In which we continue to work on an improved upper bound for .
- Lemma 27: Let be a prime and let be coprime to .
(i) If and , then .
(ii) If , then , unless . If , then this holds for , and .
We proved this by substituting in the sum, and examining the binomial expansion of .
Again we used a combination of the argument from Davenport’s book Analytic methods for Diophantine equations and Diophantine inequalities and Ben Green’s lecture notes.
Preparation for Lecture 11
Next time, we shall finish proving our improved upper bound on , and then shall prove Hua’s lemma, which will enable us to get a much better bound on for the minor arcs. You might like to remind yourself of our work so far on the minor arcs.