`-adic Properties of the Partition Function

Ramanujan’s famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend `-adically to 0. Although these congruences have inspired research in many directions, little is known about the `-adic behavior of these sequences for primes ` ≥ 13. We show that these sequences are governed by “fractal” behavior. Modulo any power of a prime ` ≥ 5, these sequences of generating functions `-adically converge to linear combinations of at most b `−1 12 c−b `2−1 24` c many special q-series. For ` ∈ {5, 7, 11} we have b `−1 12 c−b `2−1 24` c = 0, thereby giving a conceptual explanation of Ramanujan’s congruences. We use the general result to reveal the theory of “multiplicative partition congruences” that Atkin anticipated in the 1960s. His results and observations are examples of systematic infinite families of congruences which exist for all powers of primes 13 ≤ ` ≤ 31 since b `−1 12 c − b `2−1 24` c = 1.