flint-sys 0.9.0

Bindings to the FLINT C library
Documentation 
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/*
    Copyright (C) 2013 Fredrik Johansson

    This file is part of FLINT.

    FLINT 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 3 of the License, or
    (at your option) any later version.  See <https://www.gnu.org/licenses/>.
*/

#include "arb.h"
#include "acb_dirichlet.h"

static void
arb_zeta_inv_ui_euler_product(arb_t z, ulong s, slong prec)
{
    const signed char chi[1] = {1};
    _acb_dirichlet_euler_product_real_ui(z, s, chi, 1, 1, prec);
}

void
arb_bernoulli_ui_zeta(arb_t b, ulong n, slong prec)
{
    slong wp, piwp;

    arb_t t, u;

    if (n < 10 || n % 2 != 0)
        flint_throw(FLINT_ERROR, "(%s)\n", __func__);

    wp = prec + 8;
    piwp = wp + 2*FLINT_BIT_COUNT(n);

    arb_init(t);
    arb_init(u);

    /* |B_n| = 2 * n! / (2*pi)^n * zeta(n) */
    arb_fac_ui(b, n, piwp);
    arb_const_pi(t, piwp);
    arb_mul_2exp_si(t, t, 1);
    arb_pow_ui(t, t, n, piwp);

    if (n > 0.7 * wp)
    {
        arb_zeta_ui_asymp(u, n, wp);
        arb_mul(b, b, u, wp);
    }
    else
    {
        arb_zeta_inv_ui_euler_product(u, n, wp);
        arb_mul(t, t, u, wp);
    }

    arb_div(b, b, t, prec);
    arb_mul_2exp_si(b, b, 1);

    if (n % 4 == 0)
        arb_neg(b, b);

    arb_clear(t);
    arb_clear(u);
}