• Corpus ID: 245329788

Congruences like Atkin's for the partition function

@inproceedings{Ahlgren2021CongruencesLA,
  title={Congruences like Atkin's for the partition function},
  author={Scott Ahlgren and Patrick Brodie Allen and Shiang Tang},
  year={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 lie in two natural families distinguished by the square class of 1 − 24 β (mod ℓ ). In recent decades much work has been done to understand congruences of the form p ( Q m ℓn + β ) ≡ 0 (mod ℓ ). It is now known that there are many such congruences when m ≥ 4, that such congruences are scarce (if… 
2 Citations
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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,
Scarcity of congruences for the partition function
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

References

SHOWING 1-10 OF 35 REFERENCES
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
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,
SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13
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
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),
Arithmetic properties of the partition function
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
Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences
. Let p ( n ) denote the number of partitions of n . Let A,B ∈ N with A > B and (cid:2) ≥ 5 a prime, such that p ( An + B ) ≡ 0 (mod (cid:2) ) , n ∈ N . Then we will prove that (cid:2) | A and
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
Congruence properties for the partition function
  • S. Ahlgren, K. Ono
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 2001
TLDR
It is reported that such congruences are much more widespread than was previously known, and the theoretical framework that appears to explain every known Ramanujan-type congruence is described.
Modular forms of half-integral weights on SL(2, ℤ)
  • Yifan Yang
  • Mathematics
    Nagoya Mathematical Journal
  • 2014
Abstract In this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the set is isomorphic to as Hecke modules under the Shimura correspondence.
Congruences for the coefficients of weakly holomorphic modular forms
Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely
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