`acb_dirichlet_turing_method_bound`

Synopsis#include <flint/acb_dirichlet.h>

ulong acb_dirichlet_turing_method_bound(const fmpz_t p)source

Returns a number B of consecutive good Gram blocks that suffices for Turing's method [Tur1953] with the bounds of [Leh1970], [Bre1979] and [Tru2011]: if B consecutive Gram blocks with union [g_n, g_p) satisfy Rosser's rule, then N(g_n) \le n + 1 and N(g_p) \ge p + 1.

References#

[Tur1953] A. M. Turing, "Some Calculations of the Riemann Zeta-Function", Proc. of the London Mathematical Society 3(3) (1953), 99-117, https://doi.org/10.1112/plms/s3-3.1.99
[Leh1970] R. S. Lehman, "On the Distribution of Zeros of the Riemann Zeta-Function", Proc. of the London Mathematical Society 20(3) (1970), 303-320, https://doi.org/10.1112/plms/s3-20.2.303
[Bre1979] R. P. Brent, "On the Zeros of the Riemann Zeta Function in the Critical Strip", Mathematics of Computation 33 (1979), 1361-1372, https://doi.org/10.1090/S0025-5718-1979-0537983-2
[Tru2011] T. S. Trudgian, "Improvements to Turing's method", Mathematics of Computation 80 (2011), 2259-2279, https://doi.org/10.1090/S0025-5718-2011-02470-1

Cite this page

The FLINT team, “acb_dirichlet_turing_method_bound (acb_dirichlet.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_dirichlet/acb_dirichlet_turing_method_bound.html (generated 2026-10-20).

@misc{flint-doc-acb-dirichlet-acb-dirichlet-turing-method-bound,
  author       = {{The FLINT team}},
  title        = {{acb\_dirichlet\_turing\_method\_bound (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_turing_method_bound.html},
  note         = {Generated 2026-10-20}
}

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