Note: There is NO class on Friday, April 18th, on Wednesday, April 23rd, or on Friday, April 25th, due to the ETH holiday. We will return on Wednesday, April 30th, with an exercise session.
Lectures notes are available here (Version of 21.5.2025). Lectures will be recorded and videos (internal to ETH) will be posted here.
The exam will consist of four questions. Of those, one will involve partial summation, one will concern the Hardy--Littlewood conjectures and the singular series constants, and one will be a longer question about applying sieve results. Here is a practice exam.
The lecture will be accompanied by roughly biweekly exercise classes, usually during the Wednesday class. I will announce the precise dates in the lecture as well as here.
| Dates of exercise classes |
|---|
| February 26 |
| March 12 |
| March 26 |
| April 9 (note: I will be absent on April 9th; this will be a self-directed exercise session) |
| April 30 |
| May 14 |
| May 23 (note: this is a Friday. I will be absent on May 23rd; this will be a self-directed exercise session) |
| Exercise sheet | Solutions |
|---|---|
| Exercise sheet 1 | |
| Exercise sheet 2 | |
| Exercise sheet 3 | |
| Exercise sheet 4 | |
| Exercise sheet 5 | |
| Exercise sheet 6 | |
| Exercise sheet 7 |
| Day | Content |
|---|---|
| 19.2.2025 | The Cram\'er model; example Cram\'er conjectures for prime numbers, specifically distribution of gaps, maximal gaps, Riemann Hypothesis. |
| 21.2.2025 | The Hardy--Littlewood k-tuple conjectures, consequences for distribution of primes in short intervals. Singular series constants are 1 on average. |
| 28.2.2025 | Sieve theory overview; the sieve of Eratosthenes; general sieve setup and examples. |
| 5.3.2025 | Brun's pure sieve |
| 7.3.2025 | Brun's sieve and the sum of the reciprocals of twin primes; Selberg's sieve introduction and Selberg's sieve for primes |
| 14.3.2025 | Selberg's sieve upper bound for primes; general statement of Selberg's sieve. |
| 19.3.2025 | Mean values of multiplicative functions; Rankin's trick. |
| 21.3.2025 | Brun--Titchmarsh; parity problem. Started on small gaps between primes. |
| 28.3.2025 | Small gaps between primes. GPY setup and definition of Maynard weights. |
| 2.4.2025 | Small gaps between primes, continued. Stated Bombieri--Vinogradov. |
| 4.4.2025 | Finished small gaps between primes; finding an optimal function. |
| 11.4.2025 | Circle method overview and intuition for Vinogradov's three primes; S(N,alpha) evaluated at various alpha. |
| 02.5.2025 | Major arcs estimation in the circle method. |
| 07.5.2025 | Reducing the circle method argument to Vinogradov's pointwise bound. Motivating the approach, defining type I and type II sums. |
| 09.5.2025 | Vaughan's identity; type I bound. |
| 16.5.2025 | Finished type I bound; stated type II bound. |
| 21.05.2025 | Type II bounds; finished proof of Vinogradov's pointwise bound. |