*In which we finish our study of the singular series.*

- Lemma 21:
*Let be a prime and write where and are coprime. For , there exist such that*

or

*and .*

*(In fact, if is odd then will suffice.)*

- Corollary 22:
*For , there are positive numbers such that for each prime .*This was straightforward using Lemma 21 and Lemma 20. - Proposition 23 (The singular series):
*For , there is a constant such that for all .*This follows from Lemma 15, Corollary 18 and Corollary 22. - Theorem 24:
*Let be an integer. Then there is such that*

*and there is a constant such that for all .* - Corollary 25:
*Let be an integer. Then there is such that every sufficiently large integer can be written as a sum of powers.* - Lemma 26:
*Let be a prime and let be coprime to . Let . Then .*

Important note: the lecture on Friday will finish at 1130, so that we can all go to the Rouse Ball lecture in the Babbage Lecture Theatre at 1200.

#### Further reading

Much of today’s lecture is based on material in Davenport’s *Analytic methods for Diophantine equations and Diophantine inequalities* and in Ben Green’s lecture notes. There are a number of approaches that one can take, especially for results like Lemma 21.

#### Preparation for Lecture 10

Next time, we shall move on to study more generally, trying to link it back to . You might like to investigate this yourself.

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May 21, 2012 at 12:07 pm

[…] 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 […]