*In which we improve various bounds.*

- Proposition 28:
*Let be an integer and let be coprime to . Then .*We defined and used the bounds from Lemma 26 and Lemma 27, together with the multiplicativity from Lemma 16. - We 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 for odd and for even.
- Lemma 29 (Hua’s lemma):
*We have*We proved this inductively, using an inequality that we obtained in our proof of Weyl’s inequality (Proposition 3) and interpreting the resulting integral as counting integer solutions to an equation.

for any . - We saw that this enables us to improve some earlier bounds. In particular, in Proposition 5 we could show that the contribution from the minor arcs is negligible if , and in Theorem 24 and Corollary 25 the assertions hold for .

#### Further reading

Here’s Hua’s original paper. There’s an interesting biography of Hua here. I particularly enjoyed the phrase “About the only easy thing about Waring’s problem is its statement”!

#### Preparation for Lecture 12

The next (and final) lecture is going to be a surprise!

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