arb/acb_dirichlet/zeta_jet.c

/*
    Copyright (C) 2017 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"

void
_acb_dirichlet_zeta_jet(acb_t t, const acb_t h, int deflate, slong len, slong prec)
{
    acb_t a;
    acb_init(a);
    acb_one(a);

    /* use reflection formula */
    if (arf_sgn(arb_midref(acb_realref(h))) < 0)
    {
        /* zeta(s) = (2*pi)**s * sin(pi*s/2) / pi * gamma(1-s) * zeta(1-s) */
        acb_t pi, hcopy;
        acb_ptr f, s1, s2, s3, s4, u;
        slong i;

        acb_init(pi);
        acb_init(hcopy);
        f = _acb_vec_init(2);
        s1 = _acb_vec_init(len);
        s2 = _acb_vec_init(len);
        s3 = _acb_vec_init(len);
        s4 = _acb_vec_init(len);
        u = _acb_vec_init(len);
        acb_set(hcopy, h);

        acb_const_pi(pi, prec);

        /* s1 = (2*pi)**s */
        acb_mul_2exp_si(pi, pi, 1);
        _acb_poly_pow_cpx(s1, pi, h, len, prec);
        acb_mul_2exp_si(pi, pi, -1);

        /* s2 = sin(pi*s/2) / pi */
        acb_set(f, h);
        acb_one(f + 1);
        acb_mul_2exp_si(f, f, -1);
        acb_mul_2exp_si(f + 1, f + 1, -1);
        _acb_poly_sin_pi_series(s2, f, 2, len, prec);
        _acb_vec_scalar_div(s2, s2, len, pi, prec);

        /* s3 = gamma(1-s) */
        acb_sub_ui(f, hcopy, 1, prec);
        acb_neg(f, f);
        acb_set_si(f + 1, -1);
        _acb_poly_gamma_series(s3, f, 2, len, prec);

        /* s4 = zeta(1-s) */
        acb_sub_ui(f, hcopy, 1, prec);
        acb_neg(f, f);
        _acb_poly_zeta_cpx_series(s4, f, a, 0, len, prec);
        for (i = 1; i < len; i += 2)
            acb_neg(s4 + i, s4 + i);

        _acb_poly_mullow(u, s1, len, s2, len, len, prec);
        _acb_poly_mullow(s1, s3, len, s4, len, len, prec);
        _acb_poly_mullow(t, u, len, s1, len, len, prec);

        /* add 1/(1-(s+t)) = 1/(1-s) + t/(1-s)^2 + ... */
        if (deflate)
        {
            acb_sub_ui(u, hcopy, 1, prec);
            acb_neg(u, u);
            acb_inv(u, u, prec);
            for (i = 1; i < len; i++)
                acb_mul(u + i, u + i - 1, u, prec);
            _acb_vec_add(t, t, u, len, prec);
        }

        acb_clear(pi);
        acb_clear(hcopy);
        _acb_vec_clear(f, 2);
        _acb_vec_clear(s1, len);
        _acb_vec_clear(s2, len);
        _acb_vec_clear(s3, len);
        _acb_vec_clear(s4, len);
        _acb_vec_clear(u, len);
    }
    else
    {
        _acb_poly_zeta_cpx_series(t, h, a, deflate, len, prec);
    }

    acb_clear(a);
}

/* todo: should adjust precision to input accuracy */
void
acb_dirichlet_zeta_jet(acb_t res, const acb_t s, int deflate, slong len, slong prec)
{
    double cutoff;

    if (len == 1 && deflate == 0)
    {
        acb_zeta(res, s, prec);
        return;
    }

    if (deflate == 0 && (arb_contains_zero(acb_imagref(s))
                    && arb_contains_si(acb_realref(s), 1)))
    {
        _acb_vec_indeterminate(res, len);
        return;
    }

    if (len > 2 || deflate != 0)
    {
        _acb_dirichlet_zeta_jet(res, s, deflate, len, prec);
    }
    else
    {
        cutoff = 24.0 * prec * sqrt(prec);

        if (arb_is_exact(acb_realref(s)) &&
            arf_cmp_2exp_si(arb_midref(acb_realref(s)), -1) == 0)
            cutoff *= 2.5;
        else
            cutoff *= 4.0;

        if (arf_cmpabs_d(arb_midref(acb_imagref(s)), cutoff) >= 0 &&
            arf_cmpabs_d(arb_midref(acb_realref(s)), 10 + prec * 0.1) <= 0)
        {
            acb_dirichlet_zeta_jet_rs(res, s, len, prec);
        }
        else
        {
            _acb_dirichlet_zeta_jet(res, s, deflate, len, prec);
        }
    }
}