Let R(z, q) be the two-variable generating function of Dyson’s rank function. In a recent joint work with Frank Garvan, we investigated the transformation of the elements of the p-dissection of R(ζ_p, q), where ζ_p is a primitive p-th root of unity, under special congruence subgroups of SL_2(Z), leading us to interesting symmetry observations. In this work, we derive analogous transformation results for the two-variable crank generating function C(z, q) in terms of generalized eta products. We consider the action of the group Γ_0(p) on the elements of the p-dissection of C(ζ_p, q), leading us to new symmetries for the crank function. As an application, we give a new proof of the crank theorem predicted by Dyson in 1944 and resolved by Andrews and Garvan in 1988. Furthermore, we present identities expressing the elements of the crank dissection in terms of generalized eta products for primes p = 11, 13, 17 and 19.
Ivan Niven's succinct proof that pi is irrational is easy to verify, but it begins with a magical formula that appears to come out of nowhere, and whose origin remains mysterious even after one goes through the proof. The goal of this expository paper is to describe a thought process by which a mathematician might come up with the proof from scratch, without having to be a genius. Compared to previous expositions of Niven's proof, perhaps the main novelty in the present account is an explicit appeal to the theory of orthogonal polynomials, which leads naturally to the consideration of certain integrals whose relevance is otherwise not immediately obvious.
We consider the problem of the vanishing of {P}oincaré series for congruence subgroups. Denoting by $P_{k,m,N}$ the {P}oincaré series of weight $k$ and index $m$ for the group $\Gamma_0(N)$, we show that for certain choices of parameters $k,m,N$, the {P}oincaré series does not vanish. Our methods improve on previous results of Rankin (1980) and Mozzochi (1989).
We investigate the Diophantine equation x^2 + x y - y^2 = m, for m ∈ Z given, from a geometric point of view. The hyperbola given by the equation carries a known group structure, which we interpret in four different ways, firstly with the familiar parallel line construction. It turns out that the group defined by this construction corresponds to the restriction of a group operation on R^2, which is induced by the number field Q( √ 5). In this way, the group operation can be described using lean formulae. We also find a parameterisation of the hyperbola that is compatible with the group operation. This result is analogous to the fact that the parameterisation of a cubic curve using the Weierstrass ℘-function is compatible with the group structure of the elliptic curve. A fourth interpretation of the group structure is based on a geometric observation about the orientated area of triangles with one vertex in the origin and two vertices on the hyperbola. Our parameterisation allows to define intervals whose images are regions F_m on the hyperbola such that any integer solution is uniquely given by applying a linear map to ( x, y ) ∈ F_m. We also expand the considerations for solvability of Cohn of binary quadratic forms obtained from quadratic number fields and give descriptions of the integer solutions to x^2 + x y - y^2 = m using the ideals in the number field Q( √ 5).
We determine explicit formulas for the Fourier coefficients of a class of eta quotients by making use of some results from the theory of ternary quadratic forms.
We establish sharp upper bounds on shifted moments of quadratic Dirichlet L-functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.