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/*
Copyright (C) 2019 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"
/*
Claim: the error is bounded by 1/64 if n <= 1 and (1/64) (log(n)/n) if n >= 2.
A crude lower bound for g_n is 2 pi exp(W(n)), or 8*n/log(n) for n >= 8.
We want to solve pi n = -t/2 log(2 pi/t) - t/2 - pi/8 + epsilon for t (= g_n).
Using (47) in Brent [https://arxiv.org/abs/1609.03682], |epsilon| <= 1/(8 t) for for t >= 2.
Also, for x >= 3, |f'(x)| < 0.5 where f(x) = exp(W(x)).
Assume n >= 9, so that (n+1/8)/e >= 3.35. Then inverting gives
t = 2 pi exp[W( [pi n - epsilon + pi/8] / (pi e) ) + 1]
= 2 pi e exp[W((n+1/8)/e - epsilon / (pi e))]
= 2 pi e exp[W((n+1/8)/e)] + epsilon2, |epsilon2| <= 1/(8 t) <= (1/64) (log(n)/n)
One can check 0 <= n <= 8 separately.
*/
static void
gram_point_initial(arb_t x, const fmpz_t n, slong prec)
{
arb_t pi, e;
mag_t b;
arb_init(pi);
arb_init(e);
mag_init(b);
arb_const_pi(pi, prec);
arb_const_e(e, prec);
/* x = 2*pi*exp(1 + W((n+1/8)/e)) */
arb_one(x);
arb_mul_2exp_si(x, x, -3);
arb_add_fmpz(x, x, n, prec);
arb_div(x, x, e, prec);
arb_lambertw(x, x, 0, prec);
arb_add_ui(x, x, 1, prec);
arb_exp(x, x, prec);
arb_mul(x, x, pi, prec);
arb_mul_2exp_si(x, x, 1);
if (fmpz_cmp_ui(n, 1) <= 0)
{
mag_set_ui_2exp_si(b, 1, -6);
}
else
{
mag_set_fmpz(b, n);
mag_log(b, b);
mag_div_fmpz(b, b, n);
mag_mul_2exp_si(b, b, -6);
}
arb_add_error_mag(x, b);
arb_clear(pi);
arb_clear(e);
mag_clear(b);
}
void
acb_dirichlet_gram_point(arb_t res, const fmpz_t n, const dirichlet_group_t G, const dirichlet_char_t chi, slong prec)
{
slong asymp_accuracy;
/* Only implemented for n >= -1 and Riemann zeta. */
if (fmpz_cmp_si(n, -1) < 0 || G != NULL || chi != NULL)
{
arb_indeterminate(res);
return;
}
asymp_accuracy = 2 * fmpz_bits(n);
asymp_accuracy = FLINT_MIN(asymp_accuracy, prec);
gram_point_initial(res, n, asymp_accuracy + 20);
asymp_accuracy = arb_rel_accuracy_bits(res);
if (asymp_accuracy < prec)
{
acb_struct tmp[2];
arb_t f, fprime, root;
mag_t C, r;
slong * steps;
slong wp, step;
acb_init(tmp);
acb_init(tmp + 1);
arb_init(f);
arb_init(fprime);
arb_init(root);
mag_init(C);
mag_init(r);
steps = flint_malloc(sizeof(slong) * FLINT_BITS);
step = 0;
steps[step] = prec * 1.05 + 10;
while (steps[step] / 2 > asymp_accuracy)
{
steps[step + 1] = steps[step] / 2;
step++;
}
arb_set(root, res);
/* theta''(x) <= C = 1/x, x >= 1 */
arb_get_mag_lower(C, root);
if (mag_cmp_2exp_si(C, 0) >= 0)
mag_inv(C, C);
else
mag_inf(C);
arb_set(root, res);
for ( ; step >= 0; step--)
{
wp = steps[step] + 10;
wp = FLINT_MAX(wp, arb_rel_accuracy_bits(root) + 10);
/* store radius, set root to the midpoint */
mag_set(r, arb_radref(root));
mag_zero(arb_radref(root));
acb_set_arb(tmp, root);
acb_dirichlet_hardy_theta(tmp, tmp, NULL, NULL, 2, wp);
arb_set(f, acb_realref(tmp));
arb_const_pi(acb_imagref(tmp), wp);
arb_submul_fmpz(f, acb_imagref(tmp), n, wp);
arb_set(fprime, acb_realref(tmp + 1));
/* f'([m+/-r]) = f'(m) +/- f''([m +/- r]) * r */
mag_mul(r, C, r);
arb_add_error_mag(fprime, r);
arb_div(f, f, fprime, wp);
arb_sub(root, root, f, wp);
/* Verify inclusion so that C is still valid. */
if (!arb_contains(res, root))
{
flint_printf("unexpected: no containment computing Gram point\n");
arb_set(root, res);
break;
}
}
arb_set(res, root);
acb_clear(tmp);
acb_clear(tmp + 1);
arb_clear(f);
arb_clear(fprime);
arb_clear(root);
mag_clear(C);
mag_clear(r);
flint_free(steps);
}
arb_set_round(res, res, prec);
}
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