_acb_dirichlet_hardy_z_zeros_use_platt

The model compares three options.

• The Riemann–Siegel method costs about the same per zero; beyond the default precision, the last stage of the refinement takes four evaluations of $Z$ at the target precision (about 6 times the cost of a zero at the default precision at 192 bits, 12 times at 320 bits).
• The large height method with the double-word sums over $j$, at a working precision of at most $\log_2 n + 104$ bits (about $\log_2 n + 68$ bits of accuracy), has a fixed cost of a few seconds up to $n = 10^{15}$ for one multi-evaluation, which isolates thousands of zeros, then about 10 ms per zero, plus the last stage of the refinement for each zero if a higher precision is requested.
• The large height method with ball arithmetic sums at working precision prec + 48 costs 3 to 12 times as much per call and is limited in accuracy (its parameters depend only on the height) to about 253, 234, 219, 201, 187, 170 and 155 bits at $n = 10^{11}, \ldots, 10^{17}$.

At the default precision, the large height method is used from 158 zeros at $n = 10^{11}$, 51 at $10^{12}$, 14 at $10^{13}$, 10 at $10^{14}$, 3 at $10^{15}$, 2 from $10^{16}$, and (extrapolated) for a single zero from $10^{20}$. Without double-word arithmetic (x87 builds), the large height method is used from $10^{15}$, or from $10^{11}$ for more than 100 zeros. It is only used for $10^{11} \le n < 10^{23}$.

@misc{flint-doc-acb-dirichlet-acb-dirichlet-hardy-z-zeros-use-platt,
  author       = {{The FLINT team}},
  title        = {{\_acb\_dirichlet\_hardy\_z\_zeros\_use\_platt (acb\_dirichlet.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/acb_dirichlet/_acb_dirichlet_hardy_z_zeros_use_platt.html},
  note         = {Generated 2026-11-02}
}