Analytic number theory

Lectures: Tuesday 10:15-12:00 at M4, Thursday 12:15-14:00 at M3

Exercise: Friday 12:15-14:00 at M1 (starting March 17)

Exam 2: Friday 18.5. at 14:00-17:00 at M2 (Quantum).   Here is a list of formulas that you will get for the exam.

Exam 1. 

We will mainly follow the online notes of Andrew Granville (here ) and the book of Dimitris Koukoulopoulos (whose free preliminary version is available here).

A traditional textbook is Davenport's Multiplicative Number Theory.

We will cover the following topics:

1. Arithmetic functions and elementary estimates

2. Review of some complex analysis

3. The Riemann zeta function and the prime number theorem

4. Dirichlet characters and Dirichlet's theorem on primes in arithmetic progressions.

There might be room for some additional topics (let me know if you have any preferences).

Here is a list of possible extra topics.

If you have any questions about the course content, write me an e-mail or ask here: Questions and answers.

Here, you can let me know how you like the course or if you would prefer any changes:  Feedback form.

Covered material:

You can find my notes from class here. It tells you what material was covered in class and contains references for proofs (if there is no reference, it usually means that you can find it in any of your favorite analytic number theory textbook)

1. Introduction, Euler's proof of the infinitude of primes and consequences, asymptotic notation, partial summation.

2. Applications of partial summation, equivalent forms of the prime number theorem. Arithmetic functions, Dirichlet convolution.

3. Using Dirichlet's convolution identities to estimate sums of arihtmetic functions, Dirichlet's hyperbola method, Chebyshev's estimates, Mertens' estimates.

4. The sieve of Eratosthenes, Cramer's model, Dirichlet series.

5. Review of complex analysis, Perron's formula, estimating sums using Perron's formula and the residue theorem, the explicit formula for the number of primes.

   • A good source for complex analysis is the book by Stein and Shakarchi: press.princeton.edu/books/hardcover/9780691113852/complex-analysis 

   • You can also check these online notes: www.maths.ed.ac.uk/~jmf/Teaching/MT3/ComplexAnalysis.pdf 

   • Borcherd's complex analysis playlist on youtube: https://www.youtube.com/playlist?list=PL8yHsr3EFj537_iYA5QrvwhvMlpkJ1yGN

6. Truncated Perron's formula, the explicit formula for the number of primes, relation between the distribution of primes and zeros of the Riemann zeta function, plan for proving the prime number theorem, non-vanishing of zeta for re(s)=1.

7. Fourier analysis and Poisson summation, the theta function and its functional equation, Gamma function, the functional equation of zeta, the completed zeta function.

   • A video of 3Blue1Brown about the analytic continuation of zeta: www.youtube.com/watch?v=sD0NjbwqlYw 

8. Order of a function and Hadamard product, approximation for zeta'(s)/zeta(s) by a sum over zeros, zero-free region.

9. A better approximation of zeta'/zeta(s) by a sum over nearby zeros and its consequences, proof of the Prime Number Theorem.

   • Extra reading: What is the best approach to counting primes? (By Andrew Granville): arxiv.org/pdf/1406.3754.pdf 

10. Error term in the prime number theorem and the Riemann hypothesis. Introduction to primes in arithmetic progressions and Dirichlet's theorem, proof for q=3, defintion of Dirichlet characters.

    • Extra reading: prime number races (by Andrew Granville and Greg Martin): arxiv.org/pdf/math/0408319.pdf

11. Dirichlet characters, orthogonality relations, plan for the proof of Dirichlet's theorem, Dirichlet L-functions.

12. Nonvanishing of L(1, 𝜒) for complex characters, mimicking the proof of zeta that L(1+it, 𝜒) ≠ 0, non-vanishing of L(1, 𝜒) for real characters, finishing the proof. Discussion of Dirichlet's class number formula, prime number theorem in arithmetic progressions, Siegel-Walfisz theorem.

    • Extra reading: Analytic number theory (by Andrew Granville)

13. Siegel-Walfisz theorem, Exceptional zeros and size of L(1, 𝜒), Siegel's theorem, Landau's theorem, Linnik's theorem, Bombieri-Vinogradov theorem, consequences of the generalized Riemann hypothesis. Introduction to modular forms: Ramanujan's tau function and Ramanujan's conjectures, the j-function.

14. Jacobi's fromula for the number of representations as sum of 4 squares, Elliptic curves and their L-functions, Fermat's last theorem, definition of modular forms, functional equation for zeta, Dirichlet L-functions and L-functions of modular forms.

Exercises:

1. Exercise set 1.   Solutions 1.

2. Exercise set 2.   Solutions 2.   

3. Exercise set 3.   Solutions 3. 

4. Exercise set 4. There is no exercise class on April 7, you can send me your solutions by e-mail until Tuesday 11.4. For the exercises with many parts, you can get points if you solve most of them (though each part should follow from the previous one).    Solutions 4.

5. Exercise set 5.   Solutions 5.

6. Exercise set 6.   Solutions 6.

This is the last exercise set. If you need more points, you can send me the solutions to any exercise from Koukoulopoulos' book (we roughly covered chapters 1-9)