def makeCoeff(a,b,c,k):
if (a,b,c) == (0,0,0):
return 1
#calculate the discriminant
disc = 4 * a * c - b * b
#calculate the zeta functions
z1 = zeta(1 - k)
z2 = zeta(3 - (2 * k))
#make a list of nonzero coefficients of our BQF
list = []
for i in [a, b, c]:
if i != 0:
list.append(i)
sm = 0
#iterate through divisors of our BQF (this forms the core sum)
for d in range(1, abs(min(list)) + 1):
if a % d == 0 and b % d == 0 and c % d == 0:
#add term as given in McCarthy for each divisor d
POW = (d ** (k-1))
HFUN = H(k-1, disc / (d ** 2))
sm += POW * HFUN
#finish off the calculation by multiplying by 2/(Z(1 - k) * Z(3 - 32))
value = 2 * sm / (z1 * z2)
return value
def _derivative_(self, n, m, diff_param=None):
"""
The derivative of `H_{n,m}`.
EXAMPLES::
sage: k,m,n = var('k,m,n')
sage: sum(1/k, k, 1, x).diff(x)
1/6*pi^2 - harmonic_number(x, 2)
sage: harmonic_number(x, 1).diff(x)
1/6*pi^2 - harmonic_number(x, 2)
sage: harmonic_number(n, m).diff(n)
-m*(harmonic_number(n, m + 1) - zeta(m + 1))
sage: harmonic_number(n, m).diff(m)
Traceback (most recent call last):
...
ValueError: cannot differentiate harmonic_number in the second parameter
"""
from sage.functions.transcendental import zeta
if diff_param == 1:
raise ValueError(
"cannot differentiate harmonic_number in the second parameter")
if m == 1:
return harmonic_m1(n).diff()
else:
return m * (zeta(m + 1) - harmonic_number(n, m + 1))
def _derivative_(self, n, m, diff_param=None):
"""
The derivative of `H_{n,m}`.
EXAMPLES::
sage: k,m,n = var('k,m,n')
sage: sum(1/k, k, 1, x).diff(x)
1/6*pi^2 - harmonic_number(x, 2)
sage: harmonic_number(x, 1).diff(x)
1/6*pi^2 - harmonic_number(x, 2)
sage: harmonic_number(n, m).diff(n)
-m*(harmonic_number(n, m + 1) - zeta(m + 1))
sage: harmonic_number(n, m).diff(m)
Traceback (most recent call last):
...
ValueError: cannot differentiate harmonic_number in the second parameter
"""
from sage.functions.transcendental import zeta
if diff_param == 1:
raise ValueError("cannot differentiate harmonic_number in the second parameter")
if m==1:
return harmonic_m1(n).diff()
else:
return m*(zeta(m+1) - harmonic_number(n, m+1))
def H(k1,N):
#here we assign variables in a way that meshes with notation in McCarthy and Raum
k = k1 + 1
N = -N
s = 2 - k
#when N = 0 we calculate the value with the zeta func
if N == 0:
x = zeta((2 * s) - 1)
return x
#we check if N is 0 or 1 mod 4
if N % 4 == 0 or N % 4 == 1:
#find our kronecker character
D0 = fundDisc(N)
#find f
f = isqrt(int(N / D0))
#initialize our value
hSum = 0
#iterate through the divisors of f
for d in range(1, f + 1):
if f % d == 0:
#when d divides f we add the appropriate term to the sum
MOB = mobius(d)
KRON = kronecker(D0, d)
POW = d ** -s
SIG = sigmaDivisor(1 - (2 * s), int(f / d))
#print(' adding ' + str(MOB) + ' * ' + str(KRON) + ' * ' + str(POW) + ' * ' + str(SIG) + ' to H func sum.')
hSum += MOB * KRON * POW * SIG
LF = lfun(s, D0)
fCalc = hSum * LF
return fCalc
#otherwise the value is 0
else:
return 0
def _derivative_(self, z, diff_param=None):
"""
The derivative of `H_x`.
EXAMPLES::
sage: k=var('k')
sage: sum(1/k,k,1,x).diff(x)
1/6*pi^2 - harmonic_number(x, 2)
"""
from sage.functions.transcendental import zeta
return zeta(2) - harmonic_number(z, 2)
def _derivative_(self, z, diff_param=None):
"""
The derivative of `H_x`.
EXAMPLES::
sage: k=var('k')
sage: sum(1/k,k,1,x).diff(x)
1/6*pi^2 - harmonic_number(x, 2)
"""
from sage.functions.transcendental import zeta
return zeta(2)-harmonic_number(z,2)
def _eval_(self, x, D): #symbolic value
D = Integer(D)
if D % 4 > 1:
raise ValueError('Not a discriminant')
f = D.squarefree_part()
if f % 4 > 1 and not D % 4:
f *= 4
m = D // f
if f != D:
L = QuadraticLFunction()
return prod(1 - p**(-x) * kronecker(f, p)
for p in prime_divisors(m)) * L(x, f)
if D == 1:
return prod(1 - p**(-x) * kronecker(f, p)
for p in prime_divisors(m)) * zeta(x)
try:
return quadratic_L_function__exact(x, D)
except TypeError:
pass
def _evalf_(self, z, m, parent=None, algorithm=None):
"""
EXAMPLES::
sage: harmonic_number(3.,3)
zeta(3) - 0.0400198661225573
sage: harmonic_number(3.,3.)
1.16203703703704
sage: harmonic_number(3,3).n(200)
1.16203703703703703703703...
sage: harmonic_number(5,I).n()
2.36889632899995 - 3.51181956521611*I
"""
if m == 0:
if parent is None:
return z
return parent(z)
elif m == 1:
return harmonic_m1._evalf_(z, parent, algorithm)
from sage.functions.transcendental import zeta, hurwitz_zeta
return zeta(m) - hurwitz_zeta(m, z + 1)
def _evalf_(self, z, m, parent=None, algorithm=None):
"""
EXAMPLES::
sage: harmonic_number(3.,3)
zeta(3) - 0.0400198661225573
sage: harmonic_number(3.,3.)
1.16203703703704
sage: harmonic_number(3,3).n(200)
1.16203703703703703703703...
sage: harmonic_number(5,I).n()
2.36889632899995 - 3.51181956521611*I
"""
if m == 0:
if parent is None:
return z
return parent(z)
elif m == 1:
return harmonic_m1._evalf_(z, parent, algorithm)
from sage.functions.transcendental import zeta, hurwitz_zeta
return zeta(m) - hurwitz_zeta(m,z+1)
