*In which we study the local factors and their relationship to the singular series.*

- 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 proved this by expanding the right-hand side using Fourier inversion. - Definition of

and . - Lemma 15:
*(i) For and for each prime , the limit in the definition of exists. Moreover, we have , where .*

*(ii) For , there is some such that — the infinite product converges.*We proved this by writing as a suitable sum (via Fourier inversion again), and then using a telescoping sum. - Definition of .
- Lemma 16:
*The sum is multiplicative, in the sense that if and are coprime natural numbers then for any coprime to and any coprime to .*We shall prove this next time; it won’t take long.

#### Further reading

Davenport’s book (*Analytic methods for Diophantine equations and Diophantine inequalities*) reaches the same conclusion slightly more speedily, but without pausing to interpret and quite so explicitly (relating them to counting solutions to congruences).

#### Preparation for Lecture 8

Next time, we shall prove Lemma 16 and then that the function is multiplicative (in the usual sense), and we’ll then move on to show that the local factors are all positive (by showing that there are many solutions to the relevant congruences). You might like to try proving any or all of these things for yourself!

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May 16, 2012 at 12:11 pm

[…] 23 (The singular series): For , there is a constant such that for all . This follows from Lemma 15, Corollary 18 and Corollary […]

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[…] that this led to improved bounds in a number of results. In particular, in Proposition 12 and Lemma 15 we were able to relax the bound on to , and in Proposition 23 we could relax it to for odd and […]