MSc thesis, Keith Matthews, University of Queensland, 1966

  1. Polynomials in an indeterminate t over [q].
    Elementary properties of factorisation. Estimate for the divisor function.
    Congruence. Complete and reduced sets. The group G(H,[q]).
    Field property mod P. Sums involving multiplicative functions.
  2. Formal Laurent series over [q].
    Field property. Valuation function. Dirichlet diophantine approximation lemma.
  3. Trace function.
    Additive properties. Evaluation of . The functions τs,k(α) and Sξ,[q].
  4. The function e(Λ).
    Evaluation of .  Formula for r(N) in terms of S(Λ).
  5. Characters and generalised exponential sums.
    The equation xd=ξ over [q]. Gaussian sums. Estimation of |Sξ,[q]| and |SA,P|.
  6. Farey dissection.
    Properties of major arcs MF,G and minor arcs mF,G. Explicit determination of MF,G.
  7. Estimate for S(A/Q) on a minor arc.
    Analogue of Weyl's lemma and its application to S(A/Q).
  8. Estimate for r1(N), the contribution of the minor arcs.
    Analogue of Hua's lemma and its application to r1(N).
  9. Evaluation of S(A/Q) on a major arc.
    Relation of S(A/Q) to SF,G and S(Λ). An important property of S(Λ).
  10. Formula for r2(N), the contribution of the major arcs.
  11. Estimates for |SF,G|.
    Absolute convergence of the singular series.
  12. The asymptotic formula for r(N).
  13. The singular series ℭ(N).
  14. The analogue of Waring's theorem.
  15. An outline for obtaining χ(P) > 0 under the weaker condition p > k.

Last modified 6th December 2020