Arithmetic properties of the partition function

  title={Arithmetic properties of the partition function},
  author={Scott Ahlgren and Matthew Boylan},
  journal={Inventiones mathematicae},
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… 

Computing the Residue Class of Partition Numbers

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

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

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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

  • 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


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

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)

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