## UNCG Summer School in Computational Number Theory 2015

### Zeta Functions -- New Theory and Computations

From May 18 to May 22, 2015, the University of North Carolina at
Greensboro is hosting a summer school entitled **Zeta Functions -- New Theory and Computations**.

The speakers will be

- Fredrik Johansson (INRIA Bordeaux-Sud-Ouest)
- Yuri Matiyasevich (Steklov Institute of Mathematics, St. Petersburg)
- Filip Saidak (UNCG)
- Cem Yıldırım (Bogaziçi University, Istanbul)
- Peter Zvengrowski (University of Calgary)

The summer school in computational number theory fills a gap in the education of many graduate students. Most graduate courses in number theory take a mainly theoretic approach with very little emphasis on the computational aspects of the subject. The goal of the UNCG Summer School in Computational Number Theory is to complement this with a constructive-algorithmic approach. Many of the algorithms used for number theoretic computations are non-trivial, which makes it difficult to cover them in a standard course.

The subjects we plan to cover this year include: A historic approach to the Riemann Zeta function; the distribution of the zeros of zeta functions and their and their derivatives; horizontal cuts of the critical strip; efficient evaluation of zeta functions and their derivatives; related functions; gaps between primes; and twin primes.

On a typical day, external and local experts give talks in the morning, and in the afternoon students solve problems related to this material. The talks early in the week introduce the students to the subject. Talks later in the week 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 are aimed at increasing the students’ understanding of the material by working with it.

#### Schedule

All talks will take place in Room 213 in the Petty Building (campus maps and directions).

Time | Sunday 5/17 | Monday 5/18 | Tuesday 5/19 | Wednesday 5/20 | Thursday 5/21 | Friday 5/22 |
---|---|---|---|---|---|---|

9:00 | Welcome | Coffee | ||||

9:30 | Participants: Introductions (slides) | Peter Zvengrowski: History of zeta II (slides, Riemann and his zeta function) | Cem Yıldırım: Small gaps between primes: The GPY method and recent advancements over it II (abstract) | Yuri Matiyasevich: Approximation of Riemann's zeta function by finite Dirichlet series II (slides, audio) | Fredrik Johansson: High-precision methods for zeta functions III (slides) | |

10:45 | Coffee | |||||

11:15 | Filip Saidak: History of zeta I |
Cem Yıldırım: Small gaps between primes: The GPY method and recent advancements over it I (abstract) | Peter Zvengrowski: Horizontal monotonicity of zeta (Convergence of Dirichlet Series and Euler Products) | Cem Yıldırım: Some analogues of pair correlation of zeta zeros (abstract) | Yuri Matiyasevich: Approximation of Riemann's zeta function by finite Dirichlet series III (slides, audio, Turing slides, YM link to more info) | |

12:30 | Lunch | |||||

14:00 | Fredrik Johansson: High-precision methods for zeta functions I (slides) | Yuri Matiyasevich: Approximation of Riemann's zeta function by finite Dirichlet series I (slides, audio, animation ) | Excursion (Guilford Courthouse National Military Park) | Fredrik Johansson: High-precision methods for zeta functions II (slides, arb, FJ sage ex3) | Filip Saidak & Ricky Farr: Distribution of zeros of derivatives | |

15:15 | Coffee | Coffee | ||||

15:30 | Problem session (Lerch, sage ex, exercises 1 ) | Problem session (Valianatos conjecture: plot, YM exercises, YM notebook, additional exercises 2) | Problem session (FJ sage ex3) | |||

Old Town Draught House | Hillbilly Hideaway |

#### Some Resources

Some Sage can help with experimenting with zeta functions. The LMFDB contains a database of L-functions. We also have an example Arb program. Yuri Matiyasevich has a page with his talks, audio recordings, and his animation in various formats.

#### Local information

Information for visitors, including directions to UNCG and Petty Building. The PANTS 13 pages contain a list of local restaurants. Additional information (PDF)

#### Participants

- Arnab Bose (University of Lethbridge)
- Stevo Bozinovski (SC State University)
- Sneha Chaubey (University of Illinois at Urbana–Champaign)
- Kenneth Chilcoat (East Carolina University)
- Zhenchao Ge (University of Mississippi)
- Luke Giberson (Clemson University)
- Kim Klinger-Logan (University of Minnesota)
- Patrick Kühn (Universität Zürich)
- Huixi Li (Clemson University)
- Junxian Li (University of Illinois at Urbana–Champaign)
- Jingbo Liu (Wesleyan University)
- Hans Parshall (University of Georgia)
- Jonathan Sands (University of Vermont)
- Tien Trinh (Rutgers University)
- Kam-hung Yau (University of Auckland)

##### Locals

- Ricky Farr (UNCG)
- Lance Everhart (UNCG)
- Jonathan Milstead (UNCG)
- Brian Sinclair (UNCG)
- James Rudzinski (UNCG)
- Stephanie Webster (UNCG)

#### Organizers

- Brett Tangedal (UNCG)
- Dan Yasaki (UNCG)
- Filip Saidak (UNCG)
- Sebastian Pauli (UNCG)

#### Acknowledgements

The summer school in computational number theory is supported by UNCG and the NSF (DMS-1303565).