944) Some guesses in the theory of partitions, Eureka, 8, 10-15. Erd¨ os, P. and A. Iviˆc, (1989) The distribution of certain arithmetical functions at consecutive integers, Proc. Budapest Conf. Number Theory, Coll. Math. Soc. J. Bolyai, North-Holland, Amsterdam, 51, 45-91. Garvan, F. (1988) New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7, and 11, Trans. Amer. Math. Soc., 305, 47-77. Garvan F., D. Kim, and D. Stanton (1990) Cranks and t-cores, Inventiones Mathematicae, 101, 1-17. Garvan, F. and D. Stanton (1990) Sieved partition functions and q-binomial coefficients, Mathematics of Computation, 55, 299-311. Gordon, B. (1999) Private communication. Guo, L. and K. Ono, (1999) The partition function and the arithmetic of certain modular L-functions, International Mathematics Research Notices, 21, 1179-1197. Hirschhorn, M. (1980) On the residue mod 2 and mod 4 of p(n), Acta Arith., 38, 105-109. Hirschhorn, M. (1993) On the parity of p(n), II, J. Comb. Th. (A), 62, 128-138.

ARITHMETIC OF THE PARTITION FUNCTION 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

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Hirschhorn, M. and M. Subbarao (1980) On the parity of p(n), Acta Arith., 50, 355-356. Kolberg, O., (1959) Note on the parity of the partition function, Math. Scand., 7, 377-378. Lovejoy, J. and K. Ono, Ramanujan’s congruences for the partition function modulo powers of 5, preprint. Mirsky, L., (1983) The distribution of values of the partition function in residue classes, J. Math. Anal. Appl., 93, 593-598. Newman, M., (1960) Periodicity modulo m and divisibility properties of the partition function, Trans. Amer. Math. Soc., 97, 225-236. Nicolas, J.-L. and A. S´ ark¨ ozy, (1995) On the parity of partition functions, Illinois J. Math., 39, 586-597. Nicolas, J.-L., I. Z. Ruzsa and A. S´ ark¨ ozy (with an appendix by J.-P. Serre) (1998) On the parity of additive representation functions, J. Number Theory, 73, 292-317. Ono, K., (1996) Parity of the partition function in arithmetic progressions, J. reine angew. math., 472, 1-15. Ono, K., (1998) The partition function in arithmetic progressions, Mathematische Annalen, 312, 251-260. Ono, K., (2000) Distribution of the partition function modulo m, Annals of Mathematics, 151, 293-307. Parkin, T. R. and D. Shanks, (1967) On the distribution of parity in the partition function, Mathematics of Computation, 21, 466-480. Ramanujan, S., (1919) Congruences properties of partitions, Proceedings of the London Mathematical Society, 19, 207-210. Schinzel, A. and E. Wirsing, (1987) Multiplicative properties of the partition function, Proc. Indian Acad. Sci. Math. Sci., 97, 297-303. Subbarao, M., (1966) Some remarks on the partition function, Amer. Math. Monthly, 73, 851-854. Watson, G. N., (1938) Ramanujan’s vermutung u ¨ber zerf¨ allungsanzahlen, J. reine angew. math., 179, 97-128. Weaver, R. L., New congruences for the partition function, Ramanujan Journal, accepted for publication.