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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"
#define ONE_OVER_LOG2 1.4426950408889634
void
acb_dirichlet_l_euler_product(acb_t res, const acb_t s,
const dirichlet_group_t G, const dirichlet_char_t chi, slong prec)
{
arf_t left;
slong wp, powprec, left_s, acc;
ulong val, p, p_limit;
double p_needed_approx, powmag, logp, errmag;
int is_real, is_int;
acb_t t, u, v, c, negs;
acb_dirichlet_roots_t roots;
mag_t err;
if (!acb_is_finite(s))
{
acb_indeterminate(res);
return;
}
arf_init(left);
arf_set_mag(left, arb_radref(acb_realref(s)));
arf_sub(left, arb_midref(acb_realref(s)), left, 2 * MAG_BITS, ARF_RND_FLOOR);
/* Require re(s) >= 2. */
if (arf_cmp_si(left, 2) < 0)
{
acb_indeterminate(res);
arf_clear(left);
return;
}
is_real = acb_is_real(s) && dirichlet_char_is_real(G, chi);
/* L(s) ~= 1. */
if (arf_cmp_si(left, prec) > 0)
{
acb_one(res);
mag_hurwitz_zeta_uiui(arb_radref(acb_realref(res)), prec, 2);
if (!is_real)
mag_set(arb_radref(acb_imagref(res)), arb_radref(acb_realref(res)));
acb_inv(res, res, prec);
arf_clear(left);
return;
}
left_s = arf_get_si(left, ARF_RND_FLOOR);
/* Adjust precision based on possible accuracy. */
acc = acb_rel_accuracy_bits(s);
acc = FLINT_MAX(acc, 0);
acc = FLINT_MIN(acc, prec);
acc += left_s;
prec = FLINT_MIN(prec, acc + 10);
/* Heuristic. */
wp = prec + FLINT_BIT_COUNT(prec) + (prec / left_s) + 4;
/* Don't work too hard if a small s was passed as input. */
p_limit = 100 + prec * sqrt(prec);
/* Truncating at p ~= 2^(prec/s) gives an error of 2^-prec */
if (((double) prec) / left_s > 50.0)
p_needed_approx = pow(2.0, 50.0);
else
p_needed_approx = pow(2.0, ((double) prec) / left_s);
p_needed_approx = FLINT_MIN(p_limit, p_needed_approx);
/* todo: use exponent of chi instead of G? */
acb_dirichlet_roots_init(roots, G->expo,
p_needed_approx / (1.0 + log(p_needed_approx)), wp);
acb_init(t);
acb_init(u);
acb_init(v);
acb_init(c);
acb_init(negs);
is_int = acb_is_int(s);
acb_neg(negs, s);
acb_one(v);
for (p = 2; p < p_limit; p = n_nextprime(p, 1))
{
/* p^s */
logp = log(p);
powmag = left_s * logp * ONE_OVER_LOG2;
/* zeta(s,p) ~= 1/p^s + 1/((s-1) p^(s-1)) */
errmag = (log(left_s - 1.0) + (left_s - 1.0) * logp) * ONE_OVER_LOG2;
errmag = FLINT_MIN(powmag, errmag);
if (errmag > prec + 2)
break;
powprec = FLINT_MAX(wp - powmag, 8);
val = dirichlet_chi(G, chi, p);
if (val != DIRICHLET_CHI_NULL)
{
acb_dirichlet_root(c, roots, val, powprec);
acb_set_ui(t, p);
if (is_int)
{
acb_pow(t, t, s, powprec);
acb_set_round(u, v, powprec);
acb_div(t, u, t, powprec);
}
else
{
acb_pow(t, t, negs, powprec);
acb_set_round(u, v, powprec);
acb_mul(t, u, t, powprec);
}
acb_mul(t, t, c, powprec);
acb_sub(v, v, t, wp);
}
}
mag_init(err);
mag_hurwitz_zeta_uiui(err, left_s, p);
if (is_real)
arb_add_error_mag(acb_realref(v), err);
else
acb_add_error_mag(v, err);
mag_clear(err);
acb_inv(res, v, prec);
acb_dirichlet_roots_clear(roots);
acb_clear(t);
acb_clear(u);
acb_clear(v);
acb_clear(c);
acb_clear(negs);
arf_clear(left);
}
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