## Waring’s Problem: Lecture 9

In which we finish our study of the singular series.

• Lemma 21: Let $p$ be a prime and write $k = p^v k_0$ where $p$ and $k_0$ are coprime.  For $s \geq 4k$, there exist $a_1, \dotsc, a_s$ such that
$\displaystyle a_1^k + \dotsb + a_s^k \equiv N \pmod{p^{v+1}} \qquad (p \neq 2)$ or
$\displaystyle a_1^k + \dotsb + a_s^k \equiv N \pmod{2^{v+2}} \qquad (p = 2)$
and $a_1 \not\equiv 0 \pmod{p}$.
(In fact, if $k$ is odd then $s \geq 2k + 1$ will suffice.)
• Corollary 22: For $s \geq 4k$, there are positive numbers $c_p$ such that $\beta_p(N) \geq c_p$ for each prime $p$.  This was straightforward using Lemma 21 and Lemma 20.
• Proposition 23 (The singular series): For $s \geq 2^k + 1$, there is a constant $c$ such that $\mathfrak{S}(N) \geq c > 0$ for all $N$This follows from Lemma 15, Corollary 18 and Corollary 22.
• Theorem 24: Let $k \geq 2$ be an integer.  Then there is $s = s(k)$ such that
$\displaystyle \#\{(x_1, \dotsc, x_s) : x_1^k + \dotsb + x_s^k = N \} = \mathfrak{S}(N) N^{\frac{s}{k}-1} + o(N^{\frac{s}{k}-1})$

and there is a constant $c$ such that $\mathfrak{S}(N) \geq c > 0$ for all $N$.
• Corollary 25: Let $k \geq 2$ be an integer.  Then there is $s = s(k)$ such that every sufficiently large integer can be written as a sum of $s$ $k^{\mathrm{th}}$ powers.
• Lemma 26: Let $p$ be a prime and let $a$ be coprime to $p$.  Let $\delta = (k,p-1)$.  Then $|S(a,p)| \leq (\delta - 1) p^{-\frac{1}{2}}$.

Important note: the lecture on Friday will finish at 1130, so that we can all go to the Rouse Ball lecture in the Babbage Lecture Theatre at 1200.

Much of today’s lecture is based on material in Davenport’s Analytic methods for Diophantine equations and Diophantine inequalities and in Ben Green’s lecture notes.  There are a number of approaches that one can take, especially for results like Lemma 21.

#### Preparation for Lecture 10

Next time, we shall move on to study $S(a,p^j)$ more generally, trying to link it back to $S(a,p)$.  You might like to investigate this yourself.