nostalgia The first version of The ternary Goldbach problem was accepted for publication in 2015, with major revisions being optional. I decided to
  1. undertake the "serious rewrite" recommended by one of the referees,
  2. ask for at least one more round of thorough peer-review. The manuscript is now divided into parts, each of which is at a different stage of the process.
The version below dates to December 16, 2019. I am making available only the parts I feel are essentially ready. Feedback is very welcome.

Front matter

Title page, frontispice, etc.
Table of contents


Available in full, including:
Notation and preliminaries
Series and summation
Computational matters
Basic sums of arithmetical functions.

An earlier version was already refereed; some material has been added since then.

Sieves large and small

Available in full, including:
Sums used in sieve theory
A natural upper-bound sieve
The large sieve: smoothing and scattering
The L2 norm and the large sieve.

Refereed. I may later see whether a somewhat more complex-analytic explicit approach to parts of Chapters 6 or 7 is feasible and shortens the treatment. (A fully complex-analytic explicit treatment would most likely not be advantageous or even currently feasible.) There has been progress elsewhere on matters close to Chapter 7, due to (a) S. Zũñiga, (b) a research group meeting in Oaxaca. It has been shown by (b) that the sieve in Chapter 7 is optimal and in fact the only optimal one within the class of continuous sieves, even - and this is what is new - when the second-order term is considered.

Minor arcs

Type I sums
Type II sums: using small and large sieves
Minor-arc totals.

Major arcs

Available in full, including:
Major arcs: overview and results;
The Mellin transform of the twisted Gaussian;
Explicit formulas.

Refereed. Thanks are due to N. Temme for additional remarks since then.

The integral over the circle

includes: The integral over the major arcs; Optimizing smoothing functions; The integral over the minor arcs; Conclusion.


Appendix A: Norms of smoothing functions
Appendix B: Sums involving log p and φ

Bibliography and index



Small computations and bookkeeping are taken care of by SageMath code embedded within the LaTeX source file. These source files should eventually be put on arxiv, and are currently available upon request.
Larger computations were coded in C, or, in a few cases, in SageMath/Python. Here is a tar file with the source code. I will not be maintaining it in any formal sense - indeed, I think it will be better if people who are interested write their own code, following the discussions and/or pseudocode in the book. I certainly do not intend the code here to serve as any sort of model, or for it to be ready for any sort of official release. I make no claims for it save that it should give correct outputs for the inputs given in the documentation. Suggestions and bug reports are welcome.