We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd’s conjectured identities between Mahler measures and special values of L-series of elliptic curves. 1

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. We also present some new results concerning the moments of uniform random walks, in particular their derivatives. 1

When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan’s meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan’s mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan’s discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer.

My research sits at the intersection of number theory and computational math. I am particularly interested in problems relating to Mahler measure and hypergeometric functions. Such problems are closely connected to the theory of elliptic curves, and modular forms. I also have a strong interest in the mathematics of Ramanujan. 1.

On the recurrence of coefficients in the

In 1914 S. Ramanujan recorded a list of 17 series for 1/π. We survey the methods of proofs of Ramanujan’s formulae and indicate recently discovered generalizations, some of which are not yet proven. The twentieth century was full of mathematical discoveries. Here we expose two significant contributions from that time, in reverse chronological order. At first glance, the stories might be thought of a different nature. But we will try to convince the reader that they have much in common. 1 Ramanujan and Apéry: 1/π and ζ(3) In 1978 R. Apéry showed the irrationality of ζ(3) (see [5] and [21]). His rational approximations to the number in question (known nowadays as Apéry’s constant) have the form vn/un ∈ Q for n = 0, 1, 2,..., where the denominators {un} = {un}n=0,1,... and numerators {vn} = {vn}n=0,1,... satisfy the same polynomial recurrence (n + 1) 3 un+1 − (2n + 1)(17n 2 + 17n + 5)un + n 3 un−1 = 0 (1.1) with the initial data u0 = 1, u1 = 5, v0 = 0, v1 = 6.

In this note, we survey results concerning variations of the Lück-Fuglede-Kadison determinant with respect to the base group. Further, we discuss recurrences of coefficients in the determinant for certain distinguished base groups. The note is based on a talk that the second author gave at the “Segundas Jornadas de Teoría de Números”,