Checks · The unfolded zeros of the Riemann zeta function do not form a Riesz basis of exponentials

Recorded output over the first 10,000 zeros, 12 random tests

A run of verify_identity.py over the first 10,000 zeros, testing each identity 12 times, with 150 continuation steps; the file records neither these settings, which the paper states, nor its date, and the program's defaults differ (20,000 zeros, 40 tests). A_worst_resid, 5.46 × 10⁻²¹, is the largest residual of Theorem 3.1's identity, and A_rows lists the first eight tests: index, height, S just above and below the zero, the predicted value −d_n, and the residual. B_worst_resid and B_rows do the same for Proposition 3.2: the window's ends, the zeros in it, the identity's two sides and their difference. The rest are these zeros' Kadec and Avdonin statistics, with the best centering (cstar) and the largest displacement after it (Dstar_centered_sup), and part D's rows, unused by the paper.

Output of: Verification program for the displacement identities (checks/verify_identity.py), shown beside it on that page

Written by: verify_identity.py, a program by Claude Fable 5.1 (Anthropic)
Size: 3,685 bytes
SHA-256: b0dbb2bc6c582a3e5af0c6be25cc8329f402d2924a61531e72ff97e79f6abd5f

The paper's page Every file published with it This file on GitHub

Tables

A_worst_resid	5.4565149617825e-21
B_worst_resid	3.1420249976497117e-34
first_cross_index	9
first_cross_gamma	48.00515088116716
sup_abs_d	0.9484712971877421
argmax_abs_d	8571
cstar	-0.04761543832514535
Dstar_centered_sup	0.9008558588625968
mean_d	2.4925939240449597e-06
rms_d	0.2887772086241074
D_peak_index	8571
D_peak_Sbar	0.9484712971877421

A_rows

1034	1458.3937773686873	0.8796060435372085	-0.12039393912377122	0.3796060522067186	-5.4565149617825e-21
1538	2022.5456176846883	0.30762100246007157	-0.6923789791599672	-0.1923789883499478	-3.934520560191532e-21
1932	2443.7754363962	0.656760936168323	-0.3432390448495141	0.1567609456594044	-3.2563332731471017e-21
2202	2725.882989203482	0.4089786234473197	-0.5910213572227694	-0.09102136688772482	-2.9193282600025475e-21
3440	3966.1387149930442	0.38832760637560315	-0.6116723731008219	-0.11167238336260939	-2.0064218043281557e-21
4180	4677.93021914714	0.3563425942613315	-0.6436573846896825	-0.14365739521417548	-1.7011256714192052e-21
6220	6565.2258967254265	0.3006092534162361	-0.6993907244559265	-0.19939073551984518	-1.2121056145150045e-21
7365	7589.356254115857	0.11189115942397589	-0.8881088179867652	-0.3881088292813947	-1.0485404690012282e-21

B_rows

779.6808826812714	908.1548999471372	100	-0.21185498178847517	-0.21185498178847517	-1.4092501268554344e-36
133.86240046946332	336.24116026681247	115	0.3663859383303543	0.3663859383303543	-7.875467767398734e-38
9618.087290524816	9640.714476666857	27	-0.5603630331348909	-0.5603630331348909	3.1420249976497117e-34
716.1922562410142	896.4385978112027	139	-0.083246212549847	-0.083246212549847	2.293103640472491e-37
4082.324321442317	4187.840666699038	109	0.6307374477234058	0.6307374477234058	6.071859948829058e-35

D_M

1	0.9484712971877421
2	0.673987960256783
4	0.34433772588831457
8	0.1459547397794406
16	0.10263498809093101
32	0.04382089380729032
64	0.021762030674296422
128	0.01153554548046435
256	0.006021135605918106
512	0.002804113440288177
1024	0.001457061653705308
2048	0.0007169921879641276
4096	0.0003234124191063116

D_M_centered

1	0.9008558588625968
2	0.6502031325398118
4	0.3419596081650264
8	0.14303466205253024
16	0.10109006126969201
32	0.04291524575610985
64	0.021673636991767553
128	0.011314759310655662
256	0.005611283759366805
512	0.002729151898030215
1024	0.0014342657339760309
2048	0.0007142299304169166
4096	0.00031843146303661864

D_rows(N: min Sbar in last N, block mean, A-N)

2	[0.39950462332582387, 0.673987960256783, -1.051528702812258]
4	[-0.23858728697931353, 0.29791766770217837, -3.051528702812258]
8	[-0.23858728697931353, 0.08741246852737836, -7.0515287028122575]
16	[-0.23858728697931353, 0.08856096460715922, -15.051528702812258]
32	[-0.54538572873013, 0.017807635941357768, -31.05152870281226]

JSON

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