Routines for computing special values of \(L\)-functions#

sage.quadratic_forms.special_values.QuadraticBernoulliNumber(k, d)[source]#

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 srange(-163, -3, 4)                                # needs sage.libs.pari
....:             if d.is_fundamental_discriminant()]

>>> from sage.all import *
>>> test_set = [d for d in srange(-Integer(163), -Integer(3), Integer(4))                                # needs sage.libs.pari
...             if d.is_fundamental_discriminant()]

In general, we have \(B_{1, \chi_d} = -2 h/w\) for odd negative fundamental discriminants:

sage: all(QuadraticBernoulliNumber(1, d)                                        # needs sage.libs.pari
....:       == -len(BinaryQF_reduced_representatives(d))
....:     for d in test_set)
True

>>> from sage.all import *
>>> all(QuadraticBernoulliNumber(Integer(1), d)                                        # needs sage.libs.pari
...       == -len(BinaryQF_reduced_representatives(d))
...     for d in test_set)
True

REFERENCES:

sage.quadratic_forms.special_values.gamma__exact(n)[source]#

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: # needs sage.symbolic
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: # needs sage.symbolic
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)

>>> from sage.all import *
>>> gamma__exact(Integer(4))
6
>>> gamma__exact(Integer(3))
2
>>> gamma__exact(Integer(2))
1
>>> gamma__exact(Integer(1))
1

>>> # needs sage.symbolic
>>> gamma__exact(Integer(1)/Integer(2))
sqrt(pi)
>>> gamma__exact(Integer(3)/Integer(2))
1/2*sqrt(pi)
>>> gamma__exact(Integer(5)/Integer(2))
3/4*sqrt(pi)
>>> gamma__exact(Integer(7)/Integer(2))
15/8*sqrt(pi)

>>> # needs sage.symbolic
>>> gamma__exact(-Integer(1)/Integer(2))
-2*sqrt(pi)
>>> gamma__exact(-Integer(3)/Integer(2))
4/3*sqrt(pi)
>>> gamma__exact(-Integer(5)/Integer(2))
-8/15*sqrt(pi)
>>> gamma__exact(-Integer(7)/Integer(2))
16/105*sqrt(pi)
sage.quadratic_forms.special_values.quadratic_L_function__exact(n, d)[source]#

Return 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)                                        # needs sage.libs.pari sage.symbolic
1/4*pi
sage: quadratic_L_function__exact(-4, -4)                                       # needs sage.libs.pari
5/2
sage: quadratic_L_function__exact(2, 1)                                         # needs sage.libs.pari sage.symbolic
1/6*pi^2

>>> from sage.all import *
>>> quadratic_L_function__exact(Integer(1), -Integer(4))                                        # needs sage.libs.pari sage.symbolic
1/4*pi
>>> quadratic_L_function__exact(-Integer(4), -Integer(4))                                       # needs sage.libs.pari
5/2
>>> quadratic_L_function__exact(Integer(2), Integer(1))                                         # needs sage.libs.pari sage.symbolic
1/6*pi^2

REFERENCES:

sage.quadratic_forms.special_values.quadratic_L_function__numerical(n, d, num_terms=1000)[source]#

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):                                                        # needs sage.symbolic
....:     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

>>> from sage.all import *
>>> RR = RealField(Integer(100))
>>> for i in range(Integer(5)):                                                        # needs sage.symbolic
...     print("L({}, (-4/.)): {}".format(Integer(1)+Integer(2)*i,
...             RR(quadratic_L_function__exact(Integer(1)+Integer(2)*i, -Integer(4)))
...                - quadratic_L_function__numerical(RR(Integer(1)+Integer(2)*i), -Integer(4), Integer(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

>>> from sage.all import *
>>> quadratic_L_function__numerical(-Integer(3), -Integer(4), Integer(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), needs sage.symbolic
....:     if abs(RR(quadratic_L_function__numerical(1, d, 10000)
....:                - quadratic_L_function__exact(1, d))) > 0.001:
....:         print("We have a problem at d = {}: exact = {}, numerical = {}".format(d,
....:                   RR(quadratic_L_function__exact(1, d)),
....:                   RR(quadratic_L_function__numerical(1, d))))

>>> from sage.all import *
>>> for d in range(-Integer(20),Integer(0)):            # long time (2s on sage.math 2014), needs sage.symbolic
...     if abs(RR(quadratic_L_function__numerical(Integer(1), d, Integer(10000))
...                - quadratic_L_function__exact(Integer(1), d))) > RealNumber('0.001'):
...         print("We have a problem at d = {}: exact = {}, numerical = {}".format(d,
...                   RR(quadratic_L_function__exact(Integer(1), d)),
...                   RR(quadratic_L_function__numerical(Integer(1), d))))
sage.quadratic_forms.special_values.zeta__exact(n)[source]#

Return 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 [Iwa1972], 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):                                                     # needs sage.symbolic
....:     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

>>> from sage.all import *
>>> RR = RealField(Integer(100))
>>> for i in range(Integer(1),Integer(10)):                                                     # needs sage.symbolic
...     print("zeta({}): {}".format(Integer(1) - Integer(2)*i,
...                                 RR(zeta__exact(Integer(1)-Integer(2)*i)) - zeta(RR(Integer(1)-Integer(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)                 # needs sage.symbolic
....:     for i in range(1,10))
True

>>> from sage.all import *
>>> all(abs(RR(zeta__exact(Integer(2)*i)) - zeta(RR(Integer(2)*i))) < Integer(10)**(-Integer(28))                 # needs sage.symbolic
...     for i in range(Integer(1),Integer(10)))
True

REFERENCES: