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 NThis 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.

Further reading

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.

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One Response to “Waring’s Problem: Lecture 9”

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

    […] 28: Let be an integer and let be coprime to .  Then .  We defined and used the bounds from Lemma 26 and Lemma 27, together with the multiplicativity from Lemma […]

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