Ramanujan's partition generating functions modulo l

doi:10.1007/s11139-025-01241-0

2025-11-04

Kathrin Bringmann

William Craig

Ken Ono *

DOI：10.1007/s11139-025-01241-0

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For the partition function p(n), Ramanujan proved the striking identities P-5(q):=Sigma(n >= 0)p(5n+4)q(n)=5 Pi(n >= 1)(q(5);q(5))(infinity)(5)/(q;q)(infinity)(6), P-7(q):=Sigma(n >= 0)p(7n+5)q(n)=7 Pi(n >= 1)(q(7);q(7))(infinity)(3)/(q;q)(infinity)(4)+49q Pi(n >= 1)(q(7);q(7))(infinity)(7)(q;q)(infinity)(8), where (q;q)(infinity):=Pi(n >= 1)(1-q(n)). As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes l >= 5, closed form expressions of the power series P-l(q):=Sigma(n >= 0)p(ln-delta(l))q(n)(mod l), where delta(l):=l(2)-1/24. In this paper, we prove that P-l(q)equivalent to c(l)T(l)(q)/q(l);q(l))(infinity)(mod l), where c(l) is an element of Z} is explicit and T-l(q) is the generating function for the Hecke traces of l-ramified values of special Dirichlet series for weight l-1 cusp forms on SL2(Z). This is a new proof of Ramanujan's congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10.

Keyword:

Modular forms

Partition function

Ramanujan's partition congruences