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
(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.
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.