## 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 $n^k$.

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.