The partition function and Hecke operators

@article{Ono2011ThePF,
  title={The partition function and Hecke operators},
  author={Ken Ono},
  journal={Advances in Mathematics},
  year={2011},
  volume={228},
  pages={527-534}
}
  • K. Ono
  • Published 10 September 2011
  • Mathematics
  • Advances in Mathematics
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Hecke-type congruences for Andrews' spt-function modulo 16 and 32
Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt(n).…

References

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Let p(n) denote the partition function. In this article, we will show that congruences of the formexist for all primes m and l satisfying m≥13 and l≠2,3,m, where B is a suitably chosen integer…
Congruence properties for the partition function
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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.
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As has been known since the work of F.K.C. Rankin and H.P.F. Swinnerton-Dyer [8], the values of j(τ) at the zeros in H of the Eisenstein series Ek(τ) of any weight k on SL2(Z) always lie in the…
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),…
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…
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…
Proof of a conjecture of Ramanujan
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  • Mathematics
    Glasgow Mathematical Journal
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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…
Extension of Ramanujan's congruences for the partition function modulo powers of five
Ramanujans Vermutung über Zerfällungszahlen.
Congruence Properties and Density Problems for the Fourier Coefficients of Modular Forms.
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