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.

9. The quasi-Riemann theorem, and what was built on it

• OpenAI, "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) > 7/8" (OpenAI Math Release preprint, 2026-09-30), with the companion "The Quasi-Riemann Hypothesis" (alternate 11/12 proof, 2026-10-05, written with human assistance) and "Uniform exclusion of Landau-Siegel zeros" (2026-10-01). Family 003 of github.com/openai/math at commit adc7f124. Claims every Dirichlet L-function, zeta included, and every finite-order Hecke L-function over Q(sqrt(-3)) is zero-free in Re s > 7/8, via Patterson's cubic theta function on Kubota's metaplectic cover and a "continuation from a common signal" criterion. Lean statements against Mathlib's riemannZeta and DirichletCharacter.LFunction are in lean/ComparatorChallenges/; replayed here on 2026-10-08: the build, #print axioms (the standard three) and Comparator with NanoDa on, all accepting (docs/38 section 7). [FREE], Apache-2.0. For: docs/38, which records what was checked in this tree and what was not; cite the bound with its exact formal statement (section 3), as kernel-checked, and never as RH.

Used by the four hunts that build on it (docs/39), each cited where it is used:

• B. Chen, V. Gupta, Y. C. Li, "Large value estimates for Dirichlet polynomials with characters and zero density of Dirichlet L-functions", arXiv:2507.08296 (v2, 2026-07-27): sum over chi mod q of N(sigma, T, chi) << (qT)^(7(1-sigma)/3 + eps), improving Huxley's 12/5; preprint. For: hunt #123, Linnik's exponent 7/3; their Corollary 1.4 states the conditional implication.
• T. Xylouris, Bonn thesis (2011) and Chebyshevskii Sbornik (2018): Linnik's constant L = 5, then L < 5, unconditional. The bar for hunt #123.
• M. Watkins, "Class numbers of imaginary quadratic fields", Math. Comp. 73 (2004): complete lists for h <= 100, unconditional. S. Holmin, N. Jones, P. Kurlberg, C. McLeman, K. Petersen, arXiv:1510.04387: complete lists for odd h < 10^6 under GRH. J. Friedlander, H. Iwaniec, arXiv:1701.03771, Theorem 3: Littlewood's argument under a 3/4 hypothesis. Y. Lamzouri, X. Li, K. Soundararajan, Math. Comp. 84 (2015): the explicit GRH form. For: hunt #124.
• E. S. Lee, arXiv:2602.14340 (2026), Theorem 1.2: a prime between n^86 and (n+1)^86 for every n, unconditional; earlier M. Cully-Hugill, D. R. Johnston (k = 90, 2025; k = 140, 2023). D. Platt, T. Trudgian, Bull. LMS 53 (2021): RH holds up to height 3 * 10^12. C. Bellotti, P.-J. Wong, arXiv:2412.15470: explicit N(T). J. Buthe, arXiv:1511.02032: the analytic method for bounding psi(x). For: hunt #126.
• H. L. Montgomery, R. C. Vaughan, Multiplicative Number Theory I, Theorem 13.12, after Rodosskii (1954): least nonresidue from a zero-free strip, constants not explicit. E. Bach, Math. Comp. (1990): explicit GRH witness bounds. For: hunt #125.

10. Jensen polynomials: the hyperbolicity frontier

The zeta/li.py lane and docs/12 §8.1. Checked against the primary sources on 2026-10-18, after this tree read the frontier from GORZ 2019 alone and proposed an experiment the 2022 paper had already done.

• Griffin, M., K. Ono, L. Rolen and D. Zagier, "Jensen polynomials for the Riemann zeta function and other sequences" (PNAS 116 (2019), no. 23, 11103–11110; arXiv:1902.07321). For each fixed degree d, J^{d,n} is hyperbolic for all sufficiently large n, and its renormalised roots tend to those of the Hermite polynomial H_d. [FREE] on arXiv. For: the 8·ξ(½+z) normalisation zeta/li.py uses, and jensen_roots_vs_gue.
• Chasse, M., "Laguerre multiplier sequences and sector properties of entire functions" (Complex Variables and Elliptic Equations 58 (2013), no. 7, 875–885). Theorem 1.8: J^{d,n} is hyperbolic for every n ≥ 0 when d ≤ 2·10^17, from RH verified to a finite height. For: the first all-n range, which GORTTW Theorem 1.2 generalises.
• Griffin, M. J., K. Ono, L. Rolen, J. Thorner, Z. Tripp and I. Wagner, "Jensen polynomials for the Riemann xi-function" (Advances in Mathematics 397 (2022), 108186; arXiv:1910.01227v3, 17 December 2020). Theorem 1.1: hyperbolic for n ≥ c·e^{d/2} (v3; the published version states n ≥ c·e^d), c not computed. Theorem 1.2: RH to height T gives hyperbolicity for all n ≥ 0 when d ≤ ⌊T⌋². Corollary 1.3, with Platt's T = 3.06·10^10: d ≤ 9.36·10^20. Remark 3 calls a converse from partial hyperbolicity to the zeros highly unlikely. [FREE] on arXiv. For: why no finite unresolved set exists at any degree a computation can reach.
• Holland, J., "A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's ξ-function" (arXiv:2608.08682v1, 9 August 2026; preprint). Theorem 1.1: there is an absolute K > 0 such that n³log²(n+2) ≥ K·d⁵ implies J^{d,n} has d distinct negative real zeros; §12 makes no attempt to optimise K. Proof by a real-rooted comparison polynomial built from Jacobi and finite-free factors. [FREE] on arXiv. For: the current frontier, and the target hunts/jensen_wedge (hunt #135) is a delta against.
• Farmer, D. W., "Jensen polynomials are not a plausible route to proving the Riemann Hypothesis" (Advances in Mathematics 411 (2022), 108781; arXiv:2008.07206). Argues that the Hermite connection of GORZ is evidence neither for RH nor for random-matrix statistics of the zeros, that Jensen polynomials are unlikely to be useful for attacking RH, and proposes criteria for judging whether an RH equivalence is likely to be useful. [FREE] on arXiv. For: the honest ceiling on this whole lane, read alongside docs/12 §8.1.
• O'Sullivan, C., "Zeros of Jensen polynomials and asymptotics for the Riemann xi function" (Research in the Mathematical Sciences 8 (2021), Paper No. 46; arXiv:2007.13582). Detailed asymptotic expansions of the xi Taylor coefficients; Theorem 1.3 makes a Hermite-modified family P_{d,n} (not J^{d,n}) hyperbolic for n/log²n ≥ d^{3/4}/2 once d is large enough, with no explicit threshold. Holland cites it. [FREE] on arXiv. For: the coefficient asymptotics behind every wedge.
• Ning, W., "An Effective d³-Scale Criterion for the Hyperbolicity of Jensen Polynomials Associated with the Riemann ξ-Function" (Zenodo preprint, 29 April 2026, doi 10.5281/zenodo.19891455; not peer reviewed). Theorem 9.4: an absolute C₀ with n + d ≥ C₀·d³ sufficient, so a polynomial wedge d ≲ n^{1/3} that predates Holland; C₀ is not computed. [FREE]. For: hunts/jensen_wedge/NOVELTY.md, claim C.
• Sarnowski, M., "Paper 134Z: Multi-Jacobi Protected Spines and Higher-Order Matching in the Riemann ξ Jensen Hierarchy" and "Paper 134AA: Jacobi Cancellation Skeletons and All-Large-Shift Geometry of Multi-Jacobi Protected Spines" (Zenodo, both 20 August 2026, doi 10.5281/zenodo.22023893 and 10.5281/zenodo.22025447; not peer reviewed; part of a 45-deposit series). A rank-r rational quotient spine matches 2r + 2 coefficients; it tends to the diagonal [r/r] Padé approximant of (1 + y)^{1 − ε_N}, ε_N ~ 2/L_N, with a Jacobi cancellation skeleton at ε = 0. At r = 2 this is the six-coefficient two-Jacobi match of hunts/jensen_wedge §11.3 and §11.4, constants included. Paper 134AE (doi 10.5281/zenodo.22026880) adds the Beta-Stieltjes representation. Diagnostics are non-interval and no new hyperbolicity region is claimed. [FREE]. For: the prior instance of hunt #135's claim B (hunts/jensen_wedge/NOVELTY.md).
• Savva, J., "A Two-Thirds Hyperbolicity Wedge for Jensen Polynomials of Riemann's Xi-Function" (Zenodo preprint, Version 1.1, 3 September 2026, doi 10.5281/zenodo.22293642; repository aimathpapers/Jensen-2-Thirds). Theorem 1.1: an absolute K > 0 with n²log(n+2) ≥ K·d³ sufficient for J^{d,n} to have d distinct negative zeros, by Holland's ₃F₂ family with D free matching six coefficients. The author states substantial AI assistance and no human or peer review; the Lean development is conditional on typed literature inputs. K is not computed. Not checked here. [FREE]. For: the prior instance of hunt #135's claim A (hunts/jensen_wedge/NOVELTY.md).
• Planat, M., "Asymptotic Hyperbolicity of Jensen Polynomials and the Finite-Strip Obstruction to the Riemann Hypothesis" (Mathematics 14 (2026), 1884, doi 10.3390/math14111884) and "Parity Bifurcation, PIII(D6) Topology, and a Stieltjes Framework to Jensen Polynomial Hyperbolicity" (Mathematics 14 (2026), 2240, doi 10.3390/math14132240). The first states hyperbolicity for n ≥ C₀d⁴, C₀ ≈ 0.020; its normalisation drops the 1/(2n)! of the Taylor coefficients in places, and its positive onset at d ≥ 9 contradicts GORTTW Corollary 1.3 for the standard family. Neither cites GORTTW. [FREE]. For: a cautionary read; see hunts/jensen_wedge/NOVELTY.md §4 before citing either.

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.