â"ñùú ,èáùá 'å ,éùéîç íåé

Thursday, 9th January 2003, 4:00 pm

Mathematics Building, Lecture Hall 2

Stephen Miller

(Rutgers)

Summation Formulae: From Poisson and Voronoi to Automorphic Forms

** Abstract: **
The Poisson summation formula gives an exact formula for the
sum of a function over the integers, namely as the sum of its Fourier
transform over the integers. In the early 1900's Voronoi conjectured the
existence of similar summation formulae for weighted sums of *a(n) f(n)*,
where *a(n)* is an arithmetic quantity. Voronoi himself worked out two
examples, when *a(n)* is the number of divisors of *n*, and when
*a(n)* is *r_2(n)*, the number of ways to represent *n* as
the sum of two squares. His
formulae were used to reduce the error term in Gauss' circle problem (the
number of lattice points in a circle of area *X*) from the trivial bound
*O(X^(1/2))*, to *O(X^(1/3))*.

Although Voronoi proved his formulae by applying Poisson summation, they are
nowadays thought of in terms of modular forms on the complex upper half
plane, and L-functions. Generalizations to modular forms have been
applied to a variety of problems in arithmetic and analysis. After
describing the history, I will discuss a generalization of the
Voronoi-type formulae to coefficients of automorphic forms on *GL(n)*,
and describe a recent application by Sarnak and Watson to bounding the
*L^4* norms of eigenfunctions.

(Joint work with Wilfried Schmid, Harvard University.)

Light refreshments will be served in the faculty lounge at 3:30.

List of talks, 2002-03

List of talks, 2001-02

List of talks, 2000-01

List of talks, 1998-99

List of talks, 1997-98