# Research: Number Theory

## Summer School 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

Time Monday Tuesday Wednesday Thursday Friday
9:00 Welcome Coffee Coffee Coffee Coffee
9:30 Filip Saidak: The Zeta Function I: before Riemann Filip Saidak: The Zeta Function II: Riemann Filip Saidak: The Riemann zeta Function III: after Riemann Filip Saidak: Monotonicity Filip Saidak: Open Questions
10:30 Coffee
10:45 Rick Farr: Euler Maclaurin Summation Sebastian Pauli: Evaluating the Riemann Zeta Function Sebastian Pauli: Zeros of Derivatives of the Riemann Zeta Function Brett Tangedal: Special Values David Farmer: L-functions and Random Matrix Theory
11:45 Coffee
12:00 David 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 polynomials David Farmer: The explicit formula (.nb) Problem Session
13:00 Lunch Lunch Lunch Lunch Lunch
14:30 Sebastian Pauli: Intro to Computing Problem Session Excursion Problem Session
15:00 Problem Session Problem Session Problem Session
19:00 Grashoppers

#### 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.