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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"
void arb_zeta_inv_ui_euler_product(arb_t z, ulong s, slong 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);
}