The prime numbers are the atoms of arithmetic, and their distribution among the integers is one of the deepest and most beautiful subjects in all of mathematics. Euclid proved there are infinitely many; Gauss and Legendre guessed how densely they thin out; Riemann, in a single nine-page paper, tied their distribution to the zeros of a function of a complex variable — and left behind the most famous unsolved problem in mathematics. This master's-level course tells that whole story, and carries it right up to the twenty-first-century frontier.

We begin with the analytic toolkit — asymptotics, summation by parts, the Gamma function, contour integration — and elementary prime estimates (Chebyshev, Mertens). We build the Riemann zeta function rigorously, prove the Prime Number Theorem by complex analysis, and develop Dirichlet L-functions to count primes in arithmetic progressions. Then the modern machinery: the zeros of zeta and the Riemann Hypothesis, sieve methods, the spectacular recent work on gaps between primes (Zhang, Maynard–Tao), the circle method and Green–Tao, the automorphic L-function frontier, and the computational state of the art.

It assumes fluency with real analysis and, especially, complex analysis — contour integration, the residue theorem and analytic continuation are used constantly. It builds directly on the intuitive Prime Number Theorem and Riemann Hypothesis pages from the undergraduate number-theory course, and turns their pictures into proofs.

This course is being written. Lessons already available are linked; topics still to come are shown as greyed todo placeholders so the full syllabus is visible.

Module 0 — The analytic toolkit

1. Asymptotic analysis: the language of estimates
2. Abel summation and summation by parts
3. The Gamma function
4. Dirichlet series and the abscissa of convergence
5. Perron's formula
6. Poisson summation and theta functions
7. Tauberian theorems

Module 1 — Elementary prime estimates

1. The von Mangoldt function and ψ(x)
2. Chebyshev's bounds
3. Mertens' theorems
4. Bertrand's postulate
5. The equivalence of π, θ and ψ
6. The logarithmic integral Li(x)

Module 2 — The Riemann zeta function, rigorously

1. Analytic continuation of ζ(s)
2. The functional equation
3. The completed zeta function ξ(s)
4. The Hadamard product for ζ
5. The logarithmic derivative ζ′/ζ
6. The zeros of zeta

Module 3 — The Prime Number Theorem

1. The zero-free region
2. The explicit formula
3. Proving the Prime Number Theorem
4. The error term in the PNT
5. The Newman–Zagier Tauberian proof
6. Li(x) vs π(x) and Littlewood's sign change

Module 4 — Dirichlet L-functions & primes in progressions

1. Dirichlet characters
2. Character orthogonality
3. Dirichlet L-functions
4. The non-vanishing of L(1, χ)
5. Primes in arithmetic progressions
6. Siegel zeros and Siegel–Walfisz
7. The Generalised Riemann Hypothesis

Module 5 — The zeros of zeta & the Riemann Hypothesis

1. Counting zeros: the Riemann–von Mangoldt formula
2. Hardy's theorem: infinitely many zeros on the line
3. Zero-density theorems
4. RH and the prime count
5. The Lindelöf hypothesis
6. Pair correlation and random matrices

Module 6 — Sieve methods

1. The Legendre sieve and its limits
2. Brun's sieve and the twin-prime constant
3. The Selberg sieve
4. The large sieve inequality
5. The Bombieri–Vinogradov theorem
6. The parity problem

Module 7 — Gaps between primes

1. The Cramér model
2. The Hardy–Littlewood k-tuple conjectures
3. Small gaps: the GPY method
4. Bounded gaps: Zhang's theorem
5. Maynard–Tao and Polymath
6. Large gaps between primes

Module 8 — Additive prime number theory

1. The Hardy–Littlewood circle method
2. Vinogradov's three-primes theorem
3. Goldbach's conjecture
4. Waring's problem
5. The Green–Tao theorem
6. Additive combinatorics and Szemerédi

Module 9 — L-functions & the automorphic frontier

1. General L-functions and the Selberg class
2. Modular forms: a first look
3. Hecke eigenforms and their L-functions
4. The Rankin–Selberg method
5. The Sato–Tate distribution
6. Elliptic-curve L-functions
7. The Chebotarev density theorem

Module 10 — The computational & algorithmic frontier

1. Deterministic primality: the AKS test
2. Probabilistic primality: Miller–Rabin
3. Elliptic-curve primality proving
4. Factoring and the number field sieve
5. Computing zeros: the Riemann–Siegel formula
6. The state of the art