“RAMANUJAN’S CONGRUENCES AND DYSON’S CRANK”

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11. Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famous congruences for P (n), the number of partitions of n indeed, Ramanujan’s formulas lay unread until 1976 when Dyson found In the Trainty College Library of Cambridge University among papers from the estate of the late G.N.Watson. In 1986, F.Garvan wrote his Pennsylvania state Ph.D. Thesis Precisely on the formulas of Ramanujan relative to the crank. In view of this theoretical description, the story of the crank is a long romantic tale and the crank functions are intimately connected to all partitions congruences. In 2005, Mahlburg stated that the crank functions themselves obey Ramanujan type congruences.

Non-Existence of Ramanujan Congruences in Modular Forms of Level Four

Ramanujan famously found congruences like p(5n+4) ≡ 0 mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Г1(4) that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.

RAMANUJAN CONGRUENCES FOR SIEGEL MODULAR FORMS

We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.

RAMANUJAN CONGRUENCES FOR A CLASS OF ETA QUOTIENTS

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 ( mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences cN(n) satisfy, where cN(n) is defined by [Formula: see text]. This last result answers a question of H.-C. Chan.