AUTHORS:
TODO:
Bases: sage.structure.sage_object.SageObject
Dokchitser’s -functions Calculator
Create a Dokchitser -series with
Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec)
where
RIEMANN ZETA FUNCTION:
We compute with the Riemann Zeta function.
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L
Dokchitser L-series of conductor 1 and weight 1
sage: L(1)
...
ArithmeticError: ### user error: L*(s) has a pole at s=1.000000000000000000
sage: L(2)
1.64493406684823
sage: L(2, 1.1)
1.64493406684823
sage: L.derivative(2)
-0.937548254315844
sage: h = RR('0.0000000000001')
sage: (zeta(2+h) - zeta(2.))/h
-0.937028232783632
sage: L.taylor_series(2, k=5)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307235*z^4 + O(z^5)
RANK 1 ELLIPTIC CURVE:
We compute with the -series of a rank curve.
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L(1)
0
sage: L.derivative(1)
0.305999773834052
sage: L.derivative(1,2)
0.373095594536324
sage: L.num_coeffs()
48
sage: L.taylor_series(1,4)
0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4)
sage: L.check_functional_equation()
6.11218974800000e-18 # 32-bit
6.04442711160669e-18 # 64-bit
RANK 2 ELLIPTIC CURVE:
We compute the leading coefficient and Taylor expansion of the -series of a rank curve.
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs ()
156
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
-1.28158145691931e-23 + (7.26268290635587e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit
-2.69129566562797e-23 + (1.52514901968783e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit
RAMANUJAN DELTA L-FUNCTION:
The coefficients are given by Ramanujan’s tau function:
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
We redefine the default bound on the coefficients: Deligne’s estimate on tau(n) is better than the default coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster).
sage: L.num_coeffs()
12
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L.num_coeffs()
11
Now we’re ready to evaluate, etc.
sage: L(1)
0.0374412812685155
sage: L(1, 1.1)
0.0374412812685155
sage: L.taylor_series(1,3)
0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)
Verifies how well numerically the functional equation is satisfied, and also determines the residues if self.poles != [] and residues=’automatic’.
More specifically: for (default 1.2), self.check_functional_equation(T) should ideally return 0 (to the current precision).
EXAMPLES:
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation ()
-2.71050543200000e-20 # 32-bit
-2.71050543121376e-20 # 64-bit
If we choose the sign in functional equation for the function incorrectly, the functional equation doesn’t check out.
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation ()
-9.73967861488124
Return the -th derivative of the -series at .
Warning
If is greater than the order of vanishing of at you may get nonsense.
EXAMPLES:
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.derivative(1,E.rank())
1.51863300057685
Return the gp interpreter that is used to implement this Dokchitser L-function.
EXAMPLES:
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L(2)
0.546048036215014
sage: L.gp()
GP/PARI interpreter
Set the coefficients of the -series. If is not equal to its dual, pass the coefficients of the dual as the second optional argument.
INPUT:
EXAMPLES:
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
Return number of coefficients that are needed in order to perform most relevant -function computations to the desired precision.
EXAMPLES:
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
26
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
568
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
You might have to redefine the coefficient growth function if the of the -series are not given by the following PARI function:
coefgrow(n) = if(length(Lpoles),
1.5*n^(vecmax(real(Lpoles))-1),
sqrt(4*n)^(weight-1));
INPUT:
EXAMPLE:
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L(1)
0.0374412812685155
Return the first terms of the Taylor series expansion of the -series about .
This is returned as a series in var, where you should view var as equal to . Thus this function returns the formal power series whose coefficients are .
INPUT:
EXAMPLES:
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.taylor_series(2, 3)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3)
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser()
sage: L.taylor_series(1)
0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)
We compute a Taylor series where each coefficient is to high precision.
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(200)
sage: L.taylor_series(1,3)
6.2239725530250970363983975962696997888173850098274602272589e-73 + (-3.527106203544994604921190324282024612952450859320...e-73)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) # 32-bit
6.2239725530250970363983942830358807065824196704980264311105e-73 + (-3.5271062035449946049211884466825561834034352383203420991340e-73)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) # 64-bit