In which we show that the minor arcs contribute a negligible amount, and study the behaviour at a point in the major arcs.
- Proposition 5 (The contribution from the minor arcs): Suppose that and for some . Then for sufficiently large (depending on and ), we have . We proved this using our bound from last time on , together with Parseval’s identity (which here comes down to being the statement that ). This result does not give us a good value of ; later in the course, we shall see Hua’s lemma, and this result will enable us to get a much better value of .
- Definition of .
- Lemma 6: Let be an interval. Then . We saw that the key idea here is that the value of depends only on the congruence class of modulo , so we partitioned accordingly.
- Lemma 7: Let be a differentiable function. Then . This was quite a crude bound; we did not do much work to prove it (we just split into intervals of length 1 and used an easy bound on each one). Later in the course, we shall see a better way to approximate a sum by an integral, but this will do for now.
- Lemma 8: Let be a point in the major arc . Then . We partitioned into shorter intervals on which we could treat as approximately constant, and then used Lemma 7 to replace the resulting sum by an integral.
There’s a colourful picture showing major arcs on the Wikipedia page for the Hardy-Littlewood circle method.
Preparation for Lecture 5
Next time, we shall put together the contributions from individual points on the major arcs, by integrating. You might like to try this for yourself before the lecture.