*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.

#### Further reading

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 .)

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May 9, 2012 at 12:10 pm

[…] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 5 […]

May 21, 2012 at 12:07 pm

[…] noted that this led to improved bounds in a number of results. In particular, in Proposition 12 and Lemma 15 we were able to relax the bound on to , and in Proposition 23 we could relax it to […]

May 21, 2012 at 12:07 pm

[…] noted that this led to improved bounds in a number of results. In particular, in Proposition 12 and Lemma 15 we were able to relax the bound on to , and in Proposition 23 we could relax it to […]