Special Year 2009-10: Analytic Number Theory - Seminar

I first present an algorithm to compute the truncated theta function in poly-log time. The algorithm is elementary and suited for computer implementation. The algorithm is a consequence of the periodicity of the complex exponential, and the self...

Let K/Q be an extension of number fields. The Hasse norm theorem states that when K is cyclic any non-zero element of Q can be represented as a norm from K globally if and only if it can be represented everywhere locally. In this talk I will discuss...

We discuss the question of quantitative bounds on the sup-norm of automorphic cusp forms. We present an improvement on a recent result by Blomer-Holowinsky on Hecke-Maass forms on $X_0(N)$ with large level $N$. Analogous results are then established...

In this joint work with Stephan Baier, we prove a subconvexity bound for Godement-Jacquet L-functions associated with Maass forms for SL(3,Z). The bound arrives from extending a method of M. Jutila (with new ingredients and innovations) on...

A result of Kim-Sarnak (2003) gives the best known bounds towards the Ramanujan conjecture for Maass forms. The technique employed has not, until now, been made to apply to general GL2 cusp forms over number fields whose unit group is infinite. In...

A bounded Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four...

For the last 5 years or so Terry Tao and I have been working on a programme to prove certain conjectures of Hardy and Littlewood concerning the number of primes vectors p = (p_1, . . . ,p_n) in some box which satisfy the equation Ap = b . The number...