Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function

@article{Newman1957CongruencesFT,
  title={Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function},
  author={Morris Newman},
  journal={Canadian Journal of Mathematics},
  year={1957},
  volume={9},
  pages={549 - 552}
}
  • M. Newman
  • Published 1957
  • Mathematics
  • Canadian Journal of Mathematics
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, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p 2 − l)/24. 
Further Identities and Congruences for the Coefficients of Modular Forms
  • M. Newman
  • Mathematics
    Canadian Journal of Mathematics
  • 1958
If n is a non-negative integer, define p r(n) by otherwise define p r(n) as 0. (Here and in what follows all sums will be extended from 0 to ∞ and all products from 1 to ∞ unless otherwise stated.) p
Some New Congruences Modulo 5 for the General Partition Function
TLDR
The emphasis throughout this paper is to exhibit the use of q -identities to generate the congruences for p_r(n) for the general partition function by restricting r to some sequence of negative integers.
Congruences like Atkin's for the partition function
Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p(Qln+ β) ≡ 0 (mod l) where l and Q are prime and 5 ≤ l ≤ 31; these lie in two
General congruences modulo 5 and 7 for colour partitions
For any positive integers n and r , let $$p_r(n)$$ p r ( n ) denotes the number of partitions of n where each part has r distinct colours. Many authors studied the partition function $$p_r(n)$$ p r (
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,
THE PARTITION FUNCTION AND HECKE OPERATORS
The theory of congruences for the partition function p(n) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas
The partition function and Hecke operators
PERIODICITY MODULO m AND DIVISIBILITY PROPERTIES OF THE PARTITION FUNCTION(
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
ON THE PARITY OF THE PARTITION FUNCTION
Although much is known about the partition function, little is known about its parity. For the polynomials D(x) := (Dx + 1)/24, where D ≡ 23 (mod 24), we show that there are infinitely many m (resp.
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On a conjecture of Ramanujan
BY ~lsing the Hardy-Ramanujan asymptotic formula for p(n) the number of unrestricted partitions of n, true for large values of n, Lehmer 1 has recently found that (1) Ÿ ----4353 50207 84031 73482
A table of Ramanujan1 s function r(n)
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Some Theorems about pr{n), Can
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  • 1957
Identities analagous to Ramanujan's identities involving the partition function
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