*In which we continue to work on an improved upper bound for .*

- Lemma 27:
*Let be a prime and let be coprime to .*We proved this by substituting in the sum, and examining the binomial expansion of .

(i) If and , then .

(ii) If , then , unless . If , then this holds for , and .

#### Further reading

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.

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May 21, 2012 at 12:07 pm

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