# Department of Mathematics and Statistics

## UNCG Summer School in Computational Number Theory 2012

### A Computational Approach to $L$-functions

From May 14 to May 18, 2012, the University of North Carolina at Greensboro hosted a summer school entitled A Computational Approach to $L$-functions. The external speaker was David Farmer of the American Institute of Mathematics.

On a typical day, external and local experts will give talks in the morning, and in the afternoon students will solve problems related to this material. The talks early in the week will introduce the students to the subject. Talks later in the week will cover related areas of current research and unsolved problems. The problems given to the students might be of a theoretical nature but could also involve programming problems and computer experiments. All problems will be aimed at increasing the students’ understanding of the material by working with it.

The subjects we covered included an historic approach to the Riemann zeta function, the derivatives of zeta and their zeros, evaluating zeta and its derivatives, related functions, general L-functions, random matrices, and applications of special values.

#### Schedule

TimeMondayTuesday WednesdayThursdayFriday
9:00WelcomeCoffeeCoffeeCoffeeCoffee
9:30Filip Saidak: The Zeta Function I: before RiemannFilip Saidak: The Zeta Function II: RiemannFilip Saidak: The Riemann zeta Function III: after RiemannFilip Saidak: MonotonicityFilip Saidak: Open Questions
10:30Coffee
10:45Rick Farr: Euler Maclaurin SummationSebastian Pauli: Evaluating the Riemann Zeta Function Sebastian Pauli: Zeros of Derivatives of the Riemann Zeta FunctionBrett Tangedal: Special ValuesDavid Farmer: L-functions and Random Matrix Theory (.nb)
11:45Coffee
12:00David Farmer: L-functions from an analytic perspective.David Farmer: L-functions in general and how to compute them
(L-function database)
David Farmer: Zeros of L-functions and zeros of polynomialsDavid Farmer: The explicit formula (.nb)Problem Session
13:00LunchLunchLunchLunchLunch
14:30Sebastian Pauli: Intro to ComputingProblem SessionExcursionProblem Session
15:00Problem Session Problem Session Problem Session
19:00Grashoppers

#### Organizers

The summer school was organized by Sebastian Pauli, Filip Saidak, Brett Tangedal, and Dan Yasaki.

#### Acknowledgements

This workshop is supported by the UNCG $P^3$ grant, the Helen Barton Fund from the UNCG Excellence Foundation Inc, and by the Number Theory Foundation.