In which we finish estimating the contribution from the major arcs, and start thinking about the singular series.
- Proposition 12 (The contribution from the major arcs): We started the proof last time, and finished it today.
- Definition: The singular series is
- Theorem 13: For sufficiently large (depending on ), we have
We saw that this follows immediately from Proposition 5 and Proposition 12.
- We started thinking about an outline of the section on the singular series. In particular, we wrote , where
- Lemma 14: We have
where the limit is over a sequence of tending to infinity in such a way that for each prime (e.g. ). We’ll see the proof next time.
You can read (at least) one of Hardy and Littlewood’s original papers on this subject online. The one to which I’ve just linked is the fourth in a sequence of papers, and looks in particular at the singular series.
Preparation for Lecture 7
Next time, we shall prove Lemma 14 and so interpret as being related to the number of solutions to the congruences for lots of . You could think about how that might work, to get a feeling for why is related to the local solvability.