Routines for computing special values of L-functions¶
gamma__exact()
– Exact values of the \(\Gamma\) function at integers and half-integerszeta__exact()
– Exact values of the Riemann \(\zeta\) function at critical valuesquadratic_L_function__exact()
– Exact values of the Dirichlet L-functions of quadratic characters at critical valuesquadratic_L_function__numerical()
– Numerical values of the Dirichlet L-functions of quadratic characters in the domain of convergence
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sage.quadratic_forms.special_values.
QuadraticBernoulliNumber
(k, d)¶ 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:
[Iwa1972], pp 7-16.
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sage.quadratic_forms.special_values.
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)
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sage.quadratic_forms.special_values.
quadratic_L_function__exact
(n, d)¶ 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
REFERENCES:
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sage.quadratic_forms.special_values.
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))))
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sage.quadratic_forms.special_values.
zeta__exact
(n)¶ 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 [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): ....: 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
REFERENCES: