def _d(n, j, prec, sq23pi, sqrt8):
"""
Compute the sinh term in the outer sum of the HRR formula.
The constants sqrt(2/3*pi) and sqrt(8) must be precomputed.
"""
j = from_int(j)
pi = mpf_pi(prec)
a = mpf_div(sq23pi, j, prec)
b = mpf_sub(from_int(n), from_rational(1, 24, prec), prec)
c = mpf_sqrt(b, prec)
ch, sh = mpf_cosh_sinh(mpf_mul(a, c), prec)
D = mpf_div(mpf_sqrt(j, prec), mpf_mul(mpf_mul(sqrt8, b), pi), prec)
E = mpf_sub(mpf_mul(a, ch), mpf_div(sh, c, prec), prec)
return mpf_mul(D, E)
def _d(n, j, prec, sq23pi, sqrt8):
"""
Compute the sinh term in the outer sum of the HRR formula.
The constants sqrt(2/3*pi) and sqrt(8) must be precomputed.
"""
j = from_int(j)
pi = mpf_pi(prec)
a = mpf_div(sq23pi, j, prec)
b = mpf_sub(from_int(n), from_rational(1, 24, prec), prec)
c = mpf_sqrt(b, prec)
ch, sh = mpf_cosh_sinh(mpf_mul(a, c), prec)
D = mpf_div(mpf_sqrt(j, prec), mpf_mul(mpf_mul(sqrt8, b), pi), prec)
E = mpf_sub(mpf_mul(a, ch), mpf_div(sh, c, prec), prec)
return mpf_mul(D, E)
def _a(n, k, prec):
""" Compute the inner sum in HRR formula [1]_
References
==========
.. [1] http://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf
"""
if k == 1:
return fone
k1 = k
e = 0
p = _factor[k]
while k1 % p == 0:
k1 //= p
e += 1
k2 = k//k1 # k2 = p^e
v = 1 - 24*n
pi = mpf_pi(prec)
if k1 == 1:
# k = p^e
if p == 2:
mod = 8*k
v = mod + v % mod
v = (v*pow(9, k - 1, mod)) % mod
m = _sqrt_mod_prime_power(v, 2, e + 3)[0]
arg = mpf_div(mpf_mul(
from_int(4*m), pi, prec), from_int(mod), prec)
return mpf_mul(mpf_mul(
from_int((-1)**e*jacobi_symbol(m - 1, m)),
mpf_sqrt(from_int(k), prec), prec),
mpf_sin(arg, prec), prec)
if p == 3:
mod = 3*k
v = mod + v % mod
if e > 1:
v = (v*pow(64, k//3 - 1, mod)) % mod
m = _sqrt_mod_prime_power(v, 3, e + 1)[0]
arg = mpf_div(mpf_mul(from_int(4*m), pi, prec),
from_int(mod), prec)
return mpf_mul(mpf_mul(
from_int(2*(-1)**(e + 1)*legendre_symbol(m, 3)),
mpf_sqrt(from_int(k//3), prec), prec),
mpf_sin(arg, prec), prec)
v = k + v % k
if v % p == 0:
if e == 1:
return mpf_mul(
from_int(jacobi_symbol(3, k)),
mpf_sqrt(from_int(k), prec), prec)
return fzero
if not is_quad_residue(v, p):
return fzero
_phi = p**(e - 1)*(p - 1)
v = (v*pow(576, _phi - 1, k))
m = _sqrt_mod_prime_power(v, p, e)[0]
arg = mpf_div(
mpf_mul(from_int(4*m), pi, prec),
from_int(k), prec)
return mpf_mul(mpf_mul(
from_int(2*jacobi_symbol(3, k)),
mpf_sqrt(from_int(k), prec), prec),
mpf_cos(arg, prec), prec)
if p != 2 or e >= 3:
d1, d2 = igcd(k1, 24), igcd(k2, 24)
e = 24//(d1*d2)
n1 = ((d2*e*n + (k2**2 - 1)//d1)*
pow(e*k2*k2*d2, _totient[k1] - 1, k1)) % k1
n2 = ((d1*e*n + (k1**2 - 1)//d2)*
pow(e*k1*k1*d1, _totient[k2] - 1, k2)) % k2
return mpf_mul(_a(n1, k1, prec), _a(n2, k2, prec), prec)
if e == 2:
n1 = ((8*n + 5)*pow(128, _totient[k1] - 1, k1)) % k1
n2 = (4 + ((n - 2 - (k1**2 - 1)//8)*(k1**2)) % 4) % 4
return mpf_mul(mpf_mul(
from_int(-1),
_a(n1, k1, prec), prec),
_a(n2, k2, prec))
n1 = ((8*n + 1)*pow(32, _totient[k1] - 1, k1)) % k1
n2 = (2 + (n - (k1**2 - 1)//8) % 2) % 2
return mpf_mul(_a(n1, k1, prec), _a(n2, k2, prec), prec)
def _a(n, k, prec):
""" Compute the inner sum in HRR formula [1]_
References
==========
.. [1] http://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf
"""
if k == 1:
return fone
k1 = k
e = 0
p = _factor[k]
while k1 % p == 0:
k1 //= p
e += 1
k2 = k // k1 # k2 = p^e
v = 1 - 24 * n
pi = mpf_pi(prec)
if k1 == 1:
# k = p^e
if p == 2:
mod = 8 * k
v = mod + v % mod
v = (v * pow(9, k - 1, mod)) % mod
m = _sqrt_mod_prime_power(v, 2, e + 3)[0]
arg = mpf_div(mpf_mul(from_int(4 * m), pi, prec), from_int(mod),
prec)
return mpf_mul(
mpf_mul(from_int((-1)**e * jacobi_symbol(m - 1, m)),
mpf_sqrt(from_int(k), prec), prec), mpf_sin(arg, prec),
prec)
if p == 3:
mod = 3 * k
v = mod + v % mod
if e > 1:
v = (v * pow(64, k // 3 - 1, mod)) % mod
m = _sqrt_mod_prime_power(v, 3, e + 1)[0]
arg = mpf_div(mpf_mul(from_int(4 * m), pi, prec), from_int(mod),
prec)
return mpf_mul(
mpf_mul(from_int(2 * (-1)**(e + 1) * legendre_symbol(m, 3)),
mpf_sqrt(from_int(k // 3), prec), prec),
mpf_sin(arg, prec), prec)
v = k + v % k
if v % p == 0:
if e == 1:
return mpf_mul(from_int(jacobi_symbol(3, k)),
mpf_sqrt(from_int(k), prec), prec)
return fzero
if not is_quad_residue(v, p):
return fzero
_phi = p**(e - 1) * (p - 1)
v = (v * pow(576, _phi - 1, k))
m = _sqrt_mod_prime_power(v, p, e)[0]
arg = mpf_div(mpf_mul(from_int(4 * m), pi, prec), from_int(k), prec)
return mpf_mul(
mpf_mul(from_int(2 * jacobi_symbol(3, k)),
mpf_sqrt(from_int(k), prec), prec), mpf_cos(arg, prec),
prec)
if p != 2 or e >= 3:
d1, d2 = igcd(k1, 24), igcd(k2, 24)
e = 24 // (d1 * d2)
n1 = ((d2 * e * n + (k2**2 - 1) // d1) *
pow(e * k2 * k2 * d2, _totient[k1] - 1, k1)) % k1
n2 = ((d1 * e * n + (k1**2 - 1) // d2) *
pow(e * k1 * k1 * d1, _totient[k2] - 1, k2)) % k2
return mpf_mul(_a(n1, k1, prec), _a(n2, k2, prec), prec)
if e == 2:
n1 = ((8 * n + 5) * pow(128, _totient[k1] - 1, k1)) % k1
n2 = (4 + ((n - 2 - (k1**2 - 1) // 8) * (k1**2)) % 4) % 4
return mpf_mul(mpf_mul(from_int(-1), _a(n1, k1, prec), prec),
_a(n2, k2, prec))
n1 = ((8 * n + 1) * pow(32, _totient[k1] - 1, k1)) % k1
n2 = (2 + (n - (k1**2 - 1) // 8) % 2) % 2
return mpf_mul(_a(n1, k1, prec), _a(n2, k2, prec), prec)
