# Meeting Details

Title: "Asymptotic formulae in analytic number theory" Ph.D. Thesis Defense Ayla Gafni - Adviser: R. Vaughan, Penn State http:// This dissertation is composed of two main results. The first is an asymptotic formula for $p^k(n)$, the number of partitions of a number $n$ into $k$-th powers. As an immediate consequence of this formula, we derive an asymptotic equivalence for $p^k(n)$ which was claimed without proof in a 1918 paper of Hardy and Ramanujan. The result is established using the Hardy-Littlewood circle method. As a necessary step in the proof, we obtain a non-trivial bound on exponential sums of the form $\sum_{m=1}^q e(am^k/q)$. The second result is an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth function defined on the interval $[\eta,\xi]$, then the number of rational points with denominator no larger than $Q$ that lie within a $\delta$-neighborhood of the graph of $f$ is shown to be asymptotically equivalent to $(\xi-\eta)\delta Q^2$. This result has implications to the field of metric Diophantine approximation.