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Ramanujan congruences for p-k (mod 11')

Published online by Cambridge University Press:  18 May 2009

Basil Gordon
Basil Gordon
Affiliation:
University of California, Los Angeles
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Denote by

the Euler product, and by

the partition generating function. More generally, if k is any integer, put

so that p(n) = p−1(n). In [3], Atkin proved the following theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 2 , July 1983 , pp. 107 - 123
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Apostol, T. M., Modular functions and Dirichlet series in number theory (Springer, 1976).Google Scholar
2.Atkin, A. O. L., Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14–32.Google Scholar
3.Atkin, A. O. L., Ramanujan congruences for p -k(n), Canad. J. Math. 20 (1968), 67–78.Google Scholar
4.Fine, N. J., On a system of modular functions connected with the Ramanujan identities, Tôhoku Math. J. (2) 8 (1956), 149–164.Google Scholar
5.Hughes, K., Arithmetic properties of modular forms, Ph.D. thesis, University of California, Los Angeles (1980).Google Scholar
6.Newman, M., Construction and application of a class of modular functions, Proc. London Math. Soc. (3) 7 (1957), 334–350.CrossRefGoogle Scholar