Low-lying zeros of 795,381 quadratic Dirichlet L-functions up to conductor 1e10: the ratios conjecture, consecutive families and the first zero - a reproducible numerical study III

Third study of the series (study I: doi:10.5281/zenodo.23815328; study II: doi:10.5281/zenodo.23928017). For 795,381 quadratic Dirichlet L-functions L(s, chi_D) - eleven families of fundamental discriminants from |D| ~ 1e3 to 1e10, among them every discriminant with 1e6 < |D| <= 2e6 and 60,000 discriminants drawn at random - the low-lying zeros were computed with PARI/GP 2.17.2 on two computers; the 69,452 L-functions computed on both have identical zeros. The one-level density of the unfolded zeros is compared with the Katz-Sarnak limit USp, with the finite matrix USp(2N) and with the ratios conjecture of Conrey, Farmer and Zirnbauer in the form of Conrey and Snaith (2007), evaluated for every discriminant without a free parameter. The ratios conjecture describes all families (chi2 per bin 0.54-1.19 against about 0.95 expected for independent L-functions; density in [0, 1/2] for the 607,941 L-functions with 1e6 < |D| <= 2e6: 0.6082 +- 0.0014 against 0.6084), while the USp limit gives chi2 per bin of 5 to 850 in the families of at least 6,000 L-functions and is still 12.6 standard errors away at |D| = 1e10 (0.527 +- 0.009 against 0.411). Families of consecutive discriminants show chi2 below the value for independent L-functions; a cosine decomposition of the deviations puts the deficit entirely at low frequencies (u < 1): bands of 6,074 consecutive discriminants have 0.23, 0.36 and 0.48 of the low-frequency power of random samples at |D| ~ 5e4, 1.5e6 and 1e8 (5.9, 10.2 and 4.3 standard errors), with no difference at high frequencies. The effect follows from the explicit formula: in a stretch of consecutive discriminants the averages of chi_D(p) for primes below the length of the stretch run over nearly complete periods; subsets assembled from m shorter pieces lose it gradually with m, while balancing the residue classes of D - tested for 19 moduli from 3 to 840 including the prime 131, and for the sign alone, with an independent repetition - does not reproduce it. Without the zeros, the remaining variance of the sums of chi_D(p) over a subset orders the low-frequency power of the zeros of all 60 stratified subsets and of the consecutive bands along the line fixed by random samples and bands; the size agrees only at the edge (chi2 = 82.05 for 60 points with the joint errors, p = 0.03), the stratifications lowering the power somewhat less than predicted. The expected low-frequency power of any subset design follows without random draws (exact layer 0, error from the expected count); the offset of the test by a few per cent is a bias of the two lowest bins; triples of consecutive discriminants already lie below random samples; the cosine modes k and 8 - k of one L-function are coupled through products chi_D(p)chi_D(q) of small primes; and the point where the distribution of the first zero of even characters crosses USp(2N_eff) follows the end of the interval on which N_eff is fixed, so it is not a property of the zeros. A recomputation of every drawn number by layers, with the same rules and designs, keeps the conclusions and fixes their sizes: the stratifications do lower the low-frequency power in all three large families (D mod 840: 26.6 / 13.5 / 2.9 standard errors), but the line through two points is rejected in size (chi2 = 841.8 for 60); the weights of chi_D(p) in single L-functions follow the explicit formula only with the dual term of the ratios conjecture (slope 0.995-0.999 instead of 0.52-0.60); the shift of the mean first zero of even characters depends on the calibration of N_eff, the deviation of its shape does not. A separate section states what is new, as far as we have been able to find, and what confirms known theory. For the first zero, USp(2N) with N = ln(|D|/pi)/2 fails, whereas USp(2N_eff) with N_eff fixed by the ratios conjecture gives the whole distribution for odd characters within the errors at every conductor; for even characters the first zero lies higher by a lower-order term decreasing roughly as 0.094/ln(|D|/pi). An enlargement at |D| = 10^10 to 121,557 L-functions (Section 4.8) makes the errors at this conductor about three times smaller: the effect of consecutive families persists (0.62 of the power of random samples), and the even-character term of the first zero falls faster than 1/ln(|D|/pi). Article in English and Czech (PDF, DOCX, LaTeX, Markdown), figures, data, the generator of the zeros and the scripts that reproduce every number, with an independent check; the zeros themselves are regenerated by the generator and checked against the SHA-256 of every block. Text, figures and data CC BY 4.0, code MIT. Nothing here bears on the generalised Riemann hypothesis.