Analytic Number Theory (1MA038), fall 2023

Syllabus

Schedule

Literature:

    Baker: "A Comprehensive Course in Number Theory"
    Stein and Shakarchi: "Complex analysis"
    Lecture notes, based on Davenport: "Multiplicative Number Theory", Springer Verlag.

Teacher: Andreas Strömbergsson, email: astrombe@math.uu.se

Examination: There will be three compulsory assignments during the course, and a final written exam at the end of the course. Your total grade on the course will mainly be decided from your total score on the home assignments, but you also need to pass the final written exam. Each home assignment can give 50 points, and the maximal score on the final exam is 40 points; thus the maximal total score is 150+40=190 points. As a guideline for the final grade, a total score above 150 will typically result in the grade 5; a total score between 115 and 149 in the grade 4, a total score between 80 and 114 in the grade 3, and a total score below 80 will result in fail. But note that you also need to pass the final exam (i.e., get at least 18 points on it) in order to pass the course.

Regarding the home assignments: You are free to cooperate with other students and to read whatever literature you can find about the subject. You are also welcome to ask me (the teacher) for further hints and suggestions on how to attack the problems. However, you are expected to formulate your solutions independently and it is neither allowed to copy from other students nor to copy solutions from any other source!

    Assignment 1 (due September 29, before midnight; please hand in as a pdf file by email).
    - everyone who submitted solutions to Assignment 1 should now have received an email with my grading (20231011).
    Assignment 2 (due November 17, before midnight; please hand in as a pdf file by email).
    - everyone who submitted solutions to Assignment 2 should now have received an email with my grading (20231204).
    Assignment 3 (due December 22, before midnight; please hand in as a pdf file by email).
    - everyone who submitted solutions to Assignment 3 should now have received an email with my grading (20231227).

Written exam: Fri 5/1-2024, 8.00-13.00.

List of possible exam questions: here. Some old exams: Exam Jan 2022; Exam June 2022.

Preliminary plan of lectures:

Date Time Place Topic References
1. Mon, 28/8 8-10 80115 Introduction
2. Wed, 30/8 8-10 11167 Primes in arithmetic progressions LN Sec 1, Baker 15.3-5
3. Mon, 4/9 8-10 4003 Infinite products LN Sec 2, SS Ch 5.3
4. Thu, 7/9 8-10 2003 Summation by parts; Dirichlet series LN Sec 3, Baker 13.4
5. Mon, 11/9 8-10 2003 Examples/problem solving: Problems 2.1, 2.2, 2.7. 2.8, 3.4, 3.5, 3.13 in LN solution sketches
6. Wed, 13/9 8-10 4003 Dirichlet characters (Fourier analysis on finite abelian groups) LN Sec 4.1-6, Baker 15.3
7. Mon, 18/9 8-10 11167 The distribution of the primes LN Sec 6, Baker 13.1-6
8. Wed, 20/9 8-10 12167 The prime number theorem LN Sec 7; Baker Ch 14 and 15.1; SS Ch 7
9. Thu, 28/9 8-10 11167 The prime number theorem LN Sec 7; Baker Ch 14 and 15.1; SS Ch 7
10. Thu, 5/10 10-12 11167 The prime number theorem [Half of the lecture: Prof. Dennis Hejhal will present a curiously simple proof of the PNT.] LN Sec 7; Baker Ch 14 and 15.1; SS Ch 7
11. Thu, 12/10 8-10 2003 Examples/problem solving: Problems 3.13, 4.1, 4.3, 6.1, 6.2, 6.4, 6.6, 7.1, 7.2, 7.4, 7.7 in LN.
- I will probably start by discussing problem 7.4, with particular focus on part (c), which I will discuss
together with LN Theorem 7.10. 		solution sketches
12. Thu, 19/10 10-12 12167 The Gamma function LN Sec 8; SS Ch 6.1
13. Thu, 26/10 8-10 12167 The functional equation LN Sec 9; SS Ch 6.2, Baker 14.2
14. Tue, 31/10 8-10 12167 The explicit formula for psi(x) LN Sec 13
15. Mon, 6/11 10-12 4005 Zero-free region; PNT with error term LN Sec 11, Baker 14.6, 15.2
16. Mon, 13/11 8-10 12167 Examples/problem solving: Problems 8.1, 8.3, 8.4, 8.5, 8.6, 9.1, 9.2(a), 9.5, 13.4, 15.1, 16.3 in LN. solution sketches
17. Wed 15/11 8-10 11167 Binary quadratic forms LN Sec 5, Baker Ch 5
18. Mon 20/11 8-10 12167 Dirichlet's class number formula LN Sec 5 (Baker 15.6)
19. Mon 27/11 8-10 11167 The Jacobi Theta function SS Ch 10
20. Wed 29/11 8-10 12167 Sums of squares SS Ch 10 (Baker 5.4-5)
21. Tue 5/12 8-10 12167 Examples/problem solving: Problems 5.4, 5.5, 5.6, 8.9, 9.1(b),(c) in LN. solution sketches
22. Mon 11/12 8-10 12167 The large sieve LN Sec 20,21 (Baker Ch 16)
23. Thu 14/12 8-10 12167 The large sieve LN Sec 20,21 (Baker Ch 16)
24. Tue 19/12 8-10 12167 The large sieve (and the Bombieri-Vinogradov theorem) LN Sec 20,21 (Baker Ch 16)
25. Thu 21/12 10-12 11167 The large sieve; the Bombieri-Vinogradov theorem LN Sec 20,21 (Baker Ch 16)