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/*
Copyright (C) 2019 D.H.J Polymath
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 "arb_hypgeom.h"
static void
_platt_lemma_A11_X(arb_t out,
const arb_t t0, const arb_t h, slong B, const arb_t beta, slong prec)
{
arb_t x1, x2;
arb_init(x1);
arb_init(x2);
arb_set_si(x1, B);
arb_mul_2exp_si(x1, x1, -1);
arb_add(x1, x1, t0, prec);
arb_pow(x1, x1, beta, prec);
arb_set_si(x2, B);
arb_div(x2, x2, h, prec);
arb_sqr(x2, x2, prec);
arb_mul_2exp_si(x2, x2, -3);
arb_neg(x2, x2);
arb_exp(x2, x2, prec);
arb_mul(out, x1, x2, prec);
arb_clear(x1);
arb_clear(x2);
}
static void
_platt_lemma_A11_Y(arb_t out,
const arb_t t0, const arb_t h, slong B, const arb_t beta, slong prec)
{
arb_t x1, x2, x3, x4, x5;
arb_init(x1);
arb_init(x2);
arb_init(x3);
arb_init(x4);
arb_init(x5);
arb_rsqrt_ui(x1, 2, prec);
arb_pow(x2, t0, beta, prec);
arb_one(x3);
arb_mul_2exp_si(x3, x3, -1);
arb_set_si(x4, B);
arb_div(x4, x4, h, prec);
arb_sqr(x4, x4, prec);
arb_mul_2exp_si(x4, x4, -3);
arb_hypgeom_gamma_upper(x5, x3, x4, 0, prec);
arb_mul(out, x1, x2, prec);
arb_mul(out, out, x5, prec);
arb_clear(x1);
arb_clear(x2);
arb_clear(x3);
arb_clear(x4);
arb_clear(x5);
}
static void
_platt_lemma_A11_Z(arb_t out,
const arb_t t0, const arb_t h, const arb_t beta, slong prec)
{
arb_t two;
arb_t x1, x2, x3, x4, x5;
arb_init(two);
arb_init(x1);
arb_init(x2);
arb_init(x3);
arb_init(x4);
arb_init(x5);
arb_set_ui(two, 2);
arb_sub_ui(x1, beta, 1, prec);
arb_mul_2exp_si(x1, x1, -1);
arb_pow(x1, two, x1, prec);
arb_pow(x2, h, beta, prec);
arb_add_ui(x3, beta, 1, prec);
arb_mul_2exp_si(x3, x3, -1);
arb_div(x4, t0, h, prec);
arb_sqr(x4, x4, prec);
arb_mul_2exp_si(x4, x4, -1);
arb_hypgeom_gamma_upper(x5, x3, x4, 0, prec);
arb_mul(out, x1, x2, prec);
arb_mul(out, out, x5, prec);
arb_clear(two);
arb_clear(x1);
arb_clear(x2);
arb_clear(x3);
arb_clear(x4);
arb_clear(x5);
}
static int
_platt_lemma_A11_constraint(const arb_t t0, const arb_t h, slong B,
const arb_t beta, slong prec)
{
int result;
arb_t a, b, expe;
arb_init(a);
arb_init(b);
arb_init(expe);
/* expe = exp(e) */
arb_const_e(expe, prec);
arb_exp(expe, expe, prec);
/* a = beta*h^2 / t0 */
arb_sqr(a, h, prec);
arb_mul(a, a, beta, prec);
arb_div(a, a, t0, prec);
/* b = B/2 */
arb_set_si(b, B);
arb_mul_2exp_si(b, b, -1);
result = arb_le(a, b) && arb_le(b, t0) && arb_gt(t0, expe);
arb_clear(a);
arb_clear(b);
arb_clear(expe);
return result;
}
void
acb_dirichlet_platt_lemma_A11(arb_t out, const arb_t t0, const arb_t h,
slong B, slong prec)
{
arb_t beta;
arb_init(beta);
acb_dirichlet_platt_beta(beta, t0, prec);
if (_platt_lemma_A11_constraint(t0, h, B, beta, prec))
{
arb_t X, Y, Z;
arb_t x1, x2;
arb_init(X);
arb_init(Y);
arb_init(Z);
arb_init(x1);
arb_init(x2);
_platt_lemma_A11_X(X, t0, h, B, beta, prec);
_platt_lemma_A11_Y(Y, t0, h, B, beta, prec);
_platt_lemma_A11_Z(Z, t0, h, beta, prec);
arb_set_ui(x1, 2);
arb_pow(x1, x1, beta, prec);
arb_mul(x1, x1, h, prec);
arb_div_si(x1, x1, B, prec);
arb_add(x2, Y, Z, prec);
arb_mul(x2, x2, x1, prec);
arb_add(x2, x2, X, prec);
arb_mul_ui(x2, x2, 6, prec);
arb_set(out, x2);
arb_clear(X);
arb_clear(Y);
arb_clear(Z);
arb_clear(x1);
arb_clear(x2);
}
else
{
arb_zero_pm_inf(out);
}
arb_clear(beta);
}
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