Visibility in N^k by Santiago Estupinan Salamanca, Universidad de los Andes, Colombia, Class of 1960s Speaker, Friday, April 19, Mathematics Faculty Seminar, 1 – 1:45 pm, Thompson Chemistry 206
Abstract: Imagine that you were placed at the origin of the plane amidst a forest of trees, such that the coordinates that describe the position of each tree are integers. A celebrated result by Cesàro and Sylvester in Analytic Number Theory states that the proportion of trees that would be visible from your perspective, i.e. not obstructed by another tree, is asymptotically equal to 1/z(2) where z is the famous Riemann zeta function.
This stunning connection was further developed by Goins, Harris, Kubick, and Mbirika, by defining what it means for a tree to be visible according to power functions f(x)=ax^b for b in (N) and by Harris and Omar allowing b in (Q). In this talk I will present the results of joint work with Benedetti and Harris, that generalizes these notions of visibility to N^k, while recovering the preceding results.