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:

• - Multiplicative Number Theory, Third Edition by H. Davenport (Springer 2000)
• - Multiplicative Number Theory I. Classical Theory by H.L. Montgomery and R.C. Vaughan (Cambridge 2006)
• - Course notes "245a Analytic Prime Number Theory" by T. Tao. See T. Tao's Blog.

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.