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:
- [Iwa1972]_, 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 QuadraticBernoulliNumber(k, d):
r"""
Compute k-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
Reference: Iwasawa's "Lectures on p-adic L-functions", pp7-16.
EXAMPLES::
sage: ## Makes a set of odd fund discriminants < -3
sage: Fund_odd_test_set = [D for D in range(-163, -3, 4) if is_fundamental_discriminant(D)]
sage: ## In general, we have B_{1, \chi_d} = -2h/w for odd fund disc < 0
sage: for D in Fund_odd_test_set:
... if len(BinaryQF_reduced_representatives(D)) != -QuadraticBernoulliNumber(1, D):
... print "Oops! There is an error at D = ", D
"""
## 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 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:
- [Iwa1972]_, 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 QuadraticBernoulliNumber(k, d):
r"""
Compute k-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
Reference: Iwasawa's "Lectures on p-adic L-functions", pp7-16.
EXAMPLES::
sage: ## Makes a set of odd fund discriminants < -3
sage: Fund_odd_test_set = [D for D in range(-163, -3, 4) if is_fundamental_discriminant(D)]
sage: ## In general, we have B_{1, \chi_d} = -2h/w for odd fund disc < 0
sage: for D in Fund_odd_test_set:
... if len(BinaryQF_reduced_representatives(D)) != -QuadraticBernoulliNumber(1, D):
... print "Oops! There is an error at D = ", D
"""
## 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 _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1)**s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1) ** s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
co = get_coercion_model().canonical_coercion(s, x)[0]
if is_inexact(co) and not isinstance(co, Expression):
return self._evalf_(s, x, parent=parent(co))
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1)**s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
def _eval_(self, s, x):
r"""
TESTS::
sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(3, 0.5)
8.41439832211716
"""
co = get_coercion_model().canonical_coercion(s, x)[0]
if is_inexact(co) and not isinstance(co, Expression):
return self._evalf_(s, x, parent=parent(co))
if x == 1:
return zeta(s)
if s in ZZ and s > 1:
return ((-1) ** s) * psi(s - 1, x) / factorial(s - 1)
elif s in ZZ and s < 0:
return -bernoulli_polynomial(x, -s + 1) / (-s + 1)
else:
return
