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(" + str(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(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 -1 / 2
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(" + str(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(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 -1/2
def gritsenko_lift_subspace(N, weight, precision):
"""
Return a list of data, which can be used by the Fourier expansion
generator, for the space of Gritsenko lifts.
INPUT:
- `N` -- the level of the paramodular group
- ``weight`` -- the weight of the space, that we consider
- ``precision`` -- A precision class
NOTE:
The lifts returned have to have integral Fourier coefficients.
"""
jf = JacobiFormsD1NN(QQ, JacobiFormD1NN_Gamma(
N,
weight), (4 * N**2 + precision._enveloping_discriminant_bound() - 1) //
(4 * N) + 1)
return Sequence([
gritsenko_lift_fourier_expansion(
g if i != 0 else (bernoulli(weight) /
(2 * weight)).denominator() * g, precision, True)
for (i, g) in enumerate(jf.gens())
],
universe=ParamodularFormD2FourierExpansionRing(ZZ, N),
immutable=True,
check=False)
def gritsenko_lift(self, f, k, is_integral=False):
"""
INPUT:
- `f` -- the Fourier expansion of a Jacobi form as a dictionary
"""
N = self.level()
frN = 4 * N
p1list = self.precision()._p1list()
divisor_dict = self._divisor_dict()
##TODO: Precalculate disc_coeffs or at least use a recursive definition
disc_coeffs = dict()
coeffs = dict()
f_index = lambda d, b: ((d + b**2) // frN, b)
for (
((a, b, c), l), eps, disc
) in self.__precision._iter_positive_forms_with_content_and_discriminant(
):
(_, bp, _) = apply_GL_to_form(p1list[l], (a, b, c))
try:
coeffs[((a, b, c), l)] = disc_coeffs[(disc, bp, eps)]
except KeyError:
disc_coeffs[(disc, bp, eps)] = \
sum(t**(k-1) * f[ f_index(disc//t**2, (bp // t) % (2 * N)) ]
for t in divisor_dict[eps])
if disc_coeffs[(disc, bp, eps)] != 0:
coeffs[((a, b, c), l)] = disc_coeffs[(disc, bp, eps)]
for ((a, b, c), l) in self.__precision.iter_indefinite_forms():
if l == 0:
coeffs[((a, b, c), l)] = (
sigma(c, k - 1) * f[(0, 0)] if c != 0 else
(Integer(-bernoulli(k) / Integer(2 * k) *
f[(0, 0)]) if is_integral else -bernoulli(k) /
Integer(2 * k) * f[(0, 0)]))
else:
coeffs[((a, b, c), l)] = (
sigma(c // self.level(), k - 1) * f[(0, 0)] if c != 0 else
(Integer(-bernoulli(k) / Integer(2 * k) *
f[(0, 0)]) if is_integral else -bernoulli(k) /
Integer(2 * k) * f[(0, 0)]))
return coeffs
def gritsenko_lift(self, f, k, is_integral = False) :
"""
INPUT:
- `f` -- the Fourier expansion of a Jacobi form as a dictionary
"""
N = self.level()
frN = 4 * N
p1list = self.precision()._p1list()
divisor_dict = self._divisor_dict()
##TODO: Precalculate disc_coeffs or at least use a recursive definition
disc_coeffs = dict()
coeffs = dict()
f_index = lambda d,b : ((d + b**2)//frN, b)
for (((a,b,c),l), eps, disc) in self.__precision._iter_positive_forms_with_content_and_discriminant() :
(_,bp,_) = apply_GL_to_form(p1list[l], (a,b,c))
try :
coeffs[((a,b,c),l)] = disc_coeffs[(disc, bp, eps)]
except KeyError :
disc_coeffs[(disc, bp, eps)] = \
sum( t**(k-1) * f[ f_index(disc//t**2, (bp // t) % (2 * N)) ]
for t in divisor_dict[eps] )
if disc_coeffs[(disc, bp, eps)] != 0 :
coeffs[((a,b,c),l)] = disc_coeffs[(disc, bp, eps)]
for ((a,b,c), l) in self.__precision.iter_indefinite_forms() :
if l == 0 :
coeffs[((a,b,c),l)] = ( sigma(c, k-1) * f[(0,0)]
if c != 0 else
( Integer(-bernoulli(k) / Integer(2 * k) * f[(0,0)])
if is_integral else
-bernoulli(k) / Integer(2 * k) * f[(0,0)] ) )
else :
coeffs[((a,b,c),l)] = ( sigma(c//self.level(), k-1) * f[(0,0)]
if c != 0 else
( Integer(-bernoulli(k) / Integer(2 * k) * f[(0,0)])
if is_integral else
-bernoulli(k) / Integer(2 * k) * f[(0,0)] ) )
return coeffs
def _log_StirlingNegativePowers_(var, precision):
r"""
Helper function to calculate the logarithm of Stirling's approximation
formula from the negative powers of ``var`` on, i.e., it skips the
summands `n \log n - n + (\log n)/2 + \log(2\pi)/2`.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- an integer specifying the number of exact summands.
If this is negative, then the result is `0`.
OUTPUT:
An asymptotic expansion.
TESTS::
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=-1)
0
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=0)
O(m^(-1))
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=3)
1/12*m^(-1) - 1/360*m^(-3) + 1/1260*m^(-5) + O(m^(-7))
sage: _.parent()
Asymptotic Ring <m^ZZ> over Rational Field
"""
from asymptotic_ring import AsymptoticRing
from sage.rings.rational_field import QQ
A = AsymptoticRing(growth_group='{n}^ZZ'.format(n=var),
coefficient_ring=QQ)
if precision < 0:
return A.zero()
n = A.gen()
from sage.arith.all import bernoulli
from sage.arith.srange import srange
result = sum((bernoulli(k) / k / (k-1) / n**(k-1)
for k in srange(2, 2*precision + 2, 2)),
A.zero())
return result + (1 / n**(2*precision + 1)).O()
def _log_StirlingNegativePowers_(var, precision):
r"""
Helper function to calculate the logarithm of Stirling's approximation
formula from the negative powers of ``var`` on, i.e., it skips the
summands `n \log n - n + (\log n)/2 + \log(2\pi)/2`.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- an integer specifying the number of exact summands.
If this is negative, then the result is `0`.
OUTPUT:
An asymptotic expansion.
TESTS::
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=-1)
0
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=0)
O(m^(-1))
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=3)
1/12*m^(-1) - 1/360*m^(-3) + 1/1260*m^(-5) + O(m^(-7))
sage: _.parent()
Asymptotic Ring <m^ZZ> over Rational Field
"""
from asymptotic_ring import AsymptoticRing
from sage.rings.rational_field import QQ
A = AsymptoticRing(growth_group='{n}^ZZ'.format(n=var),
coefficient_ring=QQ)
if precision < 0:
return A.zero()
n = A.gen()
from sage.arith.all import bernoulli
from sage.arith.srange import srange
result = sum((bernoulli(k) / k / (k - 1) / n**(k - 1)
for k in srange(2, 2 * precision + 2, 2)), A.zero())
return result + (1 / n**(2 * precision + 1)).O()
def __maass_lifts(self, k, precision, return_value) :
r"""
Return the Fourier expansion of all Maass forms of weight `k`.
"""
result = []
if k < 4 or k % 2 != 0 :
return []
mf = ModularForms(1,k).echelon_basis()
cf = ModularForms(1,k + 2).echelon_basis()[1:]
integrality_factor = 2*k * bernoulli(k).denominator()
for c in [(integrality_factor * mf[0],0)] \
+ [ (f,0) for f in mf[1:] ] + [ (0,g) for g in cf ] :
if return_value == "lifts" :
result.append(SiegelModularFormG2MaassLift(c[0],c[1], precision, True))
else :
result.append(c)
return result
def gritsenko_lift_subspace(N, weight, precision) :
"""
Return a list of data, which can be used by the Fourier expansion
generator, for the space of Gritsenko lifts.
INPUT:
- `N` -- the level of the paramodular group
- ``weight`` -- the weight of the space, that we consider
- ``precision`` -- A precision class
NOTE:
The lifts returned have to have integral Fourier coefficients.
"""
jf = JacobiFormsD1NN( QQ, JacobiFormD1NN_Gamma(N, weight),
(4 * N**2 + precision._enveloping_discriminant_bound() - 1)//(4 * N) + 1)
return Sequence( [ gritsenko_lift_fourier_expansion( g if i != 0 else (bernoulli(weight) / (2 * weight)).denominator() * g,
precision, True )
for (i,g) in enumerate(jf.gens()) ],
universe = ParamodularFormD2FourierExpansionRing(ZZ, N), immutable = True,
check = False )
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2*k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k-1)
if M.divides(m):
sum2 = (lamk-1) * sigma(ZZ(m/M), k-1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k-1) * ZZ(m/N)**(k-1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2 * k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k - 1)
if M.divides(m):
sum2 = (lamk - 1) * sigma(ZZ(m / M), k - 1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k - 1) * ZZ(m / N)**(k - 1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
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 [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
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:
- [Iwa1972]_
- [IR1990]_
- [Was1997]_
"""
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))
def eisenstein_series_qexp(k, prec = 10, K=QQ, var='q', normalization='linear'):
r"""
Return the `q`-expansion of the normalized weight `k` Eisenstein series on
`{\rm SL}_2(\ZZ)` to precision prec in the ring `K`. Three normalizations
are available, depending on the parameter ``normalization``; the default
normalization is the one for which the linear coefficient is 1.
INPUT:
- ``k`` - an even positive integer
- ``prec`` - (default: 10) a nonnegative integer
- ``K`` - (default: `\QQ`) a ring
- ``var`` - (default: ``'q'``) variable name to use for q-expansion
- ``normalization`` - (default: ``'linear'``) normalization to use. If this
is ``'linear'``, then the series will be normalized so that the linear
term is 1. If it is ``'constant'``, the series will be normalized to have
constant term 1. If it is ``'integral'``, then the series will be
normalized to have integer coefficients and no common factor, and linear
term that is positive. Note that ``'integral'`` will work over arbitrary
base rings, while ``'linear'`` or ``'constant'`` will fail if the
denominator (resp. numerator) of `B_k / 2k` is invertible.
ALGORITHM:
We know `E_k = \text{constant} + \sum_n \sigma_{k-1}(n) q^n`. So we
compute all the `\sigma_{k-1}(n)` simultaneously, using the fact that
`\sigma` is multiplicative.
EXAMPLES::
sage: eisenstein_series_qexp(2,5)
-1/24 + q + 3*q^2 + 4*q^3 + 7*q^4 + O(q^5)
sage: eisenstein_series_qexp(2,0)
O(q^0)
sage: eisenstein_series_qexp(2,5,GF(7))
2 + q + 3*q^2 + 4*q^3 + O(q^5)
sage: eisenstein_series_qexp(2,5,GF(7),var='T')
2 + T + 3*T^2 + 4*T^3 + O(T^5)
We illustrate the use of the ``normalization`` parameter::
sage: eisenstein_series_qexp(12, 5, normalization='integral')
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='constant')
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='linear')
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 50, K=GF(13), normalization="constant")
1 + O(q^50)
TESTS:
Test that :trac:`5102` is fixed::
sage: eisenstein_series_qexp(10, 30, GF(17))
15 + q + 3*q^2 + 15*q^3 + 7*q^4 + 13*q^5 + 11*q^6 + 11*q^7 + 15*q^8 + 7*q^9 + 5*q^10 + 7*q^11 + 3*q^12 + 14*q^13 + 16*q^14 + 8*q^15 + 14*q^16 + q^17 + 4*q^18 + 3*q^19 + 6*q^20 + 12*q^21 + 4*q^22 + 12*q^23 + 4*q^24 + 4*q^25 + 8*q^26 + 14*q^27 + 9*q^28 + 6*q^29 + O(q^30)
This shows that the bug reported at :trac:`8291` is fixed::
sage: eisenstein_series_qexp(26, 10, GF(13))
7 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 2*q^8 + O(q^10)
We check that the function behaves properly over finite-characteristic base rings::
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="integral")
566*q + 236*q^2 + 286*q^3 + 194*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="constant")
Traceback (most recent call last):
...
ValueError: The numerator of -B_k/(2*k) (=691) must be invertible in the ring Ring of integers modulo 691
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="linear")
q + 667*q^2 + 252*q^3 + 601*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="integral")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="constant")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="linear")
Traceback (most recent call last):
...
ValueError: The denominator of -B_k/(2*k) (=65520) must be invertible in the ring Ring of integers modulo 2
AUTHORS:
- William Stein: original implementation
- Craig Citro (2007-06-01): rewrote for massive speedup
- Martin Raum (2009-08-02): port to cython for speedup
- David Loeffler (2010-04-07): work around an integer overflow when `k` is large
- David Loeffler (2012-03-15): add options for alternative normalizations
(motivated by :trac:`12043`)
"""
## we use this to prevent computation if it would fail anyway.
if k <= 0 or k % 2 == 1 :
raise ValueError("k must be positive and even")
a0 = - bernoulli(k) / (2*k)
if normalization == 'linear':
a0den = a0.denominator()
try:
a0fac = K(1/a0den)
except ZeroDivisionError:
raise ValueError("The denominator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0den, K))
elif normalization == 'constant':
a0num = a0.numerator()
try:
a0fac = K(1/a0num)
except ZeroDivisionError:
raise ValueError("The numerator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0num, K))
elif normalization == 'integral':
a0fac = None
else:
raise ValueError("Normalization (=%s) must be one of 'linear', 'constant', 'integral'" % normalization)
R = PowerSeriesRing(K, var)
if K == QQ and normalization == 'linear':
ls = Ek_ZZ(k, prec)
# The following is *dramatically* faster than doing the more natural
# "R(ls)" would be:
E = ZZ[var](ls, prec=prec, check=False).change_ring(QQ)
if len(ls)>0:
E._unsafe_mutate(0, a0)
return R(E, prec)
# The following is an older slower alternative to the above three lines:
#return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=False)
else:
# This used to work with check=False, but that can only be regarded as
# an improbable lucky miracle. Enabling checking is a noticeable speed
# regression; the morally right fix would be to expose FLINT's
# fmpz_poly_to_nmod_poly command (at least for word-sized N).
if a0fac is not None:
return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
else:
return R(eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
def eisenstein_series_lseries(weight, prec=53,
max_imaginary_part=0,
max_asymp_coeffs=40):
r"""
Return the L-series of the weight `2k` Eisenstein series
on `\mathrm{SL}_2(\ZZ)`.
This actually returns an interface to Tim Dokchitser's program
for computing with the L-series of the Eisenstein series
INPUT:
- ``weight`` - even integer
- ``prec`` - integer (bits precision)
- ``max_imaginary_part`` - real number
- ``max_asymp_coeffs`` - integer
OUTPUT:
The L-series of the Eisenstein series.
EXAMPLES:
We compute with the L-series of `E_{16}` and then `E_{20}`::
sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743874
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645998
Now with higher precision::
sage: L = eisenstein_series_lseries(20, prec=200)
sage: L(2)
-5.0235535164599797471968418348135050804419155747868718371029
"""
f = eisenstein_series_qexp(weight,prec)
from sage.lfunctions.all import Dokchitser
from sage.symbolic.constants import pi
key = (prec, max_imaginary_part, max_asymp_coeffs)
j = weight
L = Dokchitser(conductor = 1,
gammaV = [0,1],
weight = j,
eps = (-1)**Integer(j/2),
poles = [j],
# Using a string for residues is a hack but it works well
# since this will make PARI/GP compute sqrt(pi) with the
# right precision.
residues = '[sqrt(Pi)*(%s)]'%((-1)**Integer(j/2)*bernoulli(j)/j),
prec = prec)
s = 'coeff = %s;'%f.list()
L.init_coeffs('coeff[k+1]',pari_precode = s,
max_imaginary_part=max_imaginary_part,
max_asymp_coeffs=max_asymp_coeffs)
L.check_functional_equation()
L.rename('L-series associated to the weight %s Eisenstein series %s on SL_2(Z)'%(j,f))
return L
def eisenstein_series_qexp(k, prec = 10, K=QQ, var='q', normalization='linear'):
r"""
Return the `q`-expansion of the normalized weight `k` Eisenstein series on
`{\rm SL}_2(\ZZ)` to precision prec in the ring `K`. Three normalizations
are available, depending on the parameter ``normalization``; the default
normalization is the one for which the linear coefficient is 1.
INPUT:
- ``k`` - an even positive integer
- ``prec`` - (default: 10) a nonnegative integer
- ``K`` - (default: `\QQ`) a ring
- ``var`` - (default: ``'q'``) variable name to use for q-expansion
- ``normalization`` - (default: ``'linear'``) normalization to use. If this
is ``'linear'``, then the series will be normalized so that the linear
term is 1. If it is ``'constant'``, the series will be normalized to have
constant term 1. If it is ``'integral'``, then the series will be
normalized to have integer coefficients and no common factor, and linear
term that is positive. Note that ``'integral'`` will work over arbitrary
base rings, while ``'linear'`` or ``'constant'`` will fail if the
denominator (resp. numerator) of `B_k / 2k` is invertible.
ALGORITHM:
We know `E_k = \text{constant} + \sum_n \sigma_{k-1}(n) q^n`. So we
compute all the `\sigma_{k-1}(n)` simultaneously, using the fact that
`\sigma` is multiplicative.
EXAMPLES::
sage: eisenstein_series_qexp(2,5)
-1/24 + q + 3*q^2 + 4*q^3 + 7*q^4 + O(q^5)
sage: eisenstein_series_qexp(2,0)
O(q^0)
sage: eisenstein_series_qexp(2,5,GF(7))
2 + q + 3*q^2 + 4*q^3 + O(q^5)
sage: eisenstein_series_qexp(2,5,GF(7),var='T')
2 + T + 3*T^2 + 4*T^3 + O(T^5)
We illustrate the use of the ``normalization`` parameter::
sage: eisenstein_series_qexp(12, 5, normalization='integral')
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='constant')
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='linear')
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 50, K=GF(13), normalization="constant")
1 + O(q^50)
TESTS:
Test that :trac:`5102` is fixed::
sage: eisenstein_series_qexp(10, 30, GF(17))
15 + q + 3*q^2 + 15*q^3 + 7*q^4 + 13*q^5 + 11*q^6 + 11*q^7 + 15*q^8 + 7*q^9 + 5*q^10 + 7*q^11 + 3*q^12 + 14*q^13 + 16*q^14 + 8*q^15 + 14*q^16 + q^17 + 4*q^18 + 3*q^19 + 6*q^20 + 12*q^21 + 4*q^22 + 12*q^23 + 4*q^24 + 4*q^25 + 8*q^26 + 14*q^27 + 9*q^28 + 6*q^29 + O(q^30)
This shows that the bug reported at :trac:`8291` is fixed::
sage: eisenstein_series_qexp(26, 10, GF(13))
7 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 2*q^8 + O(q^10)
We check that the function behaves properly over finite-characteristic base rings::
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="integral")
566*q + 236*q^2 + 286*q^3 + 194*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="constant")
Traceback (most recent call last):
...
ValueError: The numerator of -B_k/(2*k) (=691) must be invertible in the ring Ring of integers modulo 691
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="linear")
q + 667*q^2 + 252*q^3 + 601*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="integral")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="constant")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="linear")
Traceback (most recent call last):
...
ValueError: The denominator of -B_k/(2*k) (=65520) must be invertible in the ring Ring of integers modulo 2
AUTHORS:
- William Stein: original implementation
- Craig Citro (2007-06-01): rewrote for massive speedup
- Martin Raum (2009-08-02): port to cython for speedup
- David Loeffler (2010-04-07): work around an integer overflow when `k` is large
- David Loeffler (2012-03-15): add options for alternative normalizations
(motivated by :trac:`12043`)
"""
## we use this to prevent computation if it would fail anyway.
if k <= 0 or k % 2 == 1 :
raise ValueError("k must be positive and even")
a0 = - bernoulli(k) / (2*k)
if normalization == 'linear':
a0den = a0.denominator()
try:
a0fac = K(1/a0den)
except ZeroDivisionError:
raise ValueError("The denominator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0den, K))
elif normalization == 'constant':
a0num = a0.numerator()
try:
a0fac = K(1/a0num)
except ZeroDivisionError:
raise ValueError("The numerator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0num, K))
elif normalization == 'integral':
a0fac = None
else:
raise ValueError("Normalization (=%s) must be one of 'linear', 'constant', 'integral'" % normalization)
R = PowerSeriesRing(K, var)
if K == QQ and normalization == 'linear':
ls = Ek_ZZ(k, prec)
# The following is *dramatically* faster than doing the more natural
# "R(ls)" would be:
E = ZZ[var](ls, prec=prec, check=False).change_ring(QQ)
if len(ls)>0:
E._unsafe_mutate(0, a0)
return R(E, prec)
# The following is an older slower alternative to the above three lines:
#return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=False)
else:
# This used to work with check=False, but that can only be regarded as
# an improbable lucky miracle. Enabling checking is a noticeable speed
# regression; the morally right fix would be to expose FLINT's
# fmpz_poly_to_nmod_poly command (at least for word-sized N).
if a0fac is not None:
return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
else:
return R(eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
def eisenstein_series_lseries(weight, prec=53,
max_imaginary_part=0,
max_asymp_coeffs=40):
r"""
Return the L-series of the weight `2k` Eisenstein series
on `\mathrm{SL}_2(\ZZ)`.
This actually returns an interface to Tim Dokchitser's program
for computing with the L-series of the Eisenstein series
INPUT:
- ``weight`` - even integer
- ``prec`` - integer (bits precision)
- ``max_imaginary_part`` - real number
- ``max_asymp_coeffs`` - integer
OUTPUT:
The L-series of the Eisenstein series.
EXAMPLES:
We compute with the L-series of `E_{16}` and then `E_{20}`::
sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743874
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645998
Now with higher precision::
sage: L = eisenstein_series_lseries(20, prec=200)
sage: L(2)
-5.0235535164599797471968418348135050804419155747868718371029
"""
f = eisenstein_series_qexp(weight,prec)
from sage.lfunctions.all import Dokchitser
j = weight
L = Dokchitser(conductor = 1,
gammaV = [0,1],
weight = j,
eps = (-1)**Integer(j/2),
poles = [j],
# Using a string for residues is a hack but it works well
# since this will make PARI/GP compute sqrt(pi) with the
# right precision.
residues = '[sqrt(Pi)*(%s)]'%((-1)**Integer(j/2)*bernoulli(j)/j),
prec = prec)
s = 'coeff = %s;'%f.list()
L.init_coeffs('coeff[k+1]',pari_precode = s,
max_imaginary_part=max_imaginary_part,
max_asymp_coeffs=max_asymp_coeffs)
L.check_functional_equation()
L.rename('L-series associated to the weight %s Eisenstein series %s on SL_2(Z)'%(j,f))
return L
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (2s on sage.math, 2014)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (2s on sage.math, 2014)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list() # long time (2s on sage.math, 2014)
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det(
) ## ??? Warning: This is a factor of 2^n larger than it should be!
# DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = " + str(d) + "\n")
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = " + str(u) + "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n - 1) // 2
d1 = fundamental_discriminant(
((-1)**m) * 2 * d *
u) ## Replaced d by 2d here to compensate for the determinant
f = abs(
d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m + 1):
factor1 *= 2 * i - 1
for i in range(m + 1, 2 * m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n // 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" +
str(d1) + "/" + str(p) + ") is " +
str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " + str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNOSTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNOSTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNOSTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if n == 2:
return genericfactor * omit * include / 2
else:
return genericfactor * omit * include
def HarmonicNumber(var, precision=None, skip_constant_summand=False):
r"""
Return the asymptotic expansion of a harmonic number.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand ``euler_gamma`` is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.HarmonicNumber('n', precision=5)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
TESTS::
sage: ex = asymptotic_expansions.HarmonicNumber('n', precision=5)
sage: n = ex.parent().gen()
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda x: sum(1/k for k in srange(1, x+1)), [5, 10, 20])
[(5, 0.0038125360?), (10, 0.00392733?), (20, 0.0039579?)]
sage: asymptotic_expansions.HarmonicNumber('n')
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
- 1/252*n^(-6) + 1/240*n^(-8) - 1/132*n^(-10)
+ 691/32760*n^(-12) - 1/12*n^(-14) + 3617/8160*n^(-16)
- 43867/14364*n^(-18) + 174611/6600*n^(-20) - 77683/276*n^(-22)
+ 236364091/65520*n^(-24) - 657931/12*n^(-26)
+ 3392780147/3480*n^(-28) - 1723168255201/85932*n^(-30)
+ 7709321041217/16320*n^(-32)
- 151628697551/12*n^(-34) + O(n^(-36))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.HarmonicNumber(
....: 'n', precision=5, skip_constant_summand=True)
log(n) + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: for p in range(5):
....: print asymptotic_expansions.HarmonicNumber(
....: 'n', precision=p)
O(log(n))
log(n) + O(1)
log(n) + euler_gamma + O(n^(-1))
log(n) + euler_gamma + 1/2*n^(-1) + O(n^(-2))
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + O(n^(-4))
sage: asymptotic_expansions.HarmonicNumber('m', precision=5)
log(m) + euler_gamma + 1/2*m^(-1) - 1/12*m^(-2) + 1/120*m^(-4) + O(m^(-6))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = A.default_prec
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += log(n)
if precision >= 2 and not skip_constant_summand:
from sage.symbolic.constants import euler_gamma
result += coefficient_ring(euler_gamma)
if precision >= 3:
result += 1 / (2 * n)
from sage.arith.srange import srange
from sage.arith.all import bernoulli
for k in srange(2, 2 * precision - 4, 2):
result += -bernoulli(k) / k / n**k
if precision < 1:
result += (log(n)).O()
elif precision == 1:
result += A(1).O()
elif precision == 2:
result += (1 / n).O()
else:
result += (1 / n**(2 * precision - 4)).O()
return result
def maass_form( self, f, g, k = None, is_integral = False) :
r"""
Return the Siegel modular form `I(f,g)` (Notation as in [Sko]).
INPUT:
- `f` -- modular form of level `1`
- `g` -- cusp form of level `1` and weight = ``weight of f + 2``
- ``is_integral`` -- ``True`` if the result is garanteed to have integer
coefficients
"""
## we introduce an abbreviations
if is_integral :
PS = self.integral_power_series_ring()
else :
PS = self.power_series_ring()
fismodular = isinstance(f, ModularFormElement)
gismodular = isinstance(g, ModularFormElement)
## We only check the arguments if f and g are ModularFormElements.
## Otherwise we trust in the user
if fismodular and gismodular :
assert( f.weight() + 2 == g.weight() | (f==0) | (g==0)), \
"incorrect weights!"
assert( g.q_expansion(1) == 0), "second argument is not a cusp form"
qexp_prec = self._get_maass_form_qexp_prec()
if qexp_prec is None : # there are no forms below prec
return dict()
if fismodular :
k = f.weight()
if f == f.parent()(0) :
f = PS(0, qexp_prec)
else :
f = PS(f.qexp(qexp_prec), qexp_prec)
elif f == 0 :
f = PS(0, qexp_prec)
else :
f = PS(f(qexp_prec), qexp_prec)
if gismodular :
k = g.weight() - 2
if g == g.parent()(0) :
g = PS(0, qexp_prec)
else :
g = PS(g.qexp(qexp_prec), qexp_prec)
elif g == 0 :
g = PS(0, qexp_prec)
else :
g = PS(g(qexp_prec), qexp_prec)
if k is None :
raise ValueError, "if neither f nor g are not ModularFormElements " + \
"you must pass k"
fderiv = f.derivative().shift(1)
f *= Integer(k/2)
gfderiv = g - fderiv
## Form A and B - the Jacobi forms used in [Sko]'s I map.
## This is not necessary if we multiply Ifg0 and Ifg1 by etapow
# (A0,A1,B0,B1) = (a0*etapow, a1*etapow, b0*etapow, b1*etapow)
## Calculate the image of the pair of modular forms (f,g) under
## [Sko]'s isomorphism I : M_{k} \oplus S_{k+2} -> J_{k,1}.
# Multiplication of big polynomials may take > 60 GB, so wie have
# to do it in small parts; This is only implemented for integral
# coefficients.
"""
Create the Jacobi form I(f,g) as in [Sko].
It suffices to construct for all Jacobi forms phi only the part
sum_{r=0,1;n} c_phi(r^2-4n) q^n zeta^r.
When, in this code part, we speak of Jacobi form we only mean this part.
We need to compute Ifg = \sum_{r=0,1; n} c(r^2-4n) q^n zeta^r up to
4n-r^2 <= Dtop, i.e. n < precision
"""
## Create the Jacobi forms A=a*etapow and B=b*etapow in stages.
## Recall a = sum_{s != r mod 2} s^2*(-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## b = sum_{s != r mod 2} (-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## r, s run over ZZ but with opposite parities.
## For r=0, we need s odd, (s^2-1)/4 < precision, with s=2t+1 hence t^2+t < precision.
## For r=1, we need s even, s^2/4 < precision, with s=2t hence t^2 < precision.
## we use a slightly overestimated ab_prec
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict(); a0dict = dict()
b1dict = dict(); b0dict = dict()
for t in xrange(1, ab_prec + 1) :
tmp = t**2
a1dict[tmp] = -8*tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8*tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2; b0dict[0] = 2
a1 = PS(a1dict); b1 = PS(b1dict)
a0 = PS(a0dict); b0 = PS(b0dict)
## Finally: I(f,g) is given by the formula below:
## We multiply by etapow explecitely and save two multiplications
# Ifg0 = k/2*f*A0 - fderiv*B0 + g*B0 + O(q^precision)
# Ifg1 = k/2*f*A1 - fderiv*B1 + g*B1 + O(q^precision)
Ifg0 = (self._eta_power() * (f*a0 + gfderiv*b0)).list()
Ifg1 = (self._eta_power() * (f*a1 + gfderiv*b1)).list()
if len(Ifg0) < qexp_prec :
Ifg0 += [0]*(qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec :
Ifg1 += [0]*(qexp_prec - len(Ifg1))
## For applying the Maass' lifting to genus 2 modular forms.
## we put the coefficients of Ifg into a dictionary Chi
## so that we can access the coefficient corresponding to
## discriminant D by going Chi[D].
Cphi = dict([(0,0)])
for i in xrange(qexp_prec) :
Cphi[-4*i] = Ifg0[i]
Cphi[1-4*i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
"""
Create the Maas lift F := VI(f,g) as in [Sko].
"""
## The constant term is given by -Cphi[0]*B_{2k}/(4*k)
## (note in [Sko] this coeff has typos).
## For nonconstant terms,
## The Siegel coefficient of q^n * zeta^r * qdash^m is given
## by the formula \sum_{ a | gcd(n,r,m) } Cphi[D/a^2] where
## D = r^2-4*n*m is the discriminant.
## Hence in either case the coefficient
## is fully deterimined by the pair (D,gcd(n,r,m)).
## Put (D,t) -> \sum_{ a | t } Cphi[D/a^2]
## in a dictionary (hash table) maassc.
maass_coeffs = dict()
divisor_dict = self._divisor_dict()
## First calculate maass coefficients corresponding to strictly positive definite matrices:
for disc in self._negative_fundamental_discriminants() :
for s in xrange(1, isqrt((-self.__precision.discriminant()) // disc) + 1) :
## add (disc*s^2,t) as a hash key, for each t that divides s
for t in divisor_dict[s] :
maass_coeffs[(disc * s**2,t)] = \
sum( a**(k-1) * Cphi[disc * s**2 / a**2]
for a in divisor_dict[t] )
## Compute the coefficients of the Siegel form $F$:
siegel_coeffs = dict()
for (n,r,m), g in self.__precision.iter_positive_forms_with_content() :
siegel_coeffs[(n,r,m)] = maass_coeffs[(r**2 - 4*m*n, g)]
## Secondly, deal with the singular part.
## Include the coeff corresponding to (0,0,0):
## maass_coeffs = {(0,0): -bernoulli(k)/(2*k)*Cphi[0]}
siegel_coeffs[(0,0,0)] = -bernoulli(k)/(2*k)*Cphi[0]
if is_integral :
siegel_coeffs[(0,0,0)] = Integer(siegel_coeffs[(0,0,0)])
## Calculate the other discriminant-zero maass coefficients.
## Since sigma is quite cheap it is faster to estimate the bound and
## save the time for repeated calculation
for i in xrange(1, self.__precision._indefinite_content_bound()) :
## maass_coeffs[(0,i)] = sigma(i, k-1) * Cphi[0]
siegel_coeffs[(0,0,i)] = sigma(i, k-1) * Cphi[0]
return siegel_coeffs
def maass_form(self, f, g, k=None, is_integral=False):
r"""
Return the Siegel modular form `I(f,g)` (Notation as in [Sko]).
INPUT:
- `f` -- modular form of level `1`
- `g` -- cusp form of level `1` and weight = ``weight of f + 2``
- ``is_integral`` -- ``True`` if the result is guaranteed to have integer
coefficients
"""
## we introduce an abbreviations
if is_integral:
PS = self.integral_power_series_ring()
else:
PS = self.power_series_ring()
fismodular = isinstance(f, ModularFormElement)
gismodular = isinstance(g, ModularFormElement)
## We only check the arguments if f and g are ModularFormElements.
## Otherwise we trust in the user
if fismodular and gismodular:
assert(f.weight() + 2 == g.weight() | (f==0) | (g==0)), \
"incorrect weights!"
assert (
g.q_expansion(1) == 0), "second argument is not a cusp form"
qexp_prec = self._get_maass_form_qexp_prec()
if qexp_prec is None: # there are no forms below prec
return dict()
if fismodular:
k = f.weight()
if f == f.parent()(0):
f = PS(0, qexp_prec)
else:
f = PS(f.qexp(qexp_prec), qexp_prec)
elif f == 0:
f = PS(0, qexp_prec)
else:
f = PS(f(qexp_prec), qexp_prec)
if gismodular:
k = g.weight() - 2
if g == g.parent()(0):
g = PS(0, qexp_prec)
else:
g = PS(g.qexp(qexp_prec), qexp_prec)
elif g == 0:
g = PS(0, qexp_prec)
else:
g = PS(g(qexp_prec), qexp_prec)
if k is None:
raise ValueError("if neither f nor g are not ModularFormElements " + \
"you must pass k")
fderiv = f.derivative().shift(1)
f *= Integer(k / 2)
gfderiv = g - fderiv
## Form A and B - the Jacobi forms used in [Sko]'s I map.
## This is not necessary if we multiply Ifg0 and Ifg1 by etapow
# (A0,A1,B0,B1) = (a0*etapow, a1*etapow, b0*etapow, b1*etapow)
## Calculate the image of the pair of modular forms (f,g) under
## [Sko]'s isomorphism I: M_{k} \oplus S_{k+2} -> J_{k,1}.
# Multiplication of big polynomials may take > 60 GB, so wie have
# to do it in small parts; This is only implemented for integral
# coefficients.
r"""
Create the Jacobi form I(f,g) as in [Sko].
It suffices to construct for all Jacobi forms phi only the part
sum_{r=0,1;n} c_phi(r^2-4n) q^n zeta^r.
When, in this code part, we speak of Jacobi form we only mean this part.
We need to compute Ifg = \sum_{r=0,1; n} c(r^2-4n) q^n zeta^r up to
4n-r^2 <= Dtop, i.e. n < precision
"""
## Create the Jacobi forms A=a*etapow and B=b*etapow in stages.
## Recall a = sum_{s != r mod 2} s^2*(-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## b = sum_{s != r mod 2} (-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## r, s run over ZZ but with opposite parities.
## For r=0, we need s odd, (s^2-1)/4 < precision, with s=2t+1 hence t^2+t < precision.
## For r=1, we need s even, s^2/4 < precision, with s=2t hence t^2 < precision.
## we use a slightly overestimated ab_prec
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict()
a0dict = dict()
b1dict = dict()
b0dict = dict()
for t in range(1, ab_prec + 1):
tmp = t**2
a1dict[tmp] = -8 * tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8 * tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2
b0dict[0] = 2
a1 = PS(a1dict)
b1 = PS(b1dict)
a0 = PS(a0dict)
b0 = PS(b0dict)
## Finally: I(f,g) is given by the formula below:
## We multiply by etapow explicitly and save two multiplications
# Ifg0 = k/2*f*A0 - fderiv*B0 + g*B0 + O(q^precision)
# Ifg1 = k/2*f*A1 - fderiv*B1 + g*B1 + O(q^precision)
Ifg0 = (self._eta_power() * (f * a0 + gfderiv * b0)).list()
Ifg1 = (self._eta_power() * (f * a1 + gfderiv * b1)).list()
if len(Ifg0) < qexp_prec:
Ifg0 += [0] * (qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec:
Ifg1 += [0] * (qexp_prec - len(Ifg1))
## For applying the Maass' lifting to genus 2 modular forms.
## we put the coefficients of Ifg into a dictionary Chi
## so that we can access the coefficient corresponding to
## discriminant D by going Chi[D].
Cphi = dict([(0, 0)])
for i in range(qexp_prec):
Cphi[-4 * i] = Ifg0[i]
Cphi[1 - 4 * i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
"""
Create the Maas lift F:= VI(f,g) as in [Sko].
"""
## The constant term is given by -Cphi[0]*B_{2k}/(4*k)
## (note in [Sko] this coeff has typos).
## For nonconstant terms,
## The Siegel coefficient of q^n * zeta^r * qdash^m is given
## by the formula \sum_{ a | gcd(n,r,m) } Cphi[D/a^2] where
## D = r^2-4*n*m is the discriminant.
## Hence in either case the coefficient
## is fully determined by the pair (D,gcd(n,r,m)).
## Put (D,t) -> \sum_{ a | t } Cphi[D/a^2]
## in a dictionary (hash table) maassc.
maass_coeffs = dict()
divisor_dict = self._divisor_dict()
## First calculate maass coefficients corresponding to strictly positive definite matrices:
for disc in self._negative_fundamental_discriminants():
for s in range(
1,
isqrt((-self.__precision.discriminant()) // disc) + 1):
## add (disc*s^2,t) as a hash key, for each t that divides s
for t in divisor_dict[s]:
maass_coeffs[(disc * s**2,t)] = \
sum(a**(k-1) * Cphi[disc * s**2 / a**2]
for a in divisor_dict[t])
## Compute the coefficients of the Siegel form $F$:
siegel_coeffs = dict()
for (n, r,
m), g in self.__precision.iter_positive_forms_with_content():
siegel_coeffs[(n, r, m)] = maass_coeffs[(r**2 - 4 * m * n, g)]
## Secondly, deal with the singular part.
## Include the coeff corresponding to (0,0,0):
## maass_coeffs = {(0,0): -bernoulli(k)/(2*k)*Cphi[0]}
siegel_coeffs[(0, 0, 0)] = -bernoulli(k) / (2 * k) * Cphi[0]
if is_integral:
siegel_coeffs[(0, 0, 0)] = Integer(siegel_coeffs[(0, 0, 0)])
## Calculate the other discriminant-zero maass coefficients.
## Since sigma is quite cheap it is faster to estimate the bound and
## save the time for repeated calculation
for i in range(1, self.__precision._indefinite_content_bound()):
## maass_coeffs[(0,i)] = sigma(i, k-1) * Cphi[0]
siegel_coeffs[(0, 0, i)] = sigma(i, k - 1) * Cphi[0]
return siegel_coeffs
def HarmonicNumber(var, precision=None, skip_constant_summand=False):
r"""
Return the asymptotic expansion of a harmonic number.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand ``euler_gamma`` is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.HarmonicNumber('n', precision=5)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
TESTS::
sage: ex = asymptotic_expansions.HarmonicNumber('n', precision=5)
sage: n = ex.parent().gen()
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda x: sum(1/k for k in srange(1, x+1)), [5, 10, 20])
[(5, 0.0038125360?), (10, 0.00392733?), (20, 0.0039579?)]
sage: asymptotic_expansions.HarmonicNumber('n')
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
- 1/252*n^(-6) + 1/240*n^(-8) - 1/132*n^(-10)
+ 691/32760*n^(-12) - 1/12*n^(-14) + 3617/8160*n^(-16)
- 43867/14364*n^(-18) + 174611/6600*n^(-20) - 77683/276*n^(-22)
+ 236364091/65520*n^(-24) - 657931/12*n^(-26)
+ 3392780147/3480*n^(-28) - 1723168255201/85932*n^(-30)
+ 7709321041217/16320*n^(-32)
- 151628697551/12*n^(-34) + O(n^(-36))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.HarmonicNumber(
....: 'n', precision=5, skip_constant_summand=True)
log(n) + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: for p in range(5):
....: print asymptotic_expansions.HarmonicNumber(
....: 'n', precision=p)
O(log(n))
log(n) + O(1)
log(n) + euler_gamma + O(n^(-1))
log(n) + euler_gamma + 1/2*n^(-1) + O(n^(-2))
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + O(n^(-4))
sage: asymptotic_expansions.HarmonicNumber('m', precision=5)
log(m) + euler_gamma + 1/2*m^(-1) - 1/12*m^(-2) + 1/120*m^(-4) + O(m^(-6))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = A.default_prec
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += log(n)
if precision >= 2 and not skip_constant_summand:
from sage.symbolic.constants import euler_gamma
result += coefficient_ring(euler_gamma)
if precision >= 3:
result += 1 / (2 * n)
from sage.arith.srange import srange
from sage.arith.all import bernoulli
for k in srange(2, 2*precision - 4, 2):
result += -bernoulli(k) / k / n**k
if precision < 1:
result += (log(n)).O()
elif precision == 1:
result += A(1).O()
elif precision == 2:
result += (1 / n).O()
else:
result += (1 / n**(2*precision - 4)).O()
return result
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (2s on sage.math, 2014)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (2s on sage.math, 2014)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list() # long time (2s on sage.math, 2014)
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det() ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n");
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n-1) // 2
d1 = fundamental_discriminant(((-1)**m) * 2*d * u) ## Replaced d by 2d here to compensate for the determinant
f = abs(d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m+1):
factor1 *= 2*i - 1
for i in range(m+1, 2*m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n // 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" + str(d1) + "/" + str(p) + ") is " + str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " +str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
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 [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
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:
- [Iwa1972]_
- [IR1990]_
- [Was1997]_
"""
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))
