Abstracts for Number theory lunch seminar

Research done by information theorists over the past decade has
shown that lattices constructed from rings of integers in number fields,
and orders in division algebras, provide codebooks well-suited for
communication over wireless channels.  In particular, recent work has shown
that when communication over a wireless channel occurs in the presence of
an eavesdropper, one can estimate the eavesdropper's probability of
correctly intercepting the message, using the regulator and Dedekind zeta

We revisit a classical problem: how far does a random walk travel in a given
number of steps (of length 1, each taken along a uniformly random direction)?
Although such random walks are asymptotically well understood, surprisingly
little is known about the exact distribution of the distance after just a few
steps.  For instance, the average distance after two steps is (trivially)
given by 4/pi; but what is the average distance after three steps?

In this talk, we therefore focus on the arithmetic properties of short random

In my talk, I will discuss the differentiability of Fourier Series of
the form

F_k(\tau)=\sum_{n=1}^{\infty}\sigma_{k-1}(n) n^{-k-1}e^{2\pi i n \tau} for k even.

These series are related to Eisenstein Series. Using modular (and
quasi-modular) properties of Eisenstein Series, we can find functional
equations for F_k, from which we can draw some conclusions on
differentiability of F_k. This approach was introduced by Itatsu in
1981 in a paper on Differentiability of Riemann's Function.

Using intensive computer calculations, the author empirically discovered
unusual methods for calculating high-precision approximations to the non-trivial zeroes
of Riemann's zeta function, its values and values of its derivative on the whole
complex plane. So far no theoretical explanation to these phenomena is known.