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
for any . 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.
- 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 .
Preparation for Lecture 12
The next (and final) lecture is going to be a surprise!