# Arithmetic properties of the partition function

@article{Ahlgren2003ArithmeticPO, title={Arithmetic properties of the partition function}, author={Scott Ahlgren and Matthew Boylan}, journal={Inventiones mathematicae}, year={2003}, volume={153}, pages={487-502} }

Let p(n) denote the number of partitions of the positive integer n; p(n) is the number of representations of n as a non-increasing sequence of positive integers (by convention, we agree that p(0) = 1 and that p(n) = 0 if n < 0). The study of the arithmetic properties of p(n) has a long and rich history; see, for example, the works of Andrews, Atkin, Dyson, Garvan, Kim, Stanton, and Swinnerton-Dyer [An,An-G,At1,At-SwD,D,G-K-S]. These works have their origins in the groundbreaking observations of…

## 92 Citations

### Computing the Residue Class of Partition Numbers

- Mathematics
- 2016

In 1919, Ramanujan initiated the study of congruence properties of the integer partition function p(n) by showing that p(5n+ 4) ≡ 0 (mod 5) and p(7n+ 5) ≡ 0 (mod 7) hold for all integers n. These…

### SUBBARAO ’ S CONJECTURE ON THE PARITY OF THE PARTITION FUNCTION

- Mathematics
- 2010

Let p(n) denote the ordinary partition function. In 1966, Subbarao [18] conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N (resp. M) ≡ r (mod t) for which…

### Odd coefficients of weakly holomorphic modular forms

- Mathematics
- 2008

). We will consider the question ofestimating the number of integers n for which a(n) 6≡0 (mod v).For a well-studied example, let p(n) be the ordinary partition function. Manyauthors have considered…

### Theta-type congruences for colored partitions

- Mathematics
- 2022

. We investigate congruence relations of the form p r ( (cid:96)mn + t ) ≡ 0 (mod (cid:96) ) for all n , where p r ( n ) is the number of r -colored partitions of n and m, (cid:96) are distinct…

### Partition values and central critical values of certain modular L-functions

- Mathematics
- 2010

Let p(n) denote the number of partitions of a positive number n, let l ∈ {5, 7, 11} and let δ l be the least non-negative residue ot 24 ―1 modulo l. In this paper we prove congruences modulo l…

### Critical L-values of Level p Newforms (mod p)

- Mathematics
- 2009

Suppose that p≥ 5 is prime, that ℱ(z) ∈ S 2k (Γ 0 (p)) is a newform, that v is a prime above p in the field generated by the coefficients of ℱ, and that D is a fundamental discriminant. We prove…

### Congruence relations for r-colored partitions

- Mathematics
- 2022

Let l ≥ 5 be prime. For the partition function p(n) and 5 ≤ l ≤ 31, Atkin found a number of examples of primes Q ≥ 5 such that there exist congruences of the form p(lQn + β) ≡ 0 (mod l). Recently,…

### The spt-function of Andrews

- MathematicsBCC
- 2017

Recent developments in the study of spt($n), including congruence properties established by Andrews, Bringmann, Folsom, Garvan, Lovejoy and Ono et al., a constructive proof of the Andrews-Dyson-Rhoades conjecture, generalizations and variations of the spt-function, are summarized.

### Distribution of the coefficients of modular forms and the partition function

- Mathematics
- 2011

AbstractLet ℓ be an odd prime and j, s be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo an odd positive integer M. As an…

### Ramanujan-type congruences for certain generating functions

- Mathematics
- 2013

For nonnegative integers a, b, the function da,b(n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$. We establish some Ramanujan-type…

## References

SHOWING 1-10 OF 21 REFERENCES

### SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13

- Mathematics
- 1967

j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular…

### Proof of a conjecture of Ramanujan

- MathematicsGlasgow Mathematical Journal
- 1967

We write and so that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his…

### PERIODICITY MODULO m AND DIVISIBILITY PROPERTIES OF THE PARTITION FUNCTION(

- Mathematics
- 1960

has infinitely many solutions in non-negative integers n. This conjecture seems difficult and I have only scattered results. In ?2 of this paper it will be shown that the conjecture is true for m= 5…

### Dyson's crank of a partition

- Mathematics
- 1988

holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6)…

### Distribution of the partition function modulo $m$

- Mathematics
- 2000

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin,…

### The theory of partitions

- Mathematics
- 1976

1. The elementary theory of partitions 2. Infinite series generating functions 3. Restricted partitions and permutations 4. Compositions and Simon Newcomb's problem 5. The Hardy-Ramanujan-Rademacher…

### Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

- Mathematics
- 2001

When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The…

### On ℓ-adic representations and congruences for coefficients of modular forms (II)

- Mathematics
- 1977

The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan [8] and a modern one initiated by Serre [9] and Deligne [3]. To describe the classical theme, let…

### Cranks andt-cores

- Mathematics
- 1990

SummaryNew statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given…