## Waring’s Problem: Lecture 6

In which we finish estimating the contribution from the major arcs, and start thinking about the singular series.

• Proposition 12 (The contribution from the major arcs): We started the proof last time, and finished it today.
• Definition: The singular series is
$\displaystyle \mathfrak{S}(N):= \frac{\Gamma(1 + \frac{1}{k})^s}{\Gamma(\frac{s}{k})} \sum_{q=1}^{\infty} \sum_{\substack{a \pmod{q} \\ (a,q) = 1}} S(a,q)^s e(-\frac{aN}{q})$.
• Theorem 13: For sufficiently large $s$ (depending on $k$), we have
$\displaystyle \# \{ (x_1, \dotsc, x_s) : x_1^k + \dotsb + x_s^k = N \} = \mathfrak{S}(N) N^{\frac{s}{k}-1} + o(N^{\frac{s}{k}-1}).$
We saw that this follows immediately from Proposition 5 and Proposition 12.
• We started thinking about an outline of the section on the singular series.  In particular, we wrote $\mathfrak{S}(N) = \beta_{\infty}(N) \beta(N)$, where
$\displaystyle \beta_{\infty}(N) = \frac{\Gamma(1 + \frac{1}{k})^s}{\Gamma(\frac{s}{k})}$ and
$\displaystyle \beta(N) = \sum_{q=1}^{\infty} \sum_{\substack{a \pmod{q} \\ (a,q) = 1}} S(a,q)^s e(-\frac{aN}{q})$.
• Lemma 14: We have
$\displaystyle \beta(N) = \lim_{Q \to \infty} \mathbb{E}_{n_1, \dotsc, n_s \pmod{Q}} Q 1_{0 \pmod{Q}} (n_1^k + \dotsb + n_s^k - N)$,
where the limit is over a sequence of $Q$ tending to infinity in such a way that $v_p(Q) \to \infty$ for each prime $p$ (e.g. $Q(n) = n!$).  We’ll see the proof next time.

Next time, we shall prove Lemma 14 and so interpret $\beta(N)$ as being related to the number of solutions to the congruences $n_1^k + \dotsb + n_s^k \equiv N \pmod{q}$ for lots of $q$.  You could think about how that might work, to get a feeling for why $\beta(N)$ is related to the local solvability.