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/*
Copyright (C) 2016 Pascal Molin
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"
#include <math.h>
#define PI 3.14159265358
#define LOG2 0.69314718055
ulong
acb_dirichlet_theta_length_d(ulong q, double t, slong prec)
{
double a, la;
a = PI / (double)q * t * t;
la = (a < .3) ? -log(2*a*(1-a)) : .8;
la = ((double)prec * LOG2 + la) / a;
return ceil(sqrt(la)+.5);
}
ulong
acb_dirichlet_theta_length(ulong q, const arb_t t, slong prec)
{
double dt;
ulong len;
arf_t at;
arf_init(at);
arb_get_lbound_arf(at, t, 53);
dt = arf_get_d(at, ARF_RND_DOWN);
len = acb_dirichlet_theta_length_d(q, dt, prec);
arf_clear(at);
return len;
}
/* bound for sum_{k>n} k*exp(-a k^2) */
void
mag_tail_kexpk2_arb(mag_t res, const arb_t a, ulong n)
{
mag_t m;
mag_init(m);
arb_get_mag_lower(m, a);
/* a < 1/4 */
if (mag_cmp_2exp_si(m, -2) <= 0)
{
mag_t c;
mag_init(c);
mag_mul_ui_lower(res, m, n*n-n+1); /* todo: possible overflow */
mag_expinv(res, res);
/* c = 2a(1+2a) */
mag_mul_2exp_si(m, m, 1);
mag_one(c);
mag_add_lower(c, m, c);
mag_mul_lower(c, m, c);
mag_div(res, res, c);
mag_clear(c);
}
else
{
mag_mul_ui_lower(res, m, n*n-n-1); /* todo: possible overflow */
mag_expinv(res, res);
mag_mul_ui(res, res, 2);
}
mag_clear(m);
}
/* a(t) = Pi / q * t^2 */
void
_acb_dirichlet_theta_argument_at_arb(arb_t xt, ulong q, const arb_t t, slong prec)
{
arb_const_pi(xt, prec);
arb_div_ui(xt, xt, q, prec);
arb_mul(xt, xt, t, prec);
arb_mul(xt, xt, t, prec);
}
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