Computing the Riemann Hypothesis: Thorin Measures, Pólya Densities and Hitting Times, version 1.2.0

Version 1.2.0 (ready for the authors' approval: Q20 of misc/catalog/QUESTIONS.md)

Compared with version 1.1.0, this version has four changes: six new chapters after Chapter 15 and before the appendices, a shrink pass over the whole text, merged results of the collaborator (Nick) and of Tony Newton, and conditional material on the quasi Riemann hypothesis. The six chapters are all after chapters of the first author of October 2026: Chapter 16, "Pólya and the zeros of entire functions", and Chapters 17 to 21 on Yoshida's Hermitian forms, the compact windows of Zhu, three families of generalized gamma convolutions, deconvolution as a damped Weil criterion, and Thorin representations of Dirichlet L functions with the Linnik constant.

The numbers of version 1.1.0 do not all survive. The shrink pass (plan and results in misc/audit/shrink/PLAN.md, final commit 0a0bbb1) tightened the text of Chapters 1 to 15 and the appendices: by the commit's figures the KDP edition went from 999 to 801 pages and main.pdf from 826 to 662, with 0 errors, 0 undefined references and overfull hboxes of at most 2.31 pt. The pass applied 1584 hunks (298 KB) of repeats, inferable passages and restated facts, read by Sonnet and decided hunk by hunk by Opus, and the reviewers rejected about 4 percent. It renumbered the printed numbers (sections, theorems, equations, tables, figures, exercises and problems) of Chapters 1 to 15, so references to version 1.1.0 by number must be checked against the new text. It deleted Appendix D, and moved the code inventory of the appendices to code/README.md, with the references to deleted tables and figures removed. Chapters 16 to 21 are numbered as they were first written. The latest scratch build, with the merges below, has main.pdf 682 and main-kdp.pdf 827 pages, 0 errors and 0 undefined references; the page numbers differ from version 1.1.0 throughout.

What Chapter 16 adds:

• George Pólya's papers on the zeros of entire functions, from the Laguerre Pólya class of 1913 to the successive derivatives of 1943, read through the reciprocal Xi law and the basepoint laws, with a table of which of his results the book already has and which are new to the programme.
• The Pólya Schur form of RH in the variable s, with the Turán inequalities for the Taylor coefficients of Xi as its first rung, and the Jensen polynomials of Xi and their shifted forms checked by Sturm's theorem.
• Schoenberg's two sided form of the GGC criterion; a floating point certificate with explicit bounds that the density of the two sided reciprocal Xi law is log concave, and an unconditional theorem that it lies within 1.6 x 10^-10 of a Pólya frequency function of every order.
• The universal factor cosh(au) and the clock: the atoms of the even part laws sit exactly at the half integer crossings of the argument clock of the basepoint law.
• Pólya's Xi star as the archimedean model, with its zeros at half Gram points.
• Pólya's Mellin criterion applied to the Riemann kernel, and why it fails: the zero of the Mellin transform that closes the Bernstein Pick route.
• The sine squared lemma read with Pólya's work: an exact form that locates the role of the primes, an exact relation between the even part and basepoint laws, and the compensated clock condition inside the critical strip at almost every height, with the exceptional set bounded by zero density estimates.
• What the zero free half plane Re s > 7/8, announced in October 2026 and not refereed, would give: Thorin densities of the basepoint laws and the bound 9/32 on the de Bruijn Newman constant; a theorem that a zero free half plane Re s > alpha_0 makes the compensated clock condition hold at every height for every alpha > alpha_0, by pairing each zero off the critical line with its mirror image; and an equivalence between a quasi Riemann hypothesis at alpha_0 and that condition on every line to its right.
• Open problems, with the progress made on each.

The chapter comes with new scripts (code/polya_.py), the data they produce (data/polya_.json) and two figures, described in a new section of Appendix A, and with twenty new bibliography entries. The catalog of results has a unit for the chapter and the repository a summary of it.

Outside the chapter the text of Chapter 16 changed in four places: one sentence of Chapter 1 that describes the new chapter, one clause of the preface that credits it, the new section of Appendix A for the scripts of Chapter 16, and one sentence of Appendix A on the runs of the new scripts.

What Chapter 17, "Yoshida's Hermitian forms, Weil positivity and the Thorin programme", adds:

• Yoshida's Hermitian form of Weil's explicit formula on a window read inside the programme: the heat trace of the zeros and every rung of its ladder are diagonal values of the form on Hermite Gaussian vectors, and RH is positivity of the odd rungs.
• The tent coefficient as the sine squared lemma summed over the zeros, and Yoshida's matrix as, up to signs, a Löwner matrix of an odd meromorphic function that is a Pick function exactly when RH holds.
• The form damped at a basepoint to the right of every zero as a Fourier multiplier, with a sharp lower bound for its least eigenvalue on every window.
• An uncertainty barrier on what prime side computations of the tent coefficient detect above the verified height, and a double precision computation that the tent coefficient is positive up to level 47.48.
• Open problems, with the progress made on each. The chapter does not prove RH.

What Chapter 18, "Zhu's compact windows and the Landau Widom horizon", adds:

• Xuefeng Zhu's 2026 preprint on Weil positivity in compact windows read in the language of Thorin measures: a dictionary that makes the form, under RH, a sampling functional of the Thorin measure on Paley Wiener functions, and Zhu's test function through a truncated reciprocal Xi transform.
• A two parameter bound for the least eigenvalue of the damped form, and its coercivity for basepoints above 1 from the Euler product.
• The Landau Widom horizon, the least eigenvalues computed at 60 digits, and a detection experiment with a hypothetical quadruple of off line zeros. The passage from windows to heights is a heuristic, and four problems are open.

What Chapter 19, "Three families of generalized gamma convolutions and the deconvolution obstruction", adds:

• Where the difficulty of RH sits in the Esscher tilt, the basepoint family and the de Bruijn Newman heat flow: the tilt is invertible and carries no information, the basepoint family is a Poisson semigroup to the right of the zeros and fails to be one at the levels with a zero to their right, and RH is equivalent to deconvolving the Thorin density at such a basepoint into a positive measure, an unbounded operation.
• The order of the error of smoothed prime sums for Dirichlet L functions, sharp under GRH for alpha below 3/2, and a theorem that no property preserved by Gaussian averaging and implying real zeros holds for any H_t.
• The visibility curve of a hypothetical off line quadruple of zeros, in closed form in three lattice models, with about a quarter of log(h/2 pi) levels needed at height h.

What Chapter 20, "Deconvolution as a damped Weil criterion", adds:

• The deconvolved object, Weil's distribution, exists without hypothesis, and its Fourier dual is a damped Weil criterion: Weil's form damped by exp(-epsilon |x|) is nonnegative exactly when no zero has real part above 1/2 + epsilon. The case epsilon = 0 is Weil's criterion, epsilon = 1/2 holds unconditionally, and the zero free half plane Re s > 7/8 announced in October 2026, not refereed and not assumed, is epsilon = 3/8.
• A ladder of finite problems in the primes up to exp(2a) by restricting the test functions to a window, with the least eigenvalue decreasing to 2 xi'/xi(1/2 + epsilon) at an algebraic rate in a under the stated hypotheses, and faster than any power of a at epsilon = 0 under RH.
• Eigenvalue computations that confirm the rates, and open problems.

What Chapter 21, "Thorin representations, Dirichlet L functions and the Linnik constant", adds:

• The reciprocal of a completed Dedekind zeta function, tilted by u_0 at least 0, is a generalized gamma convolution exactly when every nontrivial zero lies on the critical line or is a real zero within the square root of u_0 of the centre (and there is no central zero at u_0 = 0), with Thorin measure the shifted zero counting measure.
• For cyclotomic fields the Thorin property at the centre is equivalent to GRH for the Dirichlet L functions that do not vanish at the centre, which gives only K = 7 for the Linnik constant K, the value GRH already gives: nothing in the chapter lowers K.
• The centre clock of xi (moments, transfer function, negative moments), the sine squared lemma for Dirichlet characters, real and complex characters, and mixtures over a family of characters, which are not generalized gamma convolutions in general.

Chapters 17 to 21 come with 26 new scripts (code/yo_.py, zh_.py, tf_.py, dc_.py, gl_*.py), 21 data files, nine figures and 30 new bibliography entries, described in five new sections of Appendix A. The catalog of results has a unit for each chapter and the repository a summary of each.

Outside the six chapters the text changed in the sentence of Chapter 1 that describes the new chapters and the clause of the preface that credits them, which now cover Chapters 17 to 21 as well, and in Appendix A, which has one new section for the scripts of each chapter and whose last section moves up six numbers.

What the merged packages of Nick and of Newton add (all merged after a review against the book, with repaired defects listed in misc/catalog/QUESTIONS.md, Q34 to Q37):

• The Thorin set of the reciprocal Xi function (archive vadim_thorin_xi.zip): the signed Thorin representation with explicit pole factors; the threshold alpha_0 of the sine squared lemma and the generalized gamma convolution Psi at s^(1/4); the Hardy c...

Computing the Riemann Hypothesis: Thorin Measures, Pólya Densities and Hitting Times, version 1.1.0

Version 1.1.0

This version adds two chapters after Chapter 13 and before the appendices: Chapter 14, "The Pólya density as a maximum order statistic and the Thorin measure of the clock", after the first author's manuscript of October 2026, and Chapter 15, "The Thorin phase diagram", after a working report of the authors of October 2026. The chapters, sections, theorems, equations, exercises and problems of version 1.0.0 keep their numbers; only the last section of Appendix A, on the reruns, moves up one number, and the page numbers change.

What Chapter 14 adds:

• The Pólya kernel of the Riemann Xi function as the density of the square root of a sum of two independent copies of the maximum of independent exponential variables, and the Khinchin factorisation of the tilted maximum through the zeta distribution.
• Wald couples, with the logistic law as a normal scale mixture with Kolmogorov mixing read through the maximum, and the clock in a Wald couple with the Pólya variable.
• A criterion for the Thorin condition in the variable -x-iη: the clock is a generalised gamma convolution if and only if a smoothed Thorin density is nonnegative, and a zero off the critical line shows as an atom of negative mass at an explicit point.
• An explicit generalised gamma convolution within a rigorously bounded Laplace distance of the clock, built from the zeros verified up to height 3 x 10^12 and with no assumption on the zeros above it; the bound is computed in ball arithmetic.
• The term of one zero in the Thorin density of the basepoint family in the variable λ, and a finite form of the basepoint density below α = 1 with explicit pole terms and a Fejér taper.
• A proof that the clock is not a maximum of independent exponential variables, and the Gumbel norming constant of the clock from Gnedenko's extreme value theory.
• The finite stages of the maximum and their symmetrisations, with the heights up to which their zeros lie on the critical line.
• Hyperbolic complete monotonicity of the clock to a finite order, and the equivalence of the Riemann Hypothesis with membership of the clock in the Bondesson class.
• Open problems and a table of the numerical checks behind the chapter.

The chapter comes with twelve new scripts (code/maxwald_.py), the data they produce (data/maxwald_.json) and nine figures, and with twelve new bibliography entries. The catalog of results has a unit for the chapter and the repository a summary of it.

What Chapter 15 adds:

• A classification of the computations of the book by what they can see above the height to which the zeros are verified, and the three routes that reach every height.
• The threshold theorem along the de Bruijn Newman flow: at every basepoint alpha >= 1/2 the clock of xi_t is a generalised gamma convolution exactly when no zero of xi_t lies to the right of alpha.
• The Thorin phase diagram: de Bruijn's strip theorem in Thorin form, which carries the certified points (t, alpha) along parabolas and gives the de Bruijn Newman constant as the minimum of t + 2(alpha - 1/2)^2 over them, with the known results placed on it (the critical strip, Polymath15's Lambda <= 0.22, Platt and Trudgian's Lambda <= 0.2, and a conditional half plane).
• The Thorin density at positive time, computed from Pólya's integral with no primes and no zeros, and how little it differs from that at time zero on the windows computed.
• A certificate for one point of the diagram with explicit constants, and its finite part checked in ball arithmetic below height 160.
• Two experiments on where the missing uniform step could come from: the Weil Gram matrix on compactly supported test functions and the motion of the zeros along the flow, with a collision of two zeros under the backward flow located and bounded.
• The cost of the flow route, read from the table of conditional bounds of Polymath15: the verification height that each bound on Lambda needs, and what that implies for bounds below 0.15.
• A programme, five open problems and fourteen exercises.

The chapter comes with seven new scripts (code/phase_.py), the data they produce (data/phase_.json) and five figures, described in a new section of Appendix A, and with two new bibliography entries, the authors' working report and Chapter 21 of their draft on the Langlands conjectures. The catalog of results has a unit for the chapter and the repository a summary of it.

Outside the two chapters the text changed in six places: two sentences of Chapter 1 that describe the new chapters, one sentence in the introduction to Chapter 8 that points to Chapter 14, one clause of the preface that credits the manuscript and the working report behind them, one row of Appendix A that names the scripts of Chapter 14, the new section of Appendix A for the scripts of Chapter 15, and one sentence of Appendix A on the runs of the new scripts.

Computing the Riemann Hypothesis, version 1.0.0

Version 1.0.0, edition 1: the first release of the book's text, source, code and data.

• The text, figures and PDF are licensed CC BY 4.0, and the code MIT (LICENSE, code/LICENSE).
• The code and data are published here from this version on. Until now they were in https://github.com/VadimSokolov/computing-rh-code, which points here; its folder misc/repro/ is repro/ here.
• The paperback printed on 7 October 2026 holds an earlier state of the text. This version adds the material merged on 9 to 11 October 2026 (Sections 4.3.3, 5.1, 8.4, 8.5.4 to 8.5.7, 8.7, 9.1.6, 10.1.5, 10.3, 11.2, 11.4 and 11.5) and the corrections made in review since then. A revised printing of the paperback is made from this version.
• results.yaml and RESULTS.md list 250 results of the book (185 proved, 10 conditional, 52 open, 3 counterexamples) with their status, page and origin, and summaries/ has a summary of each chapter and appendix.