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.

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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 […]

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