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.

Further reading

You can read (at least) one of Hardy and Littlewood’s original papers on this subject online.  The one to which I’ve just linked is the fourth in a sequence of papers, and looks in particular at the singular series.

Preparation for Lecture 7

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.


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