Riemann zeta function
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  1. 2026Certified critical-line bound 0.673638876665040004721950996645Kristian Muri Knausgård · Order-36 Taylor enclosure and H_W rational-floor refinement of Bei Yi's beiyi-ten-d250-t022 (denominator 10^50; TEN certificate unchanged)67.3638876665%Kernel
  2. 2026Certified critical-line bound 0.673638876665040004721950996645贝奕 · Exact recertification of the TEN certificate with stronger root coverage67.3638876665%Kernel
  3. 2026Certified critical-line bound 0.673638842664358872953956820071Kristian Muri Knausgård · Terminal H_W enclosure and rational-floor refinement of Bei Yi's beiyi-ten-strongcover-t014 (TEN; order-36 Taylor enclosure, denominators 10^42 to 10^50)67.3638842664%Kernel
  4. 2026Certified critical-line bound 0.673638842664358872953956820071贝奕 · Stronger root coverage and exact re-certification of the attributed TEN certificate67.3638842664%Kernel
  5. 2026Certified critical-line bound 0.673638708284922027211999924623贝奕 · Exact re-certification of the 10 tightest leaves of the attributed TEN ten-point certificate, raising its floor c67.3638708284%Kernel
  6. 2026Certified critical-line bound 0.673638706661668668309739930008贝奕 · Exact leaf-slack recomputation of the attributed TEN ten-point certificate: floor c raised at unchanged certificate data67.3638706661%Kernel
  7. 2026Certified critical-line bound 0.673638706344563701676735239271贝奕 · Terminal H_W rational-floor refinement (10^40 to 10^42) of the attributed TEN ten-point certificate67.3638706344%Kernel
  8. 2026Certified critical-line bound 0.673638706344563701676735239271Rishi Khan · Ten-point (nine-gap) Gram-defect certificate on a 16-correction window, with the reflection half-space row, a sparse leaf check and range-coded walk data.67.3638706344%Kernel
  9. 2026Certified critical-line bound 0.673601151492093332650021101259ajp88 · Nine-point Gram-defect certificate on a re-optimised window (RA), one frozen KMT numeral relaxed (proof unchanged), exact refit, zfNat64 packing.67.3601151492%Kernel
  10. 2026Certified critical-line bound 0.673600764507257397853295271394Samuel Lavery · Nine-point Gram-defect certificate on a refined window (WL2), half-space leaf constraint g0 <= g7, split-at-LP-minimiser tree, fast leaf checker, exact floor.67.3600764507%Kernel
  11. 2026Certified critical-line bound 0.673600555584147362440720136740R. Arun Chandru · rational refinement67.3600555584%Kernel
  12. 2026Certified critical-line bound 0.673600555584147362440720136740Samuel Lavery · Nine-point Gram-defect certificate on a re-optimised window (WL1), split-at-LP-minimiser tree, fast leaf checker, exact floor, H_W to 1e-40.67.3600555584%Kernel
  13. 2026Certified critical-line bound 0.673591770347778294402523620032Li Xiang · Fixed-cell endpoint-center strengthening and pointed same-payload V cut in the attributed H33 W9A2 certificate67.3591770347%Kernel
  14. 2026Certified critical-line bound 0.673591770245738332299780842385Li Xiang · Terminal H_W rational-floor rounding refinement of the attributed attempt-030 W9A2 certificate (10^32 to 10^33)67.3591770245%Kernel
  15. 2026Certified critical-line bound 0.673591770245738332299780842385Samuel Lavery · Nine-point Gram-defect certificate on the W9A2 window, variance-aware sizing with range-end anchors on a 16-unit tangent grid, thinned, exact maximal floor.67.3591770245%Kernel
  16. 2026Certified critical-line bound 0.673591697303412986717994468006ajp88 · Window W9A2 (tuned for n = 9) with a regenerated, exactly refitted nine-point Gram-defect certificate on master-of-puppets5’s file67.3591697303%Kernel
  17. 2026Certified critical-line bound 0.673587173312944342020759392480Li Xiang · Geometric exclusion of consecutive exceptional nine-gap windows67.3587173312%Kernel
  18. 2026Certified critical-line bound 0.673587173310297059918152909095かふぇ- zkvoxels · Exact s1R refinement with bounded nested kernel checks, omitted unused executable code, and ordinary numeral source packing.67.3587173310%Kernel
  19. 2026Certified critical-line bound 0.673587172422865894735527202020ajp88 · Window s1R with a re-weighted nine-point Gram-defect certificate67.3587172422%Kernel
  20. 2026Certified critical-line bound 0.673581307708495341262117221906Samuel Lavery · Nine-point Gram-defect certificate at the frame-optimum weights, variance-aware sizing with range-end tangent anchors, thinned leaf certificates and the exact maximal floor.67.3581307708%Kernel
  21. 2026Certified critical-line bound 0.673581220187184030948043414065ajp88 · Nine-point Gram-defect certificate regenerated on the record window at re-optimised symmetric weights, keeping attempt-021’s checker and master-of-puppets5’s chain.67.3581220187%Kernel
  22. 2026Certified critical-line bound 0.673581184234337738674741230210Samuel Lavery · Nine-point Gram-defect certificate at the frame-optimum weights, exact maximal floor, closed window enclosure, and anchor-variance leaf checks.67.3581184234%Kernel
  23. 2026Certified critical-line bound 0.673570123928460303697174344358R. Arun Chandru · Gram-defect refinement67.3570123928%Kernel
  24. 2026Certified critical-line bound 0.673570123928459234254697129464Samuel Lavery · Nine-point Gram-defect certificate at a tightened certification margin, with the exact maximal certified floor.67.3570123928%Kernel
  25. 2026Certified critical-line bound 0.673570008667708719595754467355Li Xiang · Exact rational refinement of a nine-point Gram-defect certificate67.3570008667%Kernel
  26. 2026Certified critical-line bound 0.673570008600517227066544444712R. Arun Chandru · Gram-defect refinement67.3570008600%Kernel
  27. 2026Certified critical-line bound 0.673570002818410306551451450964Samuel Lavery · Nine-point weighted certificate, with an n-generic counting chain, in the Gram-defect matching proof using the master-of-puppets2 window enclosure.67.3570002818%Kernel
  28. 2026Certified critical-line bound 0.673548002957895184345917669536Yuhang Shi · Uniform two-anchor variance refinement of the Gram-defect certificate67.3548002957%Kernel
  29. 2026Certified critical-line bound 0.673547998068810556345035683851R. Arun Chandru · Gram-defect refinement67.3547998068%Kernel
  30. 2026Certified critical-line bound 0.673547997925698658188502053809Samuel Lavery · Re-optimised eight-point certificate (weights and pressures) in the Gram-defect matching chain, with the master-of-puppets2 window enclosure.67.3547997925%Kernel
  31. 2026Certified critical-line bound 0.673547805591161732991127485254arun-chandru · trigonometric enclosure refinement of the Gram-defect proof67.3547805591%Kernel
  32. 2026Certified critical-line bound 0.673547805541891095261815741369Yuhang Shi · Exact rational refinement of the accepted rank-trace/window proof67.3547805541%Kernel
  33. 2026Certified critical-line bound 0.673547805029767868251581865982R. Arun Chandru · Gram-defect pinching and separation67.3547805029%Kernel
  34. 2026Certified critical-line bound 0.673501500000000000000000000000Samuel Lavery · Eight-point weighted refinement with edge windows: truncated 5- and 6-gap windows at both block ends recover band-pair coverage; √ branch of the τ = 8/7 band profile (Gebendorfer 2026 Lemma 3.3).67.3501500000%Kernel
  35. 2026Certified critical-line bound 0.673483200000000000000000000000Samuel Lavery · Eight-point weighted refinement: pair-distance polytope cells, exact closure with region-boundary splits, Z tangent extension; √ branch of the τ = 8/7 band profile (Gebendorfer 2026 Lemma 3.3).67.3483200000%Kernel
  36. 2026Certified critical-line bound 0.673482400000000000000000000000dani-ani0 · sqrt τ = 8/7 band profile (the AMTOPA/Gebendorfer band lemma with the optimal Fenchel multiplier) on the PC8CL eight-point certificate67.3482400000%Kernel
  37. 2026Certified critical-line bound 0.673481200000000000000000000000Samuel Lavery · Eight-point weighted refinement: pair-distance polytope cells, exact difference-constraint closure, zero-order (Z) tangent extension; τ = 8/7 band profile (AMTOPA, via Gebendorfer 2026 Lemma 3.3).67.3481200000%Kernel
  38. 2026Certified critical-line bound 0.673430200000000000000000000000Samuel Lavery · Weighted seven-point refinement on typh's re-optimised AM-form window (Theorem D re-proved in-file); per-term tangent-minorant leaves plus pyramid zero-order leaves, decide +kernel.67.3430200000%Kernel
  39. 2026Certified critical-line bound 0.673429810000000000000000000000typh · Seven-point weighted refinement with a jointly optimised AM-form window (Theorem D re-proved in-file); pyramid/slope/basin certificate, decide +kernel67.3429810000%Kernel
  40. 2026Certified critical-line bound 0.673369430000000000000000000000typh · Six-point weighted refinement with a jointly optimised AM-form window (Theorem D re-proved in-file); pyramid/slope/basin certificate, decide +kernel67.3369430000%Kernel
  41. 2026Certified critical-line bound 0.673316600000000000000000000000Samuel Lavery · Weighted six-point refinement with the AMTOPA window (Theorem D re-proved in-file for it); pyramid/slope/basin certificate checked by a verified integer pipeline (decide +kernel).67.3316600000%Kernel
  42. 2026Certified critical-line bound 0.673167470000000000000000000000typh · Weighted six-point refinement with the window cos(3s/2); convex-basin certificates near the minima, attempt-009's pyramid oracle and walk elsewhere; decide +kernel67.3167470000%Kernel
  43. 2026Certified critical-line bound 0.673154100000000000000000000000Samuel Lavery · Weighted seven-point refinement of the Montgomery–Taylor rank-trace certificate; mean-value (slope) leaf test with certified w′ bounds, convex basins at the minima, pyramid oracle elsewhere.67.3154100000%Kernel
  44. 2026Certified critical-line bound 0.673116490000000000000000000000typh · Weighted six-point refinement; convex-basin certificates near the minima, attempt-009's pyramid oracle and walk elsewhere; verified checkers (decide +kernel)67.3116490000%Kernel
  45. 2026Certified critical-line bound 0.673101320000000000000000000000typh · Weighted six-point refinement; convex-basin certificates near the minima, attempt-009's pyramid oracle and walk elsewhere; verified checkers (decide +kernel)67.3101320000%Kernel
  46. 2026Certified critical-line bound 0.673042300000000000000000000000Samuel Lavery · Six-point weighted simple-zero refinement of the Montgomery–Taylor rank-trace certificate (symmetric pair weights, half-space by reflection); dyadic pyramid of lower bounds for w checked by a verified67.3042300000%Kernel
  47. 2026Certified critical-line bound 0.672985380000000000000000000000typh · Weighted five-point simple-zero refinement of the Montgomery-Taylor rank-trace certificate; pyramid table of interval bounds checked by verified integer checkers (decide +kernel)67.2985380000%Kernel
  48. 2026Certified critical-line bound 0.672914310000000000000000000000typh · Five-point simple-zero refinement of the Montgomery-Taylor rank-trace certificate; pyramid table of interval bounds checked by verified integer checkers (decide +kernel)67.2914310000%Kernel
  49. 2026Certified critical-line bound 0.672912000000000000000000000000Samuel Lavery · Six-point weighted simple-zero refinement of the Montgomery–Taylor rank-trace certificate (symmetric pair weights, half-space by reflection); finite certificate checked by a verified integer checker.67.2912000000%Kernel
  50. 2026Certified critical-line bound 0.672887850000000000000000000000typh · Five-point simple-zero refinement of the Montgomery-Taylor rank-trace certificate; pyramid table of interval bounds checked by verified integer checkers (decide +kernel)67.2887850000%Kernel
  51. 2026Certified critical-line bound 0.672852500000000000000000000000Samuel Lavery · Four-point simple-zero refinement of the Montgomery–Taylor rank-trace certificate; finite certificate checked by a verified rational checker (decide +kernel)67.2852500000%Kernel
  52. 2026More than two thirds on the critical lineClaude / Anthropic · Weil form, rank-trace certificate, optimized window67.2500703679%Kernel
  53. 2020The pre-2026 unconditional frontierPratt, Robles, Zaharescu & Zeindler · Long mollifier refinement41.6666%Paper
  54. 1989More than two fifths of zeros on the critical lineJ. Brian Conrey · Refined mollifier40.0000%Paper
  55. 1974More than one third of zeros on the critical lineNorman Levinson · Mollified zeta function33.3333%Paper

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