/*
Bound for scaled Bessel function: 2/(2 pi x)^(1/2)
Bound for tail of integral: 2 N (k / (pi N))^(k / 2) / (k - 2).
*/
void
scaled_bessel_tail_bound(arb_t b, ulong k, const arb_t N, slong prec)
{
arb_const_pi(b, prec);
arb_mul(b, b, N, prec);
arb_ui_div(b, k, b, prec);
arb_sqrt(b, b, prec);
arb_pow_ui(b, b, k, prec);
arb_mul(b, b, N, prec);
arb_mul_ui(b, b, 2, prec);
arb_div_ui(b, b, k - 2, prec);
}
void
_acb_poly_zeta_em_bound(arb_ptr bound, const acb_t s, const acb_t a, ulong N, ulong M, slong len, slong wp)
{
arb_t K, C, AN, S2M;
arb_ptr F, R;
slong k;
arb_srcptr alpha = acb_realref(a);
arb_srcptr beta = acb_imagref(a);
arb_srcptr sigma = acb_realref(s);
arb_srcptr tau = acb_imagref(s);
arb_init(AN);
arb_init(S2M);
/* require alpha + N > 1, sigma + 2M > 1 */
arb_add_ui(AN, alpha, N - 1, wp);
arb_add_ui(S2M, sigma, 2*M - 1, wp);
if (!arb_is_positive(AN) || !arb_is_positive(S2M) || N < 1 || M < 1)
{
arb_clear(AN);
arb_clear(S2M);
for (k = 0; k < len; k++)
arb_pos_inf(bound + k);
return;
}
/* alpha + N, sigma + 2M */
arb_add_ui(AN, AN, 1, wp);
arb_add_ui(S2M, S2M, 1, wp);
R = _arb_vec_init(len);
F = _arb_vec_init(len);
arb_init(K);
arb_init(C);
/* bound for power integral */
bound_C(C, AN, beta, wp);
bound_K(K, AN, beta, tau, wp);
bound_I(R, AN, S2M, C, len, wp);
for (k = 0; k < len; k++)
{
arb_mul(R + k, R + k, K, wp);
arb_div_ui(K, K, k + 1, wp);
}
/* bound for rising factorial */
bound_rfac(F, s, 2*M, len, wp);
/* product (TODO: only need upper bound; write a function for this) */
_arb_poly_mullow(bound, F, len, R, len, len, wp);
/* bound for bernoulli polynomials, 4 / (2pi)^(2M) */
arb_const_pi(C, wp);
arb_mul_2exp_si(C, C, 1);
arb_pow_ui(C, C, 2 * M, wp);
arb_ui_div(C, 4, C, wp);
_arb_vec_scalar_mul(bound, bound, len, C, wp);
arb_clear(K);
arb_clear(C);
arb_clear(AN);
arb_clear(S2M);
_arb_vec_clear(R, len);
_arb_vec_clear(F, len);
}
