Abstract
We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b ℓ (n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∏ ∞ m=1 (1−q m)ℓ−1.
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Dandurand, B., Penniston, D. ℓ-Divisibility of ℓ-regular partition functions. Ramanujan J 19, 63–70 (2009). https://doi.org/10.1007/s11139-007-9042-8
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DOI: https://doi.org/10.1007/s11139-007-9042-8