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# Congruences for the Andrews *spt* function

Edited by George E. Andrews, Pennsylvania State University, University Park, PA, and approved November 2, 2010 (received for review October 12, 2010)

## Abstract

Ramanujan-type congruences for the Andrews *spt*(*n*) partition function have been found for prime moduli 5 ≤ *ℓ* ≤ 37 in the work of Andrews [Andrews GE, (2008) *J Reine Angew Math* 624:133–142] and Garvan [Garvan F, (2010) *Int J Number Theory* 6:1–29]. We exhibit unexpectedly simple congruences for all *ℓ*≥5. Confirming a conjecture of Garvan, we show that if *ℓ*≥5 is prime and , then (mod *ℓ*). This congruence gives (*ℓ* - 1)/2 arithmetic progressions modulo *ℓ*^{3} which support a mod *ℓ* congruence. This result follows from the surprising fact that the reduction of a certain mock theta function modulo *ℓ*, for every *ℓ*≥5, is an eigenform of the Hecke operator *T*(*ℓ*^{2}).

## 1. Introduction and Statement of Results

Andrews (1) recently introduced the function *spt*(*n*), which counts the number of smallest parts among the integer partitions of *n*. For *n* = 4 we have The smallest parts are underlined, and so we have that *spt*(4) = 10. Andrews (1) proved the following elegant Ramanujan-type congruences: Inspired by these congruences, Garvan (2) has obtained several new congruences. Recently, Folsom and Ono (3) (see also ref. 4) confirmed conjectures of Garvan and Sellers, and these results provide simple congruences modulo 2 and 3.

The situation is more complicated for primes *ℓ*≥5. It is known that there are infinitely many congruences of the form This fact follows from work of Bringmann (5) (see also refs. and 7) on *N*_{2}(*n*), the second rank moment, combined with earlier work of Ahlgren and Ono on *p*(*n*) (8–10). However, explicit examples are only known for *ℓ* ≤ 37. For example, Garvan (2) has obtained The moduli of the arithmetic progressions above involve (fourth) powers of special auxiliary primes, a feature shared by the congruences which arise from this theory. The congruences are constructed using these special primes, and these primes are guaranteed to exist by the theory of odd modular *ℓ*-adic Galois representations and the Chebotarev density theorem. To find a congruence, one is then required to search, prime by prime, for an auxiliary prime. This task is analogous to the simpler problem of finding the smallest prime *p* ≡ 1 (mod *ℓ*).

We establish unique universal congruences for *spt*(*n*) without relying on the existence of such primes. For aesthetics, we define and by [1][2]We obtain the following congruences relating , , and the Legendre symbol .

If *ℓ*≥5 is prime, then

Theorem 1.1 may be reformulated in terms of the “mock theta function” [3]We refer to *M*(*q*) as a mock theta function because it is the holomorphic part of a harmonic Maass form. Although *M*(*q*) is not an eigenform of any Hecke operators, Theorem 1.1 is equivalent to the assertion, for every prime *ℓ*≥5 that

Theorem 1.1 immediately gives the following corollary.

Suppose that *ℓ*≥5 is prime. Then the following are true

If , then

We have that

Corollary 1.2 (1) gives distinct for which Indeed, if , then Corollary 1.2 (1) implies that These congruences were conjectured by Garvan in July 2008*. Garvan’s conjecture was inspired by work done by Garrett and her students in October 2007. For *ℓ* = 11 the general result gives the five congruences:

In Section 2 we prove Theorem 1.1 and Corollary 1.2 using work of Bringmann (5), and of Bruinier and Ono (11). In Section 3 we conclude with several illuminating examples.

## 2. Proofs

We assume that the reader is familiar with basic facts about modular forms and harmonic Maass forms [for background, see refs. (12–14)]. In ref. 1, Andrews obtained the following generating function for *spt*(*n*): [4]where *q* is a formal parameter and . If we let *q*≔*e*^{2πiz}, where *z* is in the upper half of the complex plane, then we have the following important theorem^{†} of Bringmann (5) which relates this generating function to a certain harmonic Maass form.

Define the function by where is Dedekind’s eta-function, and where Then is a weight 3/2 harmonic Maass form on Γ_{0}(576) with Nebentypus .

### 2.1 Producing Modular Forms.

We use Theorem 2.1 to obtain modular forms from the harmonic Maass form . By the *q*-series manipulations in ref. 1 and [**2**], it is known that Therefore, [**1**] and [**3**] imply that *M*(*z*) = *M*(*q*) is the holomorphic part of , and so it is a mock theta function.

For each prime *ℓ*≥5, we let *T*(*ℓ*^{2}) be the index *ℓ*^{2} Hecke operator for weight 3/2 harmonic Maass forms with Nebentypus *χ*_{12}. On *q* series, these operators are defined by [5]We define *M*_{ℓ}(*z*) by [6]The following theorem is crucial to the proof of Theorem 1.1.

Suppose that *ℓ*≥5 is prime, and that Then *F*_{ℓ}(*z*) is a weight (*ℓ*^{2} + 3)/2 holomorphic modular form on .

The operator , where *y* = Im(*z*), has the property that . Because *η*(24*z*) is an eigenform of the weight 1/2 Hecke operators, Lemma 7.4 of ref. 11 implies that *M*_{ℓ}(*z*) is a weight 3/2 weakly holomorphic modular form on Γ_{0}(576) with Nebentypus *χ*_{12}. Here we used the fact that the eigenvalue of *η*(24*z*) for the index *ℓ*^{2} weight 1/2 Hecke operator is *χ*_{12}(*ℓ*)(1 + *ℓ*^{-1}).

It is straightforward to check that *M*_{ℓ}(*z*) has coefficients in , and has the property that [7]Here we have used the fact that *ℓ*^{2} ≡ 1 (mod 24). Therefore, it follows that is a weight (*ℓ*^{2} + 3)/2 weakly holomorphic modular form on Γ_{0}(576) with trivial Nebentypus whose nonzero coefficients are supported on exponents which are multiples of 24. In particular, we have that *F*_{ℓ}(*z*) = *F*_{ℓ}(*z* + 1). To prove that *F*_{ℓ}(*z*) is a weakly holomorphic modular form on , it suffices to prove that

To this end, let *W* be the Fricke involution (see ref. 13, section 3.2) which acts on weight 3/2 modular forms on Γ_{0}(576) by If *f* has Nebentypus *χ*, and if *ℓ*∤576 is prime, then it is well known that If we let *A*_{ℓ}(*z*)≔*F*_{ℓ}(24*z*), then this commutation relation implies that Using the fact that we then find that Bringmann proves that is an eigenform of *W* with multiplier arising from Dedekind’s eta-function (see ref. 5, section 4 ). A reformulation of her result shows that Combining these facts, we have that Letting *z* → *z*/24 gives Therefore, *F*_{ℓ}(*z*) is a weight (*ℓ*^{2} + 3)/2 weakly holomorphic modular form on . Because it is holomorphic at infinity, it is a holomorphic modular form, which completes the proof.

### Proof of Theorem 1.1 and Corollary 1.2.

We now prove Theorem 1.1.

By [**7**], we have that Because the coefficients of *M*_{ℓ}(*z*) are *ℓ* integral, *F*_{ℓ}(24*z*) (mod *ℓ*) is well defined. Moreover, it follows that ord_{ℓ}[*F*_{ℓ}(24*z*)]≥*ℓ*^{2} + 23. Here ord_{ℓ} denotes the smallest exponent whose coefficient is nonzero modulo *ℓ*. Therefore, we have that ord_{ℓ}[*F*_{ℓ}(*z*)]≥(*ℓ*^{2} + 23)/24. However, *F*_{ℓ}(*z*) is a weight (*ℓ*^{2} + 3)/2 holomorphic modular form on , and it is well known that every *f* in this space with *ℓ*-integral coefficients has either ord_{ℓ}(*f*) ≤ (*ℓ*^{2} + 3)/24 or ord_{ℓ}(*f*) = +∞. This fact follows from the existence of “diagonal bases” for spaces of modular forms on . Therefore we have that ord_{ℓ}[*F*_{ℓ}(*z*)] = +∞, which in turn implies that *M*_{ℓ}(*z*) ≡ 0 (mod *ℓ*). The theorem now follows from [**3**], [**5**], and [**6**].

Claim (1) follows because the right-hand side is 0 (mod *ℓ*) in Theorem 1.1. Claim (2) follows by replacing *n* by *nℓ* in Theorem 1.1 because .

## 3. Examples

Here we give examples which illustrate the results and modular forms in this paper.

### 3.1. Explicit Formulas for *M*_{5}(*z*) and *M*_{7}(*z*).

Here we compute the level 1 modular forms *F*_{5}(*z*) and *F*_{7}(*z*) in terms of Δ(*z*)∶ = *η*(*z*)^{24}, and the usual Eisenstein series

For *ℓ* = 5, we find thatTheorem 2.2 implies that *F*_{5}(*z*) is a weight 14 holomorphic modular form, and we find that Therefore, we have that For *ℓ* = 7, we find that *F*_{7}(*z*) is the weight 26 modular form which in turn implies that

### 3.2. Example of Corollary 1.2 (1).

If *ℓ*≥5 is prime, then let *O*_{ℓ}(*q*) be the series By Corollary 1.2 (1), we have that *O*_{ℓ}(*q*) ≡ 0 (mod *ℓ*). If *ℓ* = 11, then we indeed see that

### 3.3. Example of Corollary 1.2 (2).

If *ℓ*≥5 is prime, then let

Corollary 1.2 (2) then asserts that For *ℓ* = 11, we find that and that

## Acknowledgments

The author acknowledges Frank Garvan for his contributions to this work. The results in this paper were previously conjectured by him, and in some cases by Tina Garrett and her research students. The author thanks Matt Boylan, Kathrin Bringmann, Frank Garvan, Marie Jameson, Zach Kent, Karl Mahlburg, and the referees for their helpful comments which improved the exposition in this paper. The author thanks the National Science Foundation and the Candler Fund for their support.

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: ono{at}mathcs.emory.edu.Author contributions: K.O. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The author declares no conflict of interest.

The article is a PNAS Direct Submission.

↵

^{*}The*spt*-congruence in your paper was originally conjectured by me. This conjecture was inspired by work of Tina Garrett's students.

## References

- ↵
- Andrews GE

*n*. J Reine Angew Math 624:133–142. - ↵
- ↵
- Folsom A,
- Ono K

*spt*-function of Andrews. Proc Natl Acad Sci USA 105:20152–20156. - ↵
- ↵
- ↵
- Bringmann K,
- Garvan F,
- Mahlburg K

- ↵
- ↵
- ↵
- Ahlgren S,
- Ono K

- ↵
- ↵
- Bruinier JH,
- Ono K

*L*-functions, and harmonic weak Maass forms. Ann Math 172, pp 2135–2181. - ↵
- ↵
- Ono K

- ↵
- Ono K

*spt*function

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