Beyond the diagonal approximation: pair correlation, form factor and resonances of the Riemann zeros from t ~ 1e4 to 1e21 - a reproducible numerical study II

Follow-up to the study of Berry saturation (doi:10.5281/zenodo.23815328). The same 18 blocks of zeros of the Riemann zeta function from t ~ 1e4 to 1e21 (LMFDB; Odlyzko's first 2,001,052 zeros; Odlyzko's tables near the zeros number 1e12, 1e21 and 1e22) are compared with the Bogomolny-Keating formula (1996) for the pair correlation at finite height, which adds to the diagonal approximation the off-diagonal terms built on the Hardy-Littlewood conjecture and has no free parameter. Pair correlation in 60 bins up to three spacings: chi2/60 = 0.62-1.09 in all 18 blocks (1.14 on two million zeros), GUE 57.7, the formula with the zeta function on the 1-line only (Phi_o = Phi_d = 1) 47.8. The 1 % residual of Sigma2 from the first study is resolved: in blocks of 200,000 zeros Berry's diagonal formula deviates by 2.4-3.8 standard errors, the Bogomolny-Keating Sigma2 by at most 0.93, the zeta-only formula by 9-48; removing the two arithmetic factors one at a time shows that this failure is carried by the prime-power repetitions Phi_d (4.4-4.6 % without them), while the Hardy-Littlewood factor Phi_o alone is a 0.5-0.7 % effect; at short distances the two are tied by 2 sum ln^2 p/(p-1)^2 = 2.771. Crossover scale L* within 7 % of the formula for H = 7-47 (smooth crossover, no threshold). Spectral form factor: diagonal peaks at tau = ln n/H with weights Lambda(n)^2/n within 0.5 %; off-diagonal peaks at tau = 1 + ln(A/B)/H with weights A/(B phi(A)^2) for squarefree A (from the Euler product in the formula; the coefficients mu^2(A)/phi(A)^2 of Hardy's 1921 Ramanujan-sum expansion of the singular series), RMS 0.010 against 0.33 without Hardy-Littlewood. Resonances of the low zeros (Berry-Keating 1999): dips of 4.5-6.2 % in the pair correlation at distances eps = gamma_1 ... gamma_10 in units of t, reproduced within 0.001 (chi2/200 = 0.20 against 262 for GUE), and up to eps = 103 (gamma_1 ... gamma_30) with chi2/408 = 0.19 against 339 for GUE and 10.1 without the repetitions, leading-order depth 8/H^2 (the -4 of the scaled correction in Fig. 4 of Berry-Keating 1999); peaks of Sigma2(L) at L = gamma_k at five heights, reproduced by Berry's formula (RMS 0.0015-0.0054) but not by random primes (0.010-0.077). On a fine grid of L, Sigma2 shows the fast period-1 oscillation of the term e^{2 pi i x} q (Berry-Keating 1999, eq. 4.27): (5.4 +- 1.2)e-4 against 5.5e-4 in zeros6, (-1.45 +- 0.06)e-3 against -1.39e-3 for the first 100,000 zeros. Three further items of the review of Berry and Keating are checked on numbers: the decorated factorial of the Riemann-Siegel coefficients (Berry 1995; C_r(z) to 200 digits for r <= 100, residual falling as r^-0.55+-0.05, the leading term alone stays at 12-17 %), Riemann's formula for pi(x) (1,000 zeros give pi(n) correctly by rounding for every n <= 999; smooth zeros or shuffled spacings of the same density fail, RMS 0.45-0.55 against 0.012) and the explicit formula with a Gaussian test function (zeros and prime powers agree to a relative 3e-13; smooth or shuffled zeros give no lines). The low zeros of families of L-functions are the subject of study III of this series. Article in English and Czech (PDF, DOCX, LaTeX, Markdown), figures, data and the scripts that reproduce every number, with an independent check; SHA-256 of every file. Text, figures and data CC BY 4.0, code MIT. Nothing here bears on the Riemann hypothesis.