**CMI-LMS Research School: **

New Trends in Analytic Number Theory 2018

13 - 17 August 2018

University of Exeter

New Trends in Analytic Number Theory 2018

*This LMS-CMI Research School is run in partnership with the London Mathematical Society and the Clay Mathematics Institute.*

This research school will focus on three major advances in analytic number theory that have emerged in the last few years: prime number theory, number theory in function fields, and the Hardy and Littlewood Circle Method and Diophantine Geometry. Several recent results in all of these areas represent new trends in analytic number theory and the main aim of the lectures is to give a gentle introduction to these exciting news.

The Clay Mathematics Institute - London Mathematical Society Research Schools aim to introduce the brightest young mathematicians in the world to active and important areas of research through lectures by international leaders, as well as to help establish these participants as future leaders in their own research communities. This LMS-CMI Research School is run in partnership with the London Mathematical Society and the Clay Mathematics Institute and is supported by the Heilbronn Institute of Mathematics Research.

You can download a poster of the meeting here.

1. Professor Steve Gonek (University of Rochester)

Topic: Topics in Classical Analytic Number Theory

2. Professor Andrew Granville (UCL)

Topic: Pretentiousness in Analytic Number Theory

3. Professor Jon Keating (University of Bristol) and Professor Zeev Rudnick (Tel-Aviv University)

Topic: Number Theory in Function Fields

4. Professor Trevor Wooley (University of Bristol)

Topic: Hardy and Littlewood Circle Method and Vinogradov's Mean Value Theorem

The Clay Mathematics Institute - London Mathematical Society Research Schools aim to introduce the brightest young mathematicians in the world to active and important areas of research through lectures by international leaders, as well as to help establish these participants as future leaders in their own research communities. This LMS-CMI Research School is run in partnership with the London Mathematical Society and the Clay Mathematics Institute and is supported by the Heilbronn Institute of Mathematics Research.

You can download a poster of the meeting here.

__LECTURE COURSES__1. Professor Steve Gonek (University of Rochester)

Topic: Topics in Classical Analytic Number Theory

2. Professor Andrew Granville (UCL)

Topic: Pretentiousness in Analytic Number Theory

3. Professor Jon Keating (University of Bristol) and Professor Zeev Rudnick (Tel-Aviv University)

Topic: Number Theory in Function Fields

4. Professor Trevor Wooley (University of Bristol)

Topic: Hardy and Littlewood Circle Method and Vinogradov's Mean Value Theorem

__DISTINGUISHED LECTURES__- Chris Hughes (University of York)
- James Maynard (University of Oxford)
- Damaris Schindler (Utrecht University)
- Caroline Turnage-Butterbaugh (Duke University)

**REGISTRATION AND APPLICATIONS**

Apply online by 14th May 2018. Research students, post-docs, and those working in industry are invited do apply. There is a two step procedure:

1. Please click here for the application form for the CMI-LMS Research School.

2. We also request that applicants ask a referee to submit a reference letter using this form.

**FEES AND FUNDING**

These are the fees for this meeting.

Research students: £150 (no charge for subsistence costs)

Early career researchers: £250 (no charge for subsistence costs)

Other participants: £250 (plus subsistence costs)

The meeting will be held at the

**University of Exeter**, Exeter, UK. All fees are due by 2 July 2018.

Financial aid will only be available to those categorised as research students. Those who will not have completed their PhDs by the start of the Research School and who would otherwise be unable to attend can apply for financial aid.

Application forms for financial aid will be available after the main application deadline on 14 May 2018 but details on the information that will be gathered can be found here:

https://tinyurl.com/mve2fxb

**SCHEDULE**

*The schedule below is tentative and subject to change.*

Here is a pdf file with the schedule of the talks.

**TALKS**

**Speaker Title**

Chris Hughes The Story of a Theorem (or, How my Research Student got a World Record)

James Maynard Fractional parts of polynomials

Damaris Schindler Diophantine inequalities for ternary diagonal forms

Caroline Turnage-Butterbaugh An effective Chebotarev density theorem for families of fields, with applications to class groups

**LOCATION**

University of Exeter

Streatham Campus

Harrison Building

North Park Road

Exeter

EX4 4QF

All the lectures will take place at the Lecture Theatre H004 at the Harrison Building.

The venue for the meeting is the number 23 in the campus map.

**PARTICIPANTS**

Forename |
Surname |
Affiliation |

Julio |
Andrade |
University of Exeter |

LECTURE NOTES, READING LIST AND SLIDES

LECTURE NOTES, READING LIST AND SLIDES

**1) Pretentiousness in Analytic Number Theory**

Reading Material:

arXiv:1406.3754

What is the best approach to counting primes?

Andrew Granville

arXiv:1606.08021

The Liouville function in short intervals [after Matomaki and Radziwill]

Kannan Soundararajan

arXiv:1706.03755

A more intuitive proof of a sharp version of Halász's theorem

Andrew Granville, Adam J Harper, K. Soundararajan

**2) Topics in Classical Analytic Number Theory**

Lectures 1 and 2: Primes, the Riemann zeta-function, and zeros.

Background on the Riemann zeta-function and its connection with primes. The functional equation, zero free regions, the explicit formula, the prime number theorem, the Riemann hypothesis.

Lecture 3: Pair correlation of the zeros.

The Riemann hypothesis describes the horizontal distribution of zeros of the zeta-function in the critical strip. Montgomery's pair correlation conjecture tells us how the zeros are distributed vertically. We derive Montgomery’s theorem, explain the conjecture, and show various applications.

Lecture 4: Mean value theorems and their applications.

Mean value (or moment) estimates for the zeta-function and other Dirichlet series play a central role in analytic number theory. What do we know about them and what can we do with them?

Lecture 5: TBA

Reading Material:

1. H. Davenport, Multiplicative Number Theory, second edition, Springer-Verlag, New York, 1980. (Chapters 8-18)

2. H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory, Proceedings of Symposia in Pure Mathematics 24 (American Mathematical Society, Providence, RI, 1973), 181–193.

**3) Number Theory in Funtion Fields**

Abstract: We will give an introductory overview of the analogues in function fields of a range of problems in number theory. We hope to cover the following topics, depending on the background and interests of the students.

Lecture 1: Background on polynomials over a finite field, the Prime Polynomial Theorem etc.

(Reference: Number Theory in Function Fields by Michael Rosen)

Lecture 2 The hyperelliptic ensemble. A tale of two limits: the large degree limit and the large finite field limit and equidistribution.

Lecture 3. More arithmetic statistics: Moments of L-functions.

Lecture 4 Variance of arithmetic functions, equidistribution and random matrix theory.

Lecture 5. TBA

Reading Material:

- Number Theory in Function Fields by Michael Rosen
- Z. Rudnick, Some problems in analytic number thoery for polynomials over a finite field. Proceedings of the ICM vol 2 (2014), 443-460.

Problem Sheet 1

**4)**

**Efficient congruencing as p-adic decoupling: an introduction to Vinogradov’s mean value theorem**

Abstract: The estimates for moments of exponential sums provided by Vinogradov’s mean value theorem provide the best known approach to problems as diverse as the zero-free region for the Riemann zeta function, the asymptotic formula in Waring’s problem for the number of representations of a large integer as the sum of a fixed number of integral k-th powers, and quantitative equi-distribution of polynomials modulo 1. Vinogradov’s estimates from the 1930’s remained little improved into the 1990’s, and only modestly so until 2010. But very recent progress has delivered the most optimistic conjectural bounds. This course will provide an overview of these developments, a sketch of some applications, and a glimpse of what have emerged as tools offering strikingly more powerful control of mean values of exponential and character sums of utility throughout analytic number theory.

Reading Material:

- R. C. Vaughan, The Hardy-Littlewood method, 2nd edition, 1997, CUP (see especially Chapters 5 and 7).
- L. B. Pierce, The Vinogradov mean value theorem [after Wooley, and Bourgain, Demeter and Guth], arXiv:1707.00119.
- T. D. Wooley, Nested Efficient Congruencing and relatives of Vinogradov’s mean value theorem, arxiv:1708.01220

**ORGANIZERS**

Julio Andrade (University of Exeter)

Brian Conrey (American Institute of Mathematics and University of Bristol)

**ACCOMMODATION**

For those who are categorized under “Early Career Researcher” or “Other”, you must organize and pay your own accommodation. Below is a list of suggested hotels which are in Exeter along with a description of each respective hotel’s distance from the university.

1) Park View B&B, 8 Howell Road (within walking distance of the university)

2) Queens Court Hotel, Bystock Terrace (within walking distance of the university)

3) Jury's Inn, Western Way (15 minute bus ride from the university)

4) The Clock Tower, 16 New N Rd (within walking distance of the university)

5) Mercure Exeter South Gate Hotel, Southernhay E (15 minute bus ride from the university)

6) The Bendene, 15-16 Richmond Rd (within walking distance of the university)

**TRAVEL AND PRACTICAL INFORMATION**

Click here for a campus map.

The research school will start on the morning of

**13th August 2018**and we anticipate that most participants will arrive on the afternoon or evening of

**12th August 2018.**See the schedule above for further details.

Here is some general advice on to travel to Exeter via bus, rail, or Exeter Airport. Onward travel from Exeter Airport is available by bus or taxi. The bus is cheap (a few pounds), but doesn't run that that late.

The two largest airports in the UK are Heathrow and Gatwick, and partipants from North America will probably have to fly to one of these (they also have flights to many European cities). By far the cheapest onward travel option in both cases is to take the bus to Exeter Bus and Coach station. From Gatwick, you can travel by train via either Reading or Clapham Junction (cheaper, but a bit slower) to Exeter St David's train station. From Heathrow, you can take either the Heathrow Express or Heathrow connect (a bit slower but much cheaper) to London Paddington train station and then take the train from Paddington to Exeter St. David's. Rail prices and timetables (including for onward travel from airports other than Exeter) can be found at National Rail Enquiries.

The research school will be held in: Harrison Building, University of Exeter, Norh Park Road, Exeter EX4 4QF.

**TOURIST INFORMATION**

Our conference packets will include a pad of A4 paper, a pen, the conference schedule, a map of the Exeter city centre, etc.

However, we will not be including a travel brochure into the conference packets. But, you can use this website which gives good travel and tourist information.

**CONFERENCE PHOTOS**

To be add later.

**CONTACT**

If you have any questions related to this meeting please email: j.c.andrade.math@gmail.com

*This LMS-CMI Research School is supported by the Heilbronn Institute of Mathematics Research and the University of Exeter.*