Waring’s Problem: Lecture 10

In which we continue to work on an improved upper bound for |S(a,q)|.

  • Lemma 27: Let p be a prime and let a be coprime to p.
    (i) If p \nmid k and 2 \leq r \leq k, then S(a,p^r) = p^{-1}.
    (ii) If r > k, then S(a,p^r) = p^{-1} S(a,p^{r-k}), unless k = p = 2.  If k = p =2, then this holds for r \geq 4, and S(a,8) = \frac{1}{2}e(\frac{a}{8}).
    We proved this by substituting n = p^{r-v-1} x + y in the sum, and examining the binomial expansion of n^k.

Further reading

Again we used a combination of the argument from Davenport’s book Analytic methods for Diophantine equations and Diophantine inequalities and Ben Green’s lecture notes.

Preparation for Lecture 11

Next time, we shall finish proving our improved upper bound on S(a,q), and then shall prove Hua’s lemma, which will enable us to get a much better bound on s for the minor arcs.  You might like to remind yourself of our work so far on the minor arcs.

One Response to “Waring’s Problem: Lecture 10”

  1. Waring’s Problem: Lecture 11 « Theorem of the week Says:

    […] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 10 […]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: