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.

Further reading

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!

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2 Responses to “Waring’s Problem: Lecture 7”

  1. Waring’s Problem: Lecture 9 « Theorem of the week Says:

    […] 23 (The singular series): For , there is a constant such that for all .  This follows from Lemma 15, Corollary 18 and Corollary […]

  2. Waring’s Problem: Lecture 11 « Theorem of the week Says:

    […] that this led to improved bounds in a number of results.  In particular, in Proposition 12 and Lemma 15 we were able to relax the bound on to , and in Proposition 23 we could relax it to for odd and […]

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