mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
6bf072eb59
This will allow us to not loose the julia session on error. See also https://github.com/wbhart/flint2/pull/243
123 lines
3.2 KiB
C
123 lines
3.2 KiB
C
/*
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Copyright (C) 2016 Fredrik Johansson
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This file is part of Arb.
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Arb is free software: you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "acb_dirichlet.h"
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#define ONE_OVER_LOG2 1.4426950408889634
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void _acb_dirichlet_euler_product_real_ui(arb_t res, ulong s,
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const signed char * chi, int mod, int reciprocal, slong prec)
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{
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slong wp, powprec;
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double logp, powmag, errmag, limit;
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arb_t t, u;
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ulong p;
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mag_t err;
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if (s <= 1)
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{
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arb_indeterminate(res);
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return;
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}
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if (prec < 2) flint_abort(); /* assert */
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/* L(s), 1/L(s) = 1 + ... For s >= 3, zeta(s,2) < 2^(1-s). */
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if (s > (ulong) prec)
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{
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arf_one(arb_midref(res));
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mag_set_ui_2exp_si(arb_radref(res), 1, -prec);
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return;
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}
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/* L(s), 1/L(s) = 1 +/- chi(2) 2^(-s) + ...
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For s >= 2, zeta(s,3) < 2^(2-floor(3s/2)). */
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if (s > 0.7 * prec)
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{
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arb_one(res);
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if (chi[2 % mod] != 0)
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{
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arf_t t;
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arf_init(t);
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arf_set_si_2exp_si(t, chi[2 % mod], -s);
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if (reciprocal)
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arf_neg(t, t);
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arb_add_arf(res, res, t, prec);
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arf_clear(t);
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}
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arb_add_error_2exp_si(res, 2 - (3 * s) / 2);
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return;
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}
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/* Heuristic. */
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wp = prec + FLINT_BIT_COUNT(prec) + (prec / s) + 4;
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arb_init(t);
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arb_init(u);
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/* 1 - chi(2) 2^(-s) */
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arb_one(res);
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arf_set_ui_2exp_si(arb_midref(t), 1, -s); /* -s cannot overflow */
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if (chi[2 % mod] == 1)
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arb_sub(res, res, t, wp);
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else if (chi[2 % mod] == -1)
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arb_add(res, res, t, wp);
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/* Cut at some point if this algorithm just isn't a good fit... */
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/* The C * prec / log(prec) cutoff in arb_zeta_ui implies that the limit
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should be at least prec ^ (log(2) / C) for the Riemann zeta function,
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which gives prec ^ 1.2956 here. */
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limit = 100 + prec * sqrt(prec);
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for (p = 3; p < limit; p = n_nextprime(p, 1))
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{
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if (mod == 1 || chi[p % mod] != 0)
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{
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/* p^s */
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logp = log(p);
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powmag = s * logp * ONE_OVER_LOG2;
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/* zeta(s,p) ~= 1/p^s + 1/((s-1) p^(s-1)) */
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errmag = (log(s - 1.0) + (s - 1.0) * logp) * ONE_OVER_LOG2;
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errmag = FLINT_MIN(powmag, errmag);
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if (errmag > prec + 2)
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break;
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powprec = FLINT_MAX(wp - powmag, 8);
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arb_ui_pow_ui(t, p, s, powprec);
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arb_set_round(u, res, powprec);
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arb_div(t, u, t, powprec);
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if (mod == 1 || (chi[p % mod] == 1))
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arb_sub(res, res, t, wp);
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else
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arb_add(res, res, t, wp);
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}
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}
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mag_init(err);
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mag_hurwitz_zeta_uiui(err, s, p);
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arb_add_error_mag(res, err);
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mag_clear(err);
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if (reciprocal)
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arb_set_round(res, res, prec);
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else
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arb_inv(res, res, prec);
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arb_clear(t);
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arb_clear(u);
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}
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