References: an annotated reading list

Companion to the course in docs/ (each doc cites what it leans on; this is the consolidated shelf). [FREE] marks items legitimately available online at no cost. Where no link is given, search the exact title, links rot, titles do not.

1. The primary source

• Riemann, B., "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" (Monatsberichte der Berliner Akademie, November 1859). Eight pages containing the analytic continuation, the functional equation, the explicit formula in outline, and the hypothesis, stated in passing. Astonishingly compressed; read it after Edwards, not before. [FREE] English translations: David Wilkins' translation (hosted at Trinity College Dublin, and widely mirrored) and the appendix of Edwards (below). The Clay Mathematics Institute also hosts a scan of the original manuscript. For: everyone, eventually.

2. Books

• Edwards, H. M., Riemann's Zeta Function (Academic Press, 1974; cheap Dover reprint, 2001). Works through Riemann's memoir line by line, developing the analysis as needed; the model for this repo's approach. For: the reader of docs/01–04. If you buy one book, buy this one.
• Titchmarsh, E. C., revised by D. R. Heath-Brown, The Theory of the Riemann Zeta-Function (2nd ed., Oxford, 1986). The professional reference: every classical theorem with full proofs, plus Heath-Brown's end-of-chapter notes updating each topic. Dry, complete, indispensable. For: looking things up once you know what to look up.
• Iwaniec, H. and E. Kowalski, Analytic Number Theory (AMS Colloquium Publications 53, 2004). The modern graduate text: zeta in the context of the whole family of L-functions, the explicit formula done properly, zero-density methods, and the large sieve. For: seeing why "prove it for one L-function, prove it for all" is the modern frame (docs/08).

3. Surveys and problem statements

• Bombieri, E., "Problems of the Millennium: The Riemann Hypothesis" (Clay Mathematics Institute official problem description, 2000). Four pages: precise statement, history, consequences, and the standard of proof required for the prize. [FREE] on the Clay Mathematics Institute website (claymath.org). For: the exact statement of what is open.
• Conrey, J. B., "The Riemann Hypothesis" (Notices of the AMS, March 2003, 341–353). The best single survey at graduate-accessible level: evidence, equivalents, random matrices, and the state of the art circa 2003 (little of which has changed). [FREE] from the AMS Notices archive. For: the reader who has finished docs/00 and wants one professional overview.

4. Zeros as a spectrum: random matrices and Hilbert–Pólya

• Montgomery, H. L., "The pair correlation of zeros of the zeta function" (Proc. Sympos. Pure Math. 24, AMS, 1973, 181–193). Where the GUE connection begins: the conjecture R₂(r) = 1 − (sin πr/πr)², proved for restricted test functions, plus the famous Dyson teatime story. For: the source behind docs/06 and scripts/04_gue_statistics.py.
• Odlyzko, A. M., the zero computations. "On the distribution of spacings between zeros of the zeta function" (Math. Comp. 48, 1987, 273–308) and the later unpublished report "The 10²⁰-th zero of the Riemann zeta function and 175 million of its neighbors". Massive-scale numerics confirming the GUE statistics to extraordinary accuracy at extraordinary height. [FREE]: Odlyzko keeps his papers and zero tables on his University of Minnesota homepage. For: what serious numerical evidence looks like, and its limits.
• External zero-table data and formats. Odlyzko's official table index (https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html) hosts the six text datasets consumed by zeta/moments.py. LMFDB documents the source and provenance (https://www.lmfdb.org/knowledge/show/rcs.source.zeros.zeta), exposes the plain index ordinate format in its route source (https://github.com/LMFDB/lmfdb/blob/main/lmfdb/zeros/zeta/zetazeros.py), and publishes the separate bulk-binary reader (https://github.com/LMFDB/lmfdb/blob/main/lmfdb/zeros/zeta/platt_zeros.py). For: the acquisition and parser contracts in docs/13.
• Hiary, G. A. and A. M. Odlyzko, "The zeta function on the critical line: Numerical evidence for moments and random matrix theory models" (Math. Comp. 81 (2012), 1723–1752; author PDF (https://www-users.cse.umn.edu/~odlyzko/doc/zeta.moments.pdf)). Documents the high-height value evaluation and integration method behind the moments programme. The 2026-08-04 data audit found published results but no downloadable dense row table. For: docs/13 §§6, 9.
• Bober, J. and G. A. Hiary, selected high zeta computations. Their Bristol pages (https://people.maths.bris.ac.uk/~jb12407/data/zeta/index_Z11.html) publish extreme values and plots, useful as computation provenance but not as an interval-moment sample. For: the source audit in docs/13 §9.
• Keating, J. P. and N. C. Snaith, "Random Matrix Theory and ζ(1/2+it)" (Commun. Math. Phys. 214 (2000), 57–89; author PDF (https://people.maths.bris.ac.uk/~mancs/papers/RMTzeta.pdf)). Derives the random-matrix factor in the general leading conjecture for critical-line moments. For: moment_reference and docs/13 §7.
• Conrey, J. B., D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith, "Integral moments of L-functions" (Proc. London Math. Soc. 91 (2005), 33–104; arXiv:math/0206018 (https://arxiv.org/abs/math/0206018)). Gives the arithmetic Euler product and full-main-term recipe, recovering the Hardy–Littlewood and Ingham leading terms and the sixth/eighth conjectures. For: the theorem/conjecture split and coefficient convention in docs/13.
• Conrey, J. B., D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith, "Lower order terms in the full moment conjecture for the Riemann zeta function" (arXiv:math/0612843 (https://arxiv.org/abs/math/0612843)). Gives the full polynomial coefficient recipe and the published k=3 table. For: moment_polynomial(3) and the convention checks in docs/13 §7.
• Rubinstein, M. O. and S. Yamagishi, "Computing the moment polynomials of the zeta function" (arXiv:1112.2201 (https://arxiv.org/abs/1112.2201)). Gives a stable coefficient algorithm and the reported stable digits used for the degree-16 k=4 polynomial; these are not interval enclosures. For: moment_polynomial(4) and docs/13 §7.
• Berry, M. V. and J. P. Keating, "The Riemann zeros and eigenvalue asymptotics" (SIAM Review 41, 1999, 236–266); also "H = xp and the Riemann zeros" (1999). The physicists' Hilbert–Pólya: semiclassical analogies between the explicit formula and trace formulae, and the H = xp proposal. For: docs/06 §Berry–Keating; inspiration, honestly labelled as not a construction.

5. The de Bruijn–Newman frontier (the heatflow.py story)

• Rodgers, B. and T. Tao, "The de Bruijn–Newman constant is non-negative" (arXiv:1801.05914, January 2018; published in Forum of Mathematics, Pi, 2020). Proof that Λ ≥ 0, Newman's 1976 conjecture, via the rigidity of backward-heat-flow zero statistics. With RH ⟺ Λ ≤ 0 this makes RH ⟺ Λ = 0. [FREE] on arXiv. For: the theorem docs/05 and zeta/heatflow.py are built around.
• Polymath, D. H. J. (Polymath15, led by T. Tao), "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn–Newman constant" (Research in the Mathematical Sciences 6, 2019, paper 31). The collaborative computation pushing de Bruijn's Λ ≤ 1/2 down to Λ ≤ 0.22, with fully effective error control, the rigorous version of what zeta/heatflow.py does illustratively. [FREE] on arXiv. For: what "numerics with proofs attached" means; the Platt–Trudgian verification then gives the current record Λ ≤ 0.2.

6. Zero dynamics under heat flow (the hunts/flow_repair connection)

• Cuenca, C. and C. McSwiggen, "The Rectangular Finite Free Heat Flow" (arXiv:2606.06859 (https://arxiv.org/abs/2606.06859), 2026). Formalizes the ODE ż_k = 2Σ 1/(z_k − z_j), the exact dynamics the flow-repair null control integrates, as a Calogero–Moser particle system in the finite free probability framework. Proves universality of the zero-repulsion law for polynomials under heat flow. [FREE] on arXiv. For: the theoretical foundation of hunts/flow_repair/ §3's 0.04% PDE-vs-ODE agreement, the repair clock reads geometry because the dynamics are universal in the CM sense.
• Hall, B. C. and C.-W. Ho, "Zeros of random polynomials undergoing the heat flow" (arXiv:2308.11685 (https://arxiv.org/abs/2308.11685), 2023). Explores the heat-flow conjecture for random polynomials, details the zero-repulsion dynamics through random matrices and Calogero–Moser. [FREE] on arXiv. For: the random-matrix side of the same coin, why the CM dynamics explain zero motion for broad classes of entire functions, not just ζ.

7. Zero spacings: the small-gap frontier

• Inoue, S., "Small gaps between consecutive zeros of the Riemann zeta-function" (arXiv:2604.05733 (https://arxiv.org/abs/2604.05733), 2026). Breaks the long-standing Montgomery–Odlyzko practical barrier with a "resonance-correlation method": μ < 0.50895 (conditional on RH). [FREE] on arXiv. For: the state-of-the-art result framing Lehmer pairs; hunts/lehmer_pair/ is a case study in the regime this bound governs.
• Goldston, D. A., T. S. Trudgian and C. Turnage-Butterbaugh, "Limitations of the Montgomery–Odlyzko method" (2022). Proves the classical M-O method cannot show infinitely many pairs closer than 0.5042× the average spacing without new techniques, the ceiling that Inoue's resonance-correlation method then breaks. For: the "why it is hard" complement to the small-gap result; pairs with docs/08.

8. Recent unconditional progress

• Guth, L. and J. Maynard, "New large value estimates for Dirichlet polynomials" (2024). The first improvement since Ingham (1940) on the zero-density exponent: roughly, fewer possible zeros far from the critical line, improving prime-distribution consequences unconditionally. Not a route to RH, a sharpening of how far from RH we are allowed to be. [FREE] on arXiv. For: what genuine unconditional progress looks like in this subject, per docs/08.

Reading orders

• Shortest honest path: Bombieri → Conrey → Edwards chs. 1–3.
• This repo's spine: Edwards (for docs/01–04) → Montgomery + Odlyzko (for docs/06) → Rodgers–Tao + Polymath15 (for docs/05).
• The hunts: Cuenca–McSwiggen (for hunts/flow_repair null control) + Inoue (for hunts/lehmer_pair spacing context) + Goldston–Trudgian–T-B (for why the M-O method needed replacing).
• Calibration: Guth–Maynard, to see the size of a genuine step.