# Introduction to Analytic Number Theory Autumn 2018

Lecturer
Ian Nicholas Petrow
Coordinator
Seraina Wachter
Classes
Wed 8:00-10:00 (HG E 22) and Th 15:00-17:00 (HG D 1.1)
This course is an introduction to the classical analytic theory of prime numbers. Topics include: Elementary methods for counting prime numbers (Eratosthenes, Euler and Chebychev), arithmetic functions, approximation by integrals, integration by parts, Dirichlet convolution, multiplicative functions, Dirichlet series, L-series, primes in arithmetic progressions, characters of a finite abelian group, Dirichlet characters, non-vanishing of L-functions at s=1, the Riemann zeta function, Riemann's memoir, Mellin transform, functional equation of zeta, functions of bounded order, Hadamard factorization, estimation of zeros of zeta, explicit formulas, the prime number theorem, the prime number theorem in arithmetic progressions (time permitting).

## Prerequisites

Linear algebra, modern algebra, complex analysis. Some familiarity with the Fourier series and the Fourier transform.

## Schedule

New: There will be an office hour by the assistant every week on Thursday from 13h00 to 14h00 in room HG J 15.1 (or HG J 14.1).

Date Format Comment
Wednesday 19.09.2018 Lecture Introduction; prime numbers and Euler's method
Thursday 20.09.2018 Lecture Euler's method and Chebyshev's method
Wednesday 26.09.2018 Exercise Exercises. Solutions.
Thursday 27.09.2018 Lecture Approximation by an integral, integration by parts, intro to Dirichlet convolution
Wednesday 3.10.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 The Möbius inversion formula, application to prime numbers. Exercises. Solutions.
Thursday 4.10.2018 Lecture Multiplicative functions, introduction to Dirichlet series
Wednesday 10.10.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Abscissa of convergence and convolutions. Exercises. Solutions.
Thursday 11.10.2018 Lecture Dirichlet Series and multiplicative functions
Wednesday 17.10.2018 Lecture Primes in arithmetic progressions, characters of a finite abelian group
Thursday 18.10.2018 Lecture 15h15 to 16h00; exercise 16h15 to 17h00 Characters of a finite abelian group. Exercises. Solutions.
Wednesday 24.10.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Dirichlet L-functions and their analytic continuation. Exercises. Solutions.
Thursday 25.10.2018 Lecture Proof of Mertens theorem in arithmetic progressions, non-vanishing of L(1,χ).
Wednesday 31.10.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Finish L(1,χ) non-zero. Exercises. Solutions.
Thursday 01.11.2018 Lecture Riemann's memoir, Fourier transform.
Wednesday 07.11.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Fourier transform, Mellin transform. Exercises. Solutions.
Thursday 08.11.2018 Lecture Mellin transform continued. Functional equation of the zeta function.
Wednesday 14.11.2018 Exercise Exercises. Solutions.
Thursday 15.11.2018 Lecture Primitive characters, Gauss sums, begin functional equation of L(s,χ)
Wednesday 21.11.2018 Exercise Exercises. Solutions.
Thursday 22.11.2018 Lecture Finish functional equation of L(s,χ), functions of bounded order.
Wednesday 28.11.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Jensen's formula. Exercises. Solutions.
Thursday 29.11.2018 Lecture Hadamard factorization theorem.
Wednesday 05.12.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Counting zeros of the zeta function Exercises. Solutions.
Thursday 06.12.2018 Lecture Weil's explicit formula
Wednesday 12.12.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Zero counting for L(s,χ). The pnt assuming a zero-free region. Exercises. Solutions.
Thursday 13.12.2018 Lecture (by Prof. Nelson) Zero free regions for the zeta function and L(s,χ)
Wednesday 19.12.2018 Lecture 8h15 to 9h00; exercise 9h15 to 10h00 Warm up to Siegel's theorem. Exercises. Solutions.
Thursday 20.12.2018 Lecture Siegel's theorem and the prime number theorem in arithmetic progressions

The new exercises will be posted here on Fridays. We expect you to look at the problems and to prepare questions for the exercise class on the following week. You have the option to submit written solutions to the exercises on Thursday in the lecture to get comments on your work. These will be returned the following Thursday lecture. Solutions and a new exercise sheet will be posted the following Friday.

The main resource for the class is the lecture notes. These are based in large part on a course of Prof. Dr. Michel given at EPFL in 2014. The following books are also highly recommended:
Here is a list of the major changes to the course notes since Dec 20, 2018:
• - The proof of Theorem 9.3 (the quantitative zero-free region for Dirichlet L-functions) was revised.
• - Fixed a typo in the definition of the Gauss sum in section 4.2.
• - p. 17: in Ex. 2.13.3 Λ(n) changed to Λ(d).
• - End of p. 27: “has be modified” changed to “… has been modified”.
• - p. 28: "s with Re(s) > σ” changed to "s with Re(s) ≥ σ”.
• - p. 36.The index of the first sum on the page changed to “1 < n ≤ X” instead of “1 ≤ n ≤ X”
• - p. 44: e(nz) changed to e(nx)
• - p. 47: (g exp(σ x))^ (t/2π) changed to (g exp(σ x))^ (-t/2π) in two places.
• - p. 48: In the 8th line of the Proof of Thm. 6.8 the word “inequality” was changed to “equality”.
• - p. 64: In the sum after “Putting these facts together, we find …” two instances of “Re” were deleted.
• - p.69: In the proof of Thm. 9.2 ζ has a simple pole at s=1, not s-1.
A written exam of three hours will take place during the exam periods in Winter and Summer 2019. A formula sheet was provided with the examination. There will be no student-produced written aids.