Jensen polynomials for the Riemann zeta function and other sequences.

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function [Formula: see text] at its point of symmetry. This hyperbolicity has been proved for degrees [Formula: see text] We obtain an asymptotic formula for the central derivatives [Formula: see text] that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all [Formula: see text] These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.

From partitions to Hodge numbers of Hilbert schemes of surfaces.

We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, marking the birth of the 'circle method', we present a contemporary example of its legacy in topology. We deduce the equidistribution of Hodge numbers for Hilbert schemes of suitable smooth projective surfaces. This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'.

Ramanujan's mock theta functions.

In his famous deathbed letter, Ramanujan introduced the notion of a mock theta function, and he offered some alleged examples. Recent work by Zwegers [Zwegers S (2001) Contemp Math 291:268-277 and Zwegers S (2002) PhD thesis (Univ of Utrecht, Utrecht, The Netherlands)] has elucidated the theory encompassing these examples. They are holomorphic parts of special harmonic weak Maass forms. Despite this understanding, little attention has been given to Ramanujan's original definition. Here, we prove that Ramanujan's examples do indeed satisfy his original definition.

On the modularity of certain functions from the Gromov-Witten theory of elliptic orbifolds.

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulae for special indefinite theta functions of type (1,2) in terms of products of mock modular forms. This formula is also of independent interest.