We discuss some recently discovered relations between L-values of modular forms and integrals involving the complete elliptic integral K. Gentle and illustrative examples will be given. Such relations also lead to closed forms of previously intractable integrals and (chemical) lattice sums.

The double zeta values are one natural way to generalise the Riemann zeta function at the positive integers; they are defined by $\zeta(a,b) = \sum_{n=1}^\infty \sum_{m=1}^{n-1} 1/n^a/m^b$. We give a unified and completely elementary method to prove several sum formulae for the double zeta values. We also discuss an experimental method for discovering such formulae.

Moreover, we use a reflection formula and recursions involving the Riemann zeta function to obtain new relations of closely related functions, such as the Witten zeta function, alternating double zeta values, and more generally, character sums.

We resolve some recent and fascinating conjectural formulae for $1/\pi$ involving the Legendre polynomials. Our mains tools are hypergeometric series and modular forms, though no prior knowledge of modular forms is required for this talk. Using these we are able to prove some general results regarding generating functions of Legendre polynomials and draw some unexpected number theoretic connections. This is joint work with Heng Huat Chan and Wadim Zudilin. The authors dedicate this paper to Jon Borwein's 60th birthday.

The complete elliptic integrals of the first and second kinds (K(x) and E(x)) will be introduced and their key properties revised. Then, new and perhaps interesting results concerning moments and other integrals of K(x) and E(x) will be derived using elementary means. Diverse connections will be made, for instance with random walks and some experimental number theory.