# Analytic Number Theory (1MA038), fall 2021

Syllabus

Literature:

Baker: "A Comprehensive Course in Number Theory" (online resource; see here for a link).
Stein and Shakarchi: "Complex analysis" (online resource; see here for a link).
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 total score on the final exam is 40 points; thus the 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 above 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 October 4, before midnight; please hand in as a pdf file by email).
- everyone who handed in solutions to Assignment 1 should now have received an email with my grading (20211011).
Assignment 2
- everyone who handed in solutions to Assignment 2 should now have received an email with my grading (20211128).
- everyone who handed in solutions to Assignment 3 should now have received an email with my grading (20211223).
Some old exams (as examples): Exam 2016; Exam 2018.

Preliminary plan of lectures:

ZOOM alternative: The lectures should - hopefully - also be possible to follow, to some extent, on https://uu-se.zoom.us/j/65898322750.

Date Time Place Topic References
1. Wed, 1/9 8-10 12167 Introduction
2. Thu, 2/9 8-10 12167 Primes in arithmetic progressions LN Sec 1, Baker 15.3-5
3. Mon, 6/9 8-10 4003 Infinite products LN Sec 2, SS Ch 5.3
4. Thu, 9/9 8-10 4004 Summation by parts; Dirichlet series LN Sec 3, Baker 13.4
5. Mon, 13/9 8-10 11167 Theory: Finish lecture 4.
Examples/problem solving: Problems 2.1, 2.2, 2.7. 2.8 in LN
(9 Sept: I added problems 2.7 and 2.8 in LN; please click 'reload' to get the updated version of LN)
6. Wed, 15/9 8-10 12167 Examples/problem solving: Problems 2.1, 2.2, 3.4, 3.5 in LN
- Since I didn't get time to discuss all these problems, here are solution suggestions for 2.2, 3.4, 3.5.
Theory: Dirichlet characters; Fourier analysis on finite abelian groups
LN Sec 4, Baker 15.3
7. Mon, 20/9 8-10 12167 Theory: Finish on Dirichlet characters.
Some more examples
8. Thu, 23/9 8-10 12167 The distribution of the primes LN Sec 6, Baker 13.1-6
9. Mon, 27/9 8-10 12167 The prime number theorem LN Sec 7; Baker Ch 14 and 15.1; SS Ch 7
10. Thu, 30/9 8-10 12167 The prime number theorem LN Sec 7; Baker Ch 14 and 15.1; SS Ch 7
11. Mon, 4/10 8-10 12167 The prime number theorem LN Sec 7; Baker Ch 14 and 15.1; SS Ch 7
12. Thu, 7/10 8-10 4004 The Gamma function LN Sec 8; SS Ch 6.1
13. Mon, 11/10 10-12 4004 The functional equation LN Sec 9; SS Ch 6.2, Baker 14.2
14. Thu, 14/10 8-10 12167 Examples/problem solving: Problems 8.1,8.2,8.3,8.4, 9.1(a) and 9.4(d) in LN (perhaps also 9.4(a)-(c))
(13 Oct: I corrected slightly in Problems 9.4(b),(c); please click 'reload' to get the updated version of LN)
15. Mon, 18/10 10-12 12167 The explicit formula for psi(x) LN Sec 13
16. Thu, 21/10 8-10 12167 Zero-free region; PNT with error term LN Sec 11, Baker 14.6, 15.2
17. Wed, 27/10 8-10 12167 Binary quadratic forms LN Sec 5, Baker Ch 5
18. Mon, 1/11 8-10 12167 Dirichlet's class number formula LN Sec 5 (Baker 15.6)
19. Wed, 10/11 8-10 12167 Dirichlet's class number formula (cont'd)
20. Wed, 17/11 8-10 12167 The Jacobi Theta function SS Ch 10 (my lecture notes)
21. Mon, 22/11 8-10 12167 Examples/problem solving: The problem on p.5 here, and problem 5.4 in LN.
Probably I will also have time to start on the next lecture (sums of squares).
22. Fri, 26/11 8-10 12167 Sums of squares SS Ch 10 (Baker 5.4-5)
23. Mon, 6/12 8-10 12167 The large sieve LN Sec 20,21 (Baker Ch 16)
24. Wed, 8/12 8-10 12167 The large sieve LN Sec 20,21 (Baker Ch 16)
25. Wed, 15/12 8-10 12167 The large sieve; the Bombieri-Vinogradov theorem LN Sec 20,21 (Baker Ch 16)