MSc thesis, Keith Matthews, University of Queensland, 1966

• Title
• Acknowledgements
• Introduction

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 τ_s,k(α) and S_ξ,[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 x^d=ξ over [q]. Gaussian sums. Estimation of |S_ξ,[q]| and |S_A,P|.

6. Farey dissection.

   Properties of major arcs M_F,G and minor arcs m_F,G. Explicit determination of M_F,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 S_F,G and S(Λ). An important property of S(Λ).

10. Formula for r₂(N), the contribution of the major arcs.
11. Estimates for |S_F,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.

• Bibliography.
• Remarks by Harold Davenport to the author.
• Corrections and comments by Clive Selwyn Davis (found in the back of his copy of the thesis at the Alumni Book Sale!): page 1, page 2, page 3.

Last modified 15th March 2024