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/*
Copyright (C) 2016 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_dirichlet.h"
#include "acb_poly.h"
void
acb_dirichlet_zeta_rs_f_coeffs(acb_ptr c, const arb_t p, slong N, slong prec)
{
arb_ptr R, I, T, X;
slong i, len;
R = _arb_vec_init(N);
I = _arb_vec_init(N);
T = _arb_vec_init(N);
X = _arb_vec_init(2);
arb_set(X, p);
arb_one(X + 1);
/* I, R = sin,cos(pi*(X^2/2 + 3/8)) */
len = FLINT_MIN(N, 3);
_arb_poly_mullow(T, X, 2, X, 2, len, prec);
_arb_vec_scalar_mul_2exp_si(T, T, len, -1);
arb_set_d(R, 0.375);
arb_add(T, T, R, prec);
_arb_poly_sin_cos_pi_series(I, R, T, len, N, prec);
/* I -= cos(pi*x/2) * sqrt(2) */
_arb_vec_scalar_mul_2exp_si(X, X, 2, -1);
_arb_poly_cos_pi_series(T, X, 2, N, prec);
arb_sqrt_ui((arb_ptr) c, 2, prec);
_arb_vec_scalar_mul(T, T, N, (arb_ptr) c, prec);
_arb_vec_sub(I, I, T, N, prec);
_arb_vec_scalar_mul_2exp_si(X, X, 2, 1);
/* T = 1 / (2 cos(pi*x)) */
_arb_poly_cos_pi_series(T, X, 2, N, prec);
_arb_vec_scalar_mul_2exp_si(T, T, N, 1);
_arb_poly_inv_series((arb_ptr) c, T, N, N, prec);
_arb_vec_swap(T, (arb_ptr) c, N);
/* R, I *= T */
_arb_poly_mullow((arb_ptr) c, R, N, T, N, N, prec);
_arb_vec_swap(R, (arb_ptr) c, N);
_arb_poly_mullow((arb_ptr) c, I, N, T, N, N, prec);
_arb_vec_swap(I, (arb_ptr) c, N);
for (i = 0; i < N; i++)
{
arb_swap(acb_realref(c + i), R + i);
arb_swap(acb_imagref(c + i), I + i);
}
_acb_poly_inv_borel_transform(c, c, N, prec);
_arb_vec_clear(R, N);
_arb_vec_clear(I, N);
_arb_vec_clear(T, N);
_arb_vec_clear(X, 2);
}
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