• Corpus ID: 219687128

# Scarcity of congruences for the partition function

@article{Ahlgren2020ScarcityOC,
title={Scarcity of congruences for the partition function},
author={Scott Ahlgren and Olivia Beckwith and Martin Raum},
journal={arXiv: Number Theory},
year={2020}
}
• Published 13 June 2020
• Mathematics
• arXiv: Number Theory
The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0\pmod\ell$ where $Q\geq 5$ is prime and $m\geq… 8 Citations Congruences for level$1$cusp forms of half-integral weight • Robert Dicks • Mathematics Proceedings of the American Mathematical Society • 2021 Suppose that$\ell \geq 5$is prime. For a positive integer$N$with$4 \mid N$, previous works studied properties of half-integral weight modular forms on$\Gamma_0(N)$which are supported on Congruences like Atkin's for the partition function • Mathematics • 2021 . Let p ( n ) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p ( Q 3 ℓn + β ) ≡ 0 (mod ℓ ) where ℓ and Q are prime and 5 ≤ ℓ ≤ 31; these Congruence relations for r-colored partitions 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, 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 Relations among Ramanujan-Type Congruences II We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences Relations among Ramanujan-Type Congruences I. We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific Hecke congruences.More generally, for weakly Relations among Ramanujan-Type Congruences I Ramanujan-type Congruences in Integral Weights We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific Hecke congruences. More generally, for weakly Congruences of Hurwitz class numbers on square classes • Mathematics • 2022 We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish ## References SHOWING 1-10 OF 42 REFERENCES New Congruences for the Partition Function AbstractLet p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n $$\begin{gathered} p(5 + 4) \equiv 0(\bmod 5), Nonzero coefficients of half-integral weight modular forms mod$$\ell $$ℓ • Mathematics • 2017 We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime$$\ell $$ℓ. Among the applications of this Modular Forms of Half Integral Weight The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function$$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty Classification of congruences for mock theta functions and weakly holomorphic modular forms Let$f(q)$denote Ramanujan's mock theta function $f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.$ It is known that Distribution of the partition function modulo$m$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, Congruences like Atkin's for the partition function • Mathematics • 2021 . Let p ( n ) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p ( Q 3 ℓn + β ) ≡ 0 (mod ℓ ) where ℓ and Q are prime and 5 ≤ ℓ ≤ 31; these Modular Functions In Analytic Number Theory Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions,$\eta(\tau)$and$\vartheta(\tau)\$, and their applications to two
Distribution of the partition function modulo composite integers M
seem to be distinguished by the fact that they are exceptionally rare. Recently, Ono [O1] has gone some way towards quantifying the latter assertion. More recently, Ono [O2] has made great progress
Proof of a conjecture of Ramanujan
• A. Atkin
• Mathematics
Glasgow 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
Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function
• M. Newman
• Mathematics
Canadian Journal of Mathematics
• 1957
If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10,