static __inline__
long _zeta_function(const fmpz_t p, long a,
long n, long d)
{
const long b = gmc_basis_size(n, d);
long i, N;
fmpz_t f, g, max;
fmpz_init(f);
fmpz_init(g);
fmpz_init(max);
if (n == 3 && fmpz_cmp_ui(p, 2) != 0)
{
fmpz_bin_uiui(f, d-1, 3);
fmpz_bin_uiui(g, b, b / 2);
fmpz_mul_ui(g, g, 2);
N = a * (*f) + fmpz_clog(g, p);
}
else if (n % 2L == 0) /* n even implies b even */
{
fmpz_bin_uiui(f, b, b / 2);
fmpz_pow_ui(g, p, (a * (b / 2) * (n - 1) + 1) / 2);
fmpz_mul(f, f, g);
fmpz_mul_ui(f, f, 2);
N = fmpz_flog(f, p) + 1;
}
else
{
for (i = b / 2; i <= b; i++)
{
fmpz_bin_uiui(f, b, i);
fmpz_pow_ui(g, p, (a * i * (n - 1) + 1) / 2);
fmpz_mul(f, f, g);
fmpz_mul_ui(f, f, 2);
if (fmpz_cmp(max, f) < 0)
fmpz_swap(max, f);
}
N = fmpz_flog(max, p) + 1;
}
fmpz_clear(f);
fmpz_clear(g);
fmpz_clear(max);
return N;
}
/* The coefficients in 2^d * \prod_{i=1}^d (x - cos(a_i)) are
easily bounded using the binomial theorem. */
static slong
magnitude_bound(slong d)
{
slong res;
fmpz_t t;
fmpz_init(t);
fmpz_bin_uiui(t, d, d / 2);
res = fmpz_bits(t);
fmpz_clear(t);
return FLINT_ABS(res) + d;
}
void
rising_difference_polynomial(fmpz * s, fmpz * c, ulong m)
{
long i, j, k;
fmpz_t t;
fmpz_init(t);
arith_stirling_number_1u_vec(s, m, m + 1);
for (k = 0; k < m; k++)
{
for (i = 0; i <= m - k - 1; i++)
{
fmpz_zero(c + m * k + i);
for (j = i + 1; j + k <= m; j++)
{
if (j == i + 1)
{
fmpz_bin_uiui(t, 1+i+k, k);
fmpz_mul_ui(t, t, m * (i+1));
}
else
{
fmpz_mul_ui(t, t, m * (k + j));
fmpz_divexact_ui(t, t, j - i);
}
fmpz_addmul(c + m * k + i, s + j + k, t);
}
}
}
fmpz_clear(t);
}