*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 .* - 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.

#### Further reading

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.

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