*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

and

. - 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.

#### Further reading

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.

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