def _eval_(self, x):
"""
EXAMPLES::
sage: [dickman_rho(n) for n in [1..10]]
[1.00000000000000, 0.306852819440055, 0.0486083882911316, 0.00491092564776083, 0.000354724700456040, 0.0000196496963539553, 8.74566995329392e-7, 3.23206930422610e-8, 1.01624828273784e-9, 2.77017183772596e-11]
sage: dickman_rho(0)
1.00000000000000
"""
if not is_RealNumber(x):
try:
x = RR(x)
except (TypeError, ValueError):
return None #PrimitiveFunction.__call__(self, SR(x))
if x < 0:
return x.parent()(0)
elif x <= 1:
return x.parent()(1)
elif x <= 2:
return 1 - x.log()
n = x.floor()
if self._cur_prec < x.parent().prec() or not self._f.has_key(n):
self._cur_prec = rel_prec = x.parent().prec()
# Go a bit beyond so we're not constantly re-computing.
max = x.parent()(1.1) * x + 10
abs_prec = (-self.approximate(max).log2() + rel_prec +
2 * max.log2()).ceil()
self._f = {}
if sys.getrecursionlimit() < max + 10:
sys.setrecursionlimit(int(max) + 10)
self._compute_power_series(max.floor(),
abs_prec,
cache_ring=x.parent())
return self._f[n](2 * (x - n - x.parent()(0.5)))
def _coerce_(self, x):
"""
EXAMPLES::
>>> s = server(); c = client(s.port)
>>> c._coerce_(False)
0
>>> c._coerce_(True)
1
>>> c._coerce_('lkjdf')
'lkjdf'
>>> c._coerce_(2.5)
2.5
>>> c._coerce_([1,2])
'__pickleeJxrYIot...'
"""
if isinstance(x, bool):
x = int(x)
elif isinstance(x, (str, int, long, float)):
pass
elif is_Integer(x) and x.nbits()<32:
x = int(x)
elif is_RealNumber(x) and x.prec()==53:
return float(x)
elif isinstance(x, unicode):
return str(x)
else:
x = '__pickle' + base64.b64encode(zlib.compress(cPickle.dumps(x, 2)))
return x
def _eval_(self, x):
"""
EXAMPLES::
sage: [dickman_rho(n) for n in [1..10]]
[1.00000000000000, 0.306852819440055, 0.0486083882911316, 0.00491092564776083, 0.000354724700456040, 0.0000196496963539553, 8.74566995329392e-7, 3.23206930422610e-8, 1.01624828273784e-9, 2.77017183772596e-11]
sage: dickman_rho(0)
1.00000000000000
"""
if not is_RealNumber(x):
try:
x = RR(x)
except (TypeError, ValueError):
return None #PrimitiveFunction.__call__(self, SR(x))
if x < 0:
return x.parent()(0)
elif x <= 1:
return x.parent()(1)
elif x <= 2:
return 1 - x.log()
n = x.floor()
if self._cur_prec < x.parent().prec() or not self._f.has_key(n):
self._cur_prec = rel_prec = x.parent().prec()
# Go a bit beyond so we're not constantly re-computing.
max = x.parent()(1.1)*x + 10
abs_prec = (-self.approximate(max).log2() + rel_prec + 2*max.log2()).ceil()
self._f = {}
if sys.getrecursionlimit() < max + 10:
sys.setrecursionlimit(int(max) + 10)
self._compute_power_series(max.floor(), abs_prec, cache_ring=x.parent())
return self._f[n](2*(x-n-x.parent()(0.5)))
def _coerce_(self, x):
"""
EXAMPLES::
>>> s = server(); c = client(s.port)
>>> c._coerce_(False)
0
>>> c._coerce_(True)
1
>>> c._coerce_('lkjdf')
'lkjdf'
>>> c._coerce_(2.5)
2.5
>>> c._coerce_([1,2])
'__pickleeJxrYIot...'
"""
if isinstance(x, bool):
x = int(x)
elif isinstance(x, (str, int, long, float)):
pass
elif x is None:
pass
elif is_Integer(x) and x.nbits() < 32:
x = int(x)
elif is_RealNumber(x) and x.prec() == 53:
return float(x)
elif isinstance(x, unicode):
return str(x)
else:
x = '__pickle' + base64.b64encode(
zlib.compress(cPickle.dumps(x, 2)))
return x
def zeta_symmetric(s):
r"""
Completed function `\xi(s)` that satisfies
`\xi(s) = \xi(1-s)` and has zeros at the same points as the
Riemann zeta function.
INPUT:
- ``s`` - real or complex number
If s is a real number the computation is done using the MPFR
library. When the input is not real, the computation is done using
the PARI C library.
More precisely,
.. math::
xi(s) = \gamma(s/2 + 1) * (s-1) * \pi^{-s/2} * \zeta(s).
EXAMPLES::
sage: zeta_symmetric(0.7)
0.497580414651127
sage: zeta_symmetric(1-0.7)
0.497580414651127
sage: RR = RealField(200)
sage: zeta_symmetric(RR(0.7))
0.49758041465112690357779107525638385212657443284080589766062
sage: C.<i> = ComplexField()
sage: zeta_symmetric(0.5 + i*14.0)
0.000201294444235258 + 1.49077798716757e-19*I
sage: zeta_symmetric(0.5 + i*14.1)
0.0000489893483255687 + 4.40457132572236e-20*I
sage: zeta_symmetric(0.5 + i*14.2)
-0.0000868931282620101 + 7.11507675693612e-20*I
REFERENCE:
- I copied the definition of xi from
http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html
"""
if not (is_ComplexNumber(s) or is_RealNumber(s)):
s = ComplexField()(s)
R = s.parent()
if s == 1: # deal with poles, hopefully
return R(0.5)
return (s / 2 + 1).gamma() * (s - 1) * (R.pi()**(-s / 2)) * s.zeta()
def zeta_symmetric(s):
r"""
Completed function `\xi(s)` that satisfies
`\xi(s) = \xi(1-s)` and has zeros at the same points as the
Riemann zeta function.
INPUT:
- ``s`` - real or complex number
If s is a real number the computation is done using the MPFR
library. When the input is not real, the computation is done using
the PARI C library.
More precisely,
.. math::
xi(s) = \gamma(s/2 + 1) * (s-1) * \pi^{-s/2} * \zeta(s).
EXAMPLES::
sage: zeta_symmetric(0.7)
0.497580414651127
sage: zeta_symmetric(1-0.7)
0.497580414651127
sage: RR = RealField(200)
sage: zeta_symmetric(RR(0.7))
0.49758041465112690357779107525638385212657443284080589766062
sage: C.<i> = ComplexField()
sage: zeta_symmetric(0.5 + i*14.0)
0.000201294444235258 + 1.49077798716757e-19*I
sage: zeta_symmetric(0.5 + i*14.1)
0.0000489893483255687 + 4.40457132572236e-20*I
sage: zeta_symmetric(0.5 + i*14.2)
-0.0000868931282620101 + 7.11507675693612e-20*I
REFERENCE:
- I copied the definition of xi from
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
"""
if not (is_ComplexNumber(s) or is_RealNumber(s)):
s = ComplexField()(s)
R = s.parent()
if s == 1: # deal with poles, hopefully
return R(0.5)
return (s/2 + 1).gamma() * (s-1) * (R.pi()**(-s/2)) * s.zeta()
