## Waring’s Problem: Lecture 7

In which we study the local factors $\beta_p(N)$ and their relationship to the singular series.

• 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 proved this by expanding the right-hand side using Fourier inversion.
• Definition of
$\displaystyle M(p^r) = \mathbb{E}_{n_1, \dotsc, n_s \pmod{p^r}} p^r 1_{0 \pmod{p^r}}(n_1^k + \dotsb + n_s^k - N)$
and $\displaystyle \beta_p(N) = \lim_{r \to \infty}M(p^r)$.
• Lemma 15: (i) For $s \geq 2^k+1$ and for each prime $p$, the limit in the definition of $\beta_p(N)$ exists.  Moreover, we have $\beta_p(N) = 1 + O_{\epsilon}(p^{1-\frac{s}{K} + \epsilon})$, where $K = 2^{k-1}$.
(ii) For $s \geq 2^k+1$, there is some $p_0$ such that $\displaystyle \frac{1}{2} \leq \prod_{p > p_0} \beta_p(N) \leq \frac{3}{2}$ — the infinite product converges.  We proved this by writing $M(p^r)$ as a suitable sum (via Fourier inversion again), and then using a telescoping sum.
• Definition of $\displaystyle A(q) = \sum_{\substack{a \pmod{q} \\ (a,q)=1}} S(a,q)^s e(-\frac{aN}{q})$.
• Lemma 16: The sum $S(a,q)$ is multiplicative, in the sense that if $q_1$ and $q_2$ are coprime natural numbers then $S(a_1, q_1)S(a_2, q_2) = S(a_1 q_2 + a_2 q_1, q_1 q_2)$ for any $a_1$ coprime to $q_1$ and any $a_2$ coprime to $q_2$.  We shall prove this next time; it won’t take long.

Davenport’s book (Analytic methods for Diophantine equations and Diophantine inequalities) reaches the same conclusion slightly more speedily, but without pausing to interpret $\beta(N)$ and $\beta_p(N)$ quite so explicitly (relating them to counting solutions to congruences).

#### Preparation for Lecture 8

Next time, we shall prove Lemma 16 and then that the function $A$ is multiplicative (in the usual sense), and we’ll then move on to show that the local factors are all positive (by showing that there are many solutions to the relevant congruences).  You might like to try proving any or all of these things for yourself!