Prerequisites
Most problems in the programme are accessible with:
- A first course in number theory. Comfort with congruences, the Prime Number Theorem in informal form, and Dirichlet's theorem on primes in arithmetic progressions.
- Undergraduate linear algebra and discrete mathematics. Matrix operations, basic combinatorial arguments, proof by induction.
- Working knowledge of one scientific programming language. Python with NumPy, SciPy, and Matplotlib is sufficient for all computational problems; Julia, R, and Mathematica are equally viable.
- Comfort reading mathematical papers. Volumes I and II are written to be self-contained for a reader with the above prerequisites.
Project types
Computational verification & extension
Replicate or extend the numerical experiments in Volume I and Volume II. For example:
- Extend the Monte Carlo LF-ratio experiment to C > 1,000, characterizing the growth of Z with C.
- Reproduce the Lomb-Scargle zeta-alignment result with an independent implementation. Test the null hypothesis with different randomization strategies.
- Implement the Tesfa Grid operator in Julia or a compiled language, benchmark against the Python reference, release the code.
Analytical exploration
Work on the open problems (see the open problems page). Accessible starting points:
- Prove or disprove the modular generalization of the Row-6 Exclusion Theorem (OP 5).
- Derive an explicit form of the twin prime row symmetry asymptotic (OP 6).
- Investigate 8-row, 12-row, and 30-row generalizations of the Tesfa Grid (OP 8).
Expository work
Clear exposition is genuinely valuable in its own right. Useful projects:
- An explainer of the Harmonic Sieve Theorem (Theorem IX) for a signal-processing audience unfamiliar with number theory.
- A writeup of the connection between the explicit formula (von Mangoldt) and the Tesfa Wave, for a number theory audience unfamiliar with spectral methods.
- Interactive visualizations of the Tesfa Grid and its theorems.
How to begin
- Read Volume I in full. The proofs are short; the experiments are reproducible.
- Read the parts of Volume II that interest you the methodology section is self-contained.
- Pick an open problem and write a one-page proposal: your background, what you plan to do, and what a successful outcome looks like.
- Send it in. You will receive a response with suggested next steps, further reading, and any prior partial results.