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Fredrik Johansson 2016-09-10 12:14:10 +02:00 • committed by Pascal
parent f89bbc6daa
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23
doc/source/acb_dirichlet.rst
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@ -198,7 +198,7 @@ No discrete log computation is performed.
.. function:: ulong acb_dirichlet_number_primitive(const acb_dirichlet_group_t G)
return the number of primitive elements in *G*.
Return the number of primitive elements in *G*.
.. function:: ulong acb_dirichlet_ui_conductor(const acb_dirichlet_group_t G, ulong a)
@ -262,7 +262,7 @@ Roots of unity
.. function:: void acb_dirichlet_vec_nth_roots(acb_ptr z, slong order, slong prec)
compute the vector ``1,z,z^2,\dots z^{\mathrm{order}-1}`` where `z=\exp(\frac{2i\pi}{\mathrm{order}})` to precision *prec*.
Compute the vector ``1,z,z^2,\dots z^{\mathrm{order}-1}`` where `z=\exp(\frac{2i\pi}{\mathrm{order}})` to precision *prec*.
In order to avoid precision loss, this function does not simply compute powers of a primitive root.
@ -481,20 +481,23 @@ L-functions
.. function:: void acb_dirichlet_l_hurwitz(acb_t res, const acb_t s, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong prec)
Compute `L(s,\chi)` for `s\neq 1`, using decomposition in terms of Hurwitz zeta function
Compute `L(s,\chi)` using decomposition in terms of the Hurwitz zeta function
.. math::
.. math::
L(s,\chi) = q^{-s}\sum_{k=1}^{q-1} \chi(k) \zeta(s,\frac kq)
L(s,\chi) = q^{-s}\sum_{k=1}^{q-1} \chi(k) \,\zeta\!\left(s,\frac kq\right).
This formula is slow for large *q* (and not defined for `s=1`).
If `s = 1` and `\chi` is non-principal, the deflated Hurwitz zeta function
is used to avoid poles.
This formula is slow for large *q*.
.. function:: void acb_dirichlet_l_vec_hurwitz(acb_ptr res, const acb_t s, const acb_dirichlet_group_t G, slong prec)
Compute all values `L(s,\chi)` for `\chi` mod `q`, by Hurwitz formula and
discrete Fourier transform.
*res* is assumed to have length *G->phi_q* and values are stored by lexicographically ordered Conrey
index. See :func:`acb_dirichlet_dft_conrey`.
Compute all values `L(s,\chi)` for `\chi` mod `q`, by Hurwitz formula and
discrete Fourier transform.
*res* is assumed to have length *G->phi_q* and values are stored by lexicographically ordered Conrey
index. See :func:`acb_dirichlet_dft_conrey`.
Implementation notes
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