- 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. - Formal Laurent series over [q].
Field property. Valuation function. Dirichlet diophantine approximation lemma.

- Trace function.
Additive properties. Evaluation of . The functions τ

_{s,k}(α) and S_{ξ,[q]}. - The function e(Λ).
Evaluation of . Formula for r(N) in terms of S(Λ).

- Characters and generalised exponential sums.
The equation x

^{d}=ξ over [q]. Gaussian sums. Estimation of |S_{ξ,[q]}| and |S_{A,P}|. - Farey dissection.
Properties of major arcs M

_{F,G}and minor arcs m_{F,G}. Explicit determination of M_{F,G}. - Estimate for S(A/Q) on a minor arc.
Analogue of Weyl's lemma and its application to S(A/Q).

- Estimate for r
_{1}(N), the contribution of the minor arcs.Analogue of Hua's lemma and its application to r

_{1}(N). - 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). - Formula for r
_{2}(N), the contribution of the major arcs. - Estimates for |S
_{F,G}|.Absolute convergence of the singular series.

- The asymptotic formula for r(N).
- The singular series ℭ(N).
- The analogue of Waring's theorem.
- 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: page 1, page 2, page 3, page 4.

* Last modified 28th May 2011*