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special_values.py
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special_values.py
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"""
Routines for computing special values of L-functions
- :func:`gamma__exact` -- Exact values of the `\Gamma` function at integers and half-integers
- :func:`zeta__exact` -- Exact values of the Riemann `\zeta` function at critical values
- :func:`quadratic_L_function__exact` -- Exact values of the Dirichlet L-functions of quadratic characters at critical values
- :func:`quadratic_L_function__numerical` -- Numerical values of the Dirichlet L-functions of quadratic characters in the domain of convergence
"""
# python3
from __future__ import division, print_function
from sage.combinat.combinat import bernoulli_polynomial
from sage.misc.functional import denominator
from sage.rings.all import RealField
from sage.arith.all import kronecker_symbol, bernoulli, factorial, fundamental_discriminant
from sage.rings.infinity import infinity
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
from sage.rings.real_mpfr import is_RealField
from sage.symbolic.constants import pi
from sage.symbolic.pynac import I
# ---------------- The Gamma Function ------------------
def gamma__exact(n):
"""
Evaluates the exact value of the `\Gamma` function at an integer or
half-integer argument.
EXAMPLES::
sage: gamma__exact(4)
6
sage: gamma__exact(3)
2
sage: gamma__exact(2)
1
sage: gamma__exact(1)
1
sage: gamma__exact(1/2)
sqrt(pi)
sage: gamma__exact(3/2)
1/2*sqrt(pi)
sage: gamma__exact(5/2)
3/4*sqrt(pi)
sage: gamma__exact(7/2)
15/8*sqrt(pi)
sage: gamma__exact(-1/2)
-2*sqrt(pi)
sage: gamma__exact(-3/2)
4/3*sqrt(pi)
sage: gamma__exact(-5/2)
-8/15*sqrt(pi)
sage: gamma__exact(-7/2)
16/105*sqrt(pi)
TESTS::
sage: gamma__exact(1/3)
Traceback (most recent call last):
...
TypeError: you must give an integer or half-integer argument
"""
from sage.all import sqrt
n = QQ(n)
if denominator(n) == 1:
if n <= 0:
return infinity
if n > 0:
return factorial(n-1)
elif denominator(n) == 2:
ans = QQ.one()
while n != QQ((1, 2)):
if n < 0:
ans /= n
n += 1
elif n > 0:
n += -1
ans *= n
ans *= sqrt(pi)
return ans
else:
raise TypeError("you must give an integer or half-integer argument")
# ------------- The Riemann Zeta Function --------------
def zeta__exact(n):
r"""
Returns the exact value of the Riemann Zeta function
The argument must be a critical value, namely either positive even
or negative odd.
See for example [Iwasawa]_, p13, Special value of `\zeta(2k)`
EXAMPLES:
Let us test the accuracy for negative special values::
sage: RR = RealField(100)
sage: for i in range(1,10):
....: print("zeta({}): {}".format(1-2*i, RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
Let us test the accuracy for positive special values::
sage: all(abs(RR(zeta__exact(2*i))-zeta(RR(2*i))) < 10**(-28) for i in range(1,10))
True
TESTS::
sage: zeta__exact(4)
1/90*pi^4
sage: zeta__exact(-3)
1/120
sage: zeta__exact(0)
-1/2
sage: zeta__exact(5)
Traceback (most recent call last):
...
TypeError: n must be a critical value (i.e. even > 0 or odd < 0)
REFERENCES:
.. [Iwasawa] Iwasawa, *Lectures on p-adic L-functions*
.. [IreRos] Ireland and Rosen, *A Classical Introduction to Modern Number Theory*
.. [WashCyc] Washington, *Cyclotomic Fields*
"""
if n < 0:
return bernoulli(1-n)/(n-1)
elif n > 1:
if (n % 2 == 0):
return ZZ(-1)**(n//2 + 1) * ZZ(2)**(n-1) * pi**n * bernoulli(n) / factorial(n)
else:
raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
elif n == 1:
return infinity
elif n == 0:
return QQ((-1, 2))
# ---------- Dirichlet L-functions with quadratic characters ----------
def QuadraticBernoulliNumber(k, d):
r"""
Compute `k`-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
EXAMPLES:
Let us create a list of some odd negative fundamental discriminants::
sage: test_set = [d for d in range(-163, -3, 4) if is_fundamental_discriminant(d)]
In general, we have `B_{1, \chi_d} = -2 h/w` for odd negative fundamental
discriminants::
sage: all(QuadraticBernoulliNumber(1, d) == -len(BinaryQF_reduced_representatives(d)) for d in test_set)
True
REFERENCES:
- [Iwasawa]_, pp 7-16.
"""
# Ensure the character is primitive
d1 = fundamental_discriminant(d)
f = abs(d1)
# Make the (usual) k-th Bernoulli polynomial
x = PolynomialRing(QQ, 'x').gen()
bp = bernoulli_polynomial(x, k)
# Make the k-th quadratic Bernoulli number
total = sum([kronecker_symbol(d1, i) * bp(i/f) for i in range(f)])
total *= (f ** (k-1))
return total
def quadratic_L_function__exact(n, d):
r"""
Returns the exact value of a quadratic twist of the Riemann Zeta function
by `\chi_d(x) = \left(\frac{d}{x}\right)`.
The input `n` must be a critical value.
EXAMPLES::
sage: quadratic_L_function__exact(1, -4)
1/4*pi
sage: quadratic_L_function__exact(-4, -4)
5/2
sage: quadratic_L_function__exact(2, 1)
1/6*pi^2
TESTS::
sage: quadratic_L_function__exact(2, -4)
Traceback (most recent call last):
...
TypeError: n must be a critical value (i.e. odd > 0 or even <= 0)
REFERENCES:
- [Iwasawa]_, pp 16-17, Special values of `L(1-n, \chi)` and `L(n, \chi)`
- [IreRos]_
- [WashCyc]_
"""
from sage.all import SR, sqrt
if n <= 0:
return QuadraticBernoulliNumber(1-n,d)/(n-1)
elif n >= 1:
# Compute the kind of critical values (p10)
if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
delta = 0
else:
delta = 1
# Compute the positive special values (p17)
if ((n - delta) % 2 == 0):
f = abs(fundamental_discriminant(d))
if delta == 0:
GS = sqrt(f)
else:
GS = I * sqrt(f)
ans = SR(ZZ(-1)**(1+(n-delta)/2))
ans *= (2*pi/f)**n
ans *= GS # Evaluate the Gauss sum here! =0
ans *= QQ.one()/(2 * I**delta)
ans *= QuadraticBernoulliNumber(n,d)/factorial(n)
return ans
else:
if delta == 0:
raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
if delta == 1:
raise TypeError("n must be a critical value (i.e. odd > 0 or even <= 0)")
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES:
First, let us test several values for a given character::
sage: RR = RealField(100)
sage: for i in range(5):
....: print("L({}, (-4/.)): {}".format(1+2*i, RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)))
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
This procedure fails for negative special values, as the Dirichlet
series does not converge here::
sage: quadratic_L_function__numerical(-3,-4, 10000)
Traceback (most recent call last):
...
ValueError: the Dirichlet series does not converge here
Test for several characters that the result agrees with the exact
value, to a given accuracy ::
sage: for d in range(-20,0): # long time (2s on sage.math 2014)
....: if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
....: print("Oops! We have a problem at d = {}: exact = {}, numerical = {}".format(d, RR(quadratic_L_function__exact(1, d)), RR(quadratic_L_function__numerical(1, d))))
"""
# Set the correct precision if it is given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
if n < 0:
raise ValueError('the Dirichlet series does not converge here')
d1 = fundamental_discriminant(d)
ans = R.zero()
for i in range(1,num_terms):
ans += R(kronecker_symbol(d1,i) / R(i)**n)
return ans