In which we study the contribution from the major arcs.
- Lemma 9: For any real , we have
This was an exercise in integration.
- Lemma 10: For , we have
where is the Gamma function. This was also an exercise in integration.
- Lemma 11: Let be a natural number and let be coprime to . Then where . This was an easy consequence of Weyl’s inequality (Theorem 3).
- Proposition 12 (The contribution from the major arcs): Suppose that and satisfy the following conditions:
- and as ;
- for some ; and
- for some .
(For example, take and for a suitable .) Then for we have
We started proving this, using our estimate for from last time and then just chasing through the resulting errors. We’ll finish the proof next time.
The usual collection of books and lecture notes. Please leave a comment if you find another exposition that you like.
Preparation for Lecture 6
You could try to finish the proof of Proposition 12 for yourself. You could also try to get an improved bound on . (Later in the course, we shall prove that if and are coprime then .)