Zariski’s conjecture and Euler–Chow series

Adelic Singular Series and the Goldbach Conjecture

The purpose of this paper is to show how adelic ideas might be used to make progress on the Goldbach Conjecture. In particular, we present a new Schwartz function which is able to keep track of the number of prime factors of an integer. We then use this, along with the Ono/Igusa adelic methods for Diophantine equations, to present an infinite sum whose evaluation would prove or disprove the ver...

Volume Conjecture and Asymptotic Expansion of q-Series

Theta Functions, Elliptic Hypergeometric Series, and Kawanaka’s Macdonald Polynomial Conjecture

We give a new theta-function identity, a special case of which is utilised to prove Kawanaka’s Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.

On the Littlewood conjecture in fields of power series

Let k be an arbitrary field. For any fixed badly approximable power series Θ in k((X−1)), we give an explicit construction of continuum many badly approximable power series Φ for which the pair (Θ,Φ) satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.

The Merino-Welsh conjecture holds for series-parallel graphs

The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.