HEXAGONbeta

The Riemann hypothesis as a margin: a two-sided criterion verified to 2*10^29, the fourth-moment anatomy of Mobius cancellation, and one constant seen from every side

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CV — Complex Variables

Contributed by Pedro Caceres ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

This paper collects, with uniform notation, a month of new results around one way of looking at the Riemann hypothesis (RH): as the statement that explicitly computable margins stay on one side of explicit constants for ever. Part I studies the Cesaro margin M(N) between a smooth curve and the weighted prime sum 2*sum_{q<N} Lambda(q) q^{-1/2} log(N/q): RH is equivalent to M(N) >= 0 for all large N (a theorem of Suzuki), and we verify 0 < M(N) < 0.18484 for every real 1 < N <= 2.15*10^29, with explicit unconditional bounds beyond; the margin is analyzed as a walk of the primes against a curve, as an energy, and as a one-sided mean: RH holds if and only if a weighted mean of the prime-counting error stays below -B = -(1 + gamma/2 - (1/2) log 4pi) for ever. Part II dissects the Mobius side of the same coin: an exact positive-definite variance identity for sum_{d<=D} mu(d)({x/d}-1/2), empirically Gaussian statistics, and an exact anatomy of the fourth moment — the conductor-1 orbit decouples, the cross term moves only at composite squarefree conductors, two of its four sectors are O(D log^4 D) unconditionally, and an empirical jump law, first refuted, is proved exactly in its sector. Part III shows that 2B is the Baez-Duarte constant of the Nyman-Beurling criterion and recomputes the Nyman-Beurling distances from scratch through Vasyunin's cotangent formula: d_N^2 log N reaches 0.0461 against 2B = 0.04619, with no zero of zeta used anywhere in the computation. Part IV distils the author's twenty-two-part study of the cross-energy of consecutive Dirichlet partial sums: the trichotomy that singles out the critical line as the trace of the pole, the energy law (the mean energy of the prime-counting error is >= 2B unconditionally, with equality if and only if RH holds with simple zeros), the energy slope as a truncated Weil functional exact in the primes, the functional reflection that reproduces Hardy's function from the integers, and the systematic Davenport-Heilbronn and Beurling controls. Part V states precisely what none of this can do: a no-go theorem shows that weights indistinguishable from the primes' to q^(delta-1/2) satisfy every averaged statement here and still carry an off-line zero, so the exact identity Lambda = mu * log must enter beyond its first moments. The paper opens with a list of its twenty-two original contributions and a matching statement of what is due to others. We do not prove the Riemann hypothesis, and no statement in this paper should be read as progress towards it beyond what it literally says.

Provenance statement

The research programme and the choice of objects are the author's. Claude (Anthropic) was used throughout as a research and writing aid: to derive and check lemmas, to write and run the computations, to referee drafts in independent sessions, and to draft text and figures. The author checked all results and is responsible for the content. All exact rational claims were reproduced independently in separate arithmetic during review.

Tools used

Anthropic
Claude FableVersion 5
Python
PythonVersion 3.13

References

  1. J. Franel, E. Landau, Les suites de Farey et le problème des nombres premiers, Gött. Nachr. (1924), 198--206.
  2. L. Báez-Duarte, A strengthening of the Nyman--Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 14 (2003), 5--11.
  3. L. Báez-Duarte, M. Balazard, B. Landreau, E. Saias, Notes sur la fonction ζ\zeta de Riemann, 3, Adv. Math. 149 (2000), 130--144.
  4. S. Bettin, J. B. Conrey, D. W. Farmer, An optimal choice of Dirichlet polynomials for the Nyman--Beurling criterion, Proc. Steklov Inst. Math. 280 (2013), suppl. 2, S30--S36; arXiv:1211.5191.arXiv
  5. A. Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci. USA 41 (1955), 312--314.
  6. J.-F. Burnol, A lower bound in an approximation problem involving the zeros of the Riemann zeta function, Adv. Math. 170 (2002), 56--70.
  7. P. Caceres, Cross-energy of Dirichlet partial sums: the critical line as the trace of the pole, the energy of the prime-counting error, the energy slope as a Weil functional, and the functional reflection, consolidated monograph (v3), preprint, 2026.
  8. G. Csordas, Fourier transforms of positive definite kernels and the Riemann ξ\xi-function, arXiv:1309.0055, 2013.arXiv
  9. H. Davenport, Multiplicative Number Theory , 3rd ed., Springer, 2000.
  10. B. Nyman, On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces , Thesis, Uppsala, 1950.
Show all 41 referencesShow fewer references
  1. V. I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, Algebra i Analiz 7 (1995), no. 3, 118--135; English transl., St. Petersburg Math. J. 7 (1996), 405--419.
  2. A. Connes, C. Consani, Weil positivity and trace formula, the archimedean place, Selecta Math. 27 (2021), 77; arXiv:2006.13771.arXiv
  3. A. Connes, C. Consani, H. Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423, 2023.arXiv
  4. J. Büthe, An analytic method for bounding ψ(x)\psi(x), Math. Comp. 87 (2018), 1991--2009.
  5. F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), 1281--1292.
  6. A. Akbary, N. Ng, M. Shahabi, Limiting distributions of the classical error terms of prime number theory, Q. J. Math. 65 (2014), 743--780.
  7. A. S. Besicovitch, Almost Periodic Functions , Cambridge University Press, 1932.
  8. E. Bombieri, Remarks on Weil's quadratic functional in the theory of prime numbers I, Rend. Mat. Acc. Lincei (9) 11 (2000), 183--233.
  9. A. Connes, The Riemann hypothesis: past, present and a letter through time, arXiv:2602.04022, 2026.arXiv
  10. H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series I, II, J. London Math. Soc. 11 (1936), 181--185, 307--312.
  11. S. M. Gonek, On negative moments of the Riemann zeta-function, Mathematika 36 (1989), 71--88.
  12. H. G. Diamond, H. L. Montgomery, U. M. A. Vorhauer, Beurling primes with large oscillation, Math. Ann. 334 (2006), 1--36.
  13. H. Iwaniec, E. Kowalski, Analytic Number Theory , AMS Colloquium Publications 53, 2004.
  14. A. E. Ingham, The Distribution of Prime Numbers , Cambridge Tracts 30, Cambridge University Press, 1932.
  15. D. R. Johnston, Improving bounds on prime counting functions by partial verification of the Riemann hypothesis, Ramanujan J. 59 (2022), 1307--1321.
  16. M. Rubinstein, P. Sarnak, Chebyshev's bias, Experiment. Math. 3 (1994), 173--197.
  17. H. L. Montgomery, R. C. Vaughan, Multiplicative Number Theory I. Classical Theory , Cambridge University Press, 2007.
  18. H. Kadiri, A. Lumley, N. Ng, Explicit zero density for the Riemann zeta function, J. Math. Anal. Appl. 465 (2018), 22--46; arXiv:2101.12263.arXiv
  19. N. Ng, The distribution of the summatory function of the Möbius function, Proc. London Math. Soc. (3) 89 (2004), 361--389.
  20. D. Platt, T. Trudgian, The Riemann hypothesis is true up to 3⋅10123\cdot10^{12}, Bull. Lond. Math. Soc. 53 (2021), 792--797.
  21. M. Suzuki, Aspects of the screw function corresponding to the Riemann zeta-function, arXiv:2206.03682, 2022 (v4, 2023).arXiv
  22. M. Suzuki, On variants of Chebyshev's conjecture, Ramanujan J. (2025), doi:10.1007/s11139-025-01238-9, arXiv:2411.07436.arXivDOI
  23. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., Oxford University Press, 1986, Section 10.25.
  24. T. S. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line II, J. Number Theory 134 (2014), 280--292.
  25. A. Voros, Zeta Functions over Zeros of Zeta Functions , Lecture Notes of the Unione Matematica Italiana 8, Springer, 2010.
  26. A. Weil, Sur les ``formules explicites'' de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund (1952), tome supplémentaire, 252--265.
  27. D. V. Widder, The Laplace Transform , Princeton University Press, 1941.
  28. P. Caceres, The Caceres Prime Function: a square-root damped prime weight, its exact Prime Zeta structure, and a dual spectrum of the Riemann zeros, preprint, 2026.
  29. P. Caceres, Cross-energy of Dirichlet partial sums. Part 20: The Euler cross-energy, Weil positivity and a one-variable form of the Riemann hypothesis, preprint, 2026.
  30. P. Caceres, Cross-energy of Dirichlet partial sums. Part 21: The margin as W−EW-E, a Hodge index for the integers, preprint, 2026.
  31. P. Caceres, Cross-energy of Dirichlet partial sums. Part 22: The minimizer of the Weil form in a window, and the scale 2πN2\pi N, preprint, 2026.

Version history

  1. v1Submitted by Pedro CaceresInitial depositCurrentOct 30, 2026