Ramanujan Congruences for Fractional Partition Functions

2019/07/15 by Bevilacqua, Erin, Chandran, Kapil, Choi, Yunseo
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.06716

Abstract

For rational α, the fractional partition functions p_α(n) are given by the coefficients of the generating function (q;q)^α_∞. When α=-1, one obtains the usual partition function. Congruences of the form p(ℓ n + c)≡ 0 \pmodℓ for a prime ℓ and integer c were studied by Ramanujan. Such congruences exist only for ℓ∈\5,7,11\. Chan and Wang [4] recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as p_(57)/(61)(17²n-3)≡ 0 \pmod17².

Related

• Arithmetic properties of the ℓ-regular partitions
• Ramanujan congruences for a class of eta quotients
• Infinite Families of Congruences Modulo 5 for Ramanujan's General Partition Function
• Ramanujan-Type congruences for cubic partition functions
• Scarcity of congruences for the partition function