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"""
---------------------------------------------------------------------
.. sectionauthor:: Juan Arias de Reyna <[email protected]>
This module implements zeta-related functions using the Riemann-Siegel
expansion: zeta_offline(s,k=0)
* coef(J, eps): Need in the computation of Rzeta(s,k)
* Rzeta_simul(s, der=0) computes Rzeta^(k)(s) and Rzeta^(k)(1-s) simultaneously
for 0 <= k <= der. Used by zeta_offline and z_offline
* Rzeta_set(s, derivatives) computes Rzeta^(k)(s) for given derivatives, used by
z_half(t,k) and zeta_half
* z_offline(w,k): Z(w) and its derivatives of order k <= 4
* z_half(t,k): Z(t) (Riemann Siegel function) and its derivatives of order k <= 4
* zeta_offline(s): zeta(s) and its derivatives of order k<= 4
* zeta_half(1/2+it,k): zeta(s) and its derivatives of order k<= 4
* rs_zeta(s,k=0) Computes zeta^(k)(s) Unifies zeta_half and zeta_offline
* rs_z(w,k=0) Computes Z^(k)(w) Unifies z_offline and z_half
----------------------------------------------------------------------
This program uses Riemann-Siegel expansion even to compute
zeta(s) on points s = sigma + i t with sigma arbitrary not
necessarily equal to 1/2.
It is founded on a new deduction of the formula, with rigorous
and sharp bounds for the terms and rest of this expansion.
More information on the papers:
J. Arias de Reyna, High Precision Computation of Riemann's
Zeta Function by the Riemann-Siegel Formula I, II
We refer to them as I, II.
In them we shall find detailed explanation of all the
procedure.
The program uses Riemann-Siegel expansion.
This is useful when t is big, ( say t > 10000 ).
The precision is limited, roughly it can compute zeta(sigma+it)
with an error less than exp(-c t) for some constant c depending
on sigma. The program gives an error when the Riemann-Siegel
formula can not compute to the wanted precision.
"""
import math
class RSCache(object):
def __init__(ctx):
ctx._rs_cache = [0, 10, {}, {}]
from .functions import defun
#-------------------------------------------------------------------------------#
# #
# coef(ctx, J, eps, _cache=[0, 10, {} ] ) #
# #
#-------------------------------------------------------------------------------#
# This function computes the coefficients c[n] defined on (I, equation (47))
# but see also (II, section 3.14).
#
# Since these coefficients are very difficult to compute we save the values
# in a cache. So if we compute several values of the functions Rzeta(s) for
# near values of s, we do not recompute these coefficients.
#
# c[n] are the Taylor coefficients of the function:
#
# F(z):= (exp(pi*j*(z*z/2+3/8))-j* sqrt(2) cos(pi*z/2))/(2*cos(pi *z))
#
#
def _coef(ctx, J, eps):
r"""
Computes the coefficients `c_n` for `0\le n\le 2J` with error less than eps
**Definition**
The coefficients c_n are defined by
.. math ::
\begin{equation}
F(z)=\frac{e^{\pi i
\bigl(\frac{z^2}{2}+\frac38\bigr)}-i\sqrt{2}\cos\frac{\pi}{2}z}{2\cos\pi
z}=\sum_{n=0}^\infty c_{2n} z^{2n}
\end{equation}
they are computed applying the relation
.. math ::
\begin{multline}
c_{2n}=-\frac{i}{\sqrt{2}}\Bigl(\frac{\pi}{2}\Bigr)^{2n}
\sum_{k=0}^n\frac{(-1)^k}{(2k)!}
2^{2n-2k}\frac{(-1)^{n-k}E_{2n-2k}}{(2n-2k)!}+\\
+e^{3\pi i/8}\sum_{j=0}^n(-1)^j\frac{
E_{2j}}{(2j)!}\frac{i^{n-j}\pi^{n+j}}{(n-j)!2^{n-j+1}}.
\end{multline}
"""
newJ = J+2 # compute more coefficients that are needed
neweps6 = eps/2. # compute with a slight more precision that are needed
# PREPARATION FOR THE COMPUTATION OF V(N) AND W(N)
# See II Section 3.16
#
# Computing the exponent wpvw of the error II equation (81)
wpvw = max(ctx.mag(10*(newJ+3)), 4*newJ+5-ctx.mag(neweps6))
# Preparation of Euler numbers (we need until the 2*RS_NEWJ)
E = ctx._eulernum(2*newJ)
# Now we have in the cache all the needed Euler numbers.
#
# Computing the powers of pi
#
# We need to compute the powers pi**n for 1<= n <= 2*J
# with relative error less than 2**(-wpvw)
# it is easy to show that this is obtained
# taking wppi as the least d with
# 2**d>40*J and 2**d> 4.24 *newJ + 2**wpvw
# In II Section 3.9 we need also that
# wppi > wptcoef[0], and that the powers
# here computed 0<= k <= 2*newJ are more
# than those needed there that are 2*L-2.
# so we need J >= L this will be checked
# before computing tcoef[]
wppi = max(ctx.mag(40*newJ), ctx.mag(newJ)+3 +wpvw)
ctx.prec = wppi
pipower = {}
pipower[0] = ctx.one
pipower[1] = ctx.pi
for n in range(2,2*newJ+1):
pipower[n] = pipower[n-1]*ctx.pi
# COMPUTING THE COEFFICIENTS v(n) AND w(n)
# see II equation (61) and equations (81) and (82)
ctx.prec = wpvw+2
v={}
w={}
for n in range(0,newJ+1):
va = (-1)**n * ctx._eulernum(2*n)
va = ctx.mpf(va)/ctx.fac(2*n)
v[n]=va*pipower[2*n]
for n in range(0,2*newJ+1):
wa = ctx.one/ctx.fac(n)
wa=wa/(2**n)
w[n]=wa*pipower[n]
# COMPUTATION OF THE CONVOLUTIONS RS_P1 AND RS_P2
# See II Section 3.16
ctx.prec = 15
wpp1a = 9 - ctx.mag(neweps6)
P1 = {}
for n in range(0,newJ+1):
ctx.prec = 15
wpp1 = max(ctx.mag(10*(n+4)),4*n+wpp1a)
ctx.prec = wpp1
sump = 0
for k in range(0,n+1):
sump += ((-1)**k) * v[k]*w[2*n-2*k]
P1[n]=((-1)**(n+1))*ctx.j*sump
P2={}
for n in range(0,newJ+1):
ctx.prec = 15
wpp2 = max(ctx.mag(10*(n+4)),4*n+wpp1a)
ctx.prec = wpp2
sump = 0
for k in range(0,n+1):
sump += (ctx.j**(n-k)) * v[k]*w[n-k]
P2[n]=sump
# COMPUTING THE COEFFICIENTS c[2n]
# See II Section 3.14
ctx.prec = 15
wpc0 = 5 - ctx.mag(neweps6)
wpc = max(6,4*newJ+wpc0)
ctx.prec = wpc
mu = ctx.sqrt(ctx.mpf('2'))/2
nu = ctx.expjpi(3./8)/2
c={}
for n in range(0,newJ):
ctx.prec = 15
wpc = max(6,4*n+wpc0)
ctx.prec = wpc
c[2*n] = mu*P1[n]+nu*P2[n]
for n in range(1,2*newJ,2):
c[n] = 0
return [newJ, neweps6, c, pipower]
def coef(ctx, J, eps):
_cache = ctx._rs_cache
if J <= _cache[0] and eps >= _cache[1]:
return _cache[2], _cache[3]
orig = ctx._mp.prec
try:
data = _coef(ctx._mp, J, eps)
finally:
ctx._mp.prec = orig
if ctx is not ctx._mp:
data[2] = dict((k,ctx.convert(v)) for (k,v) in data[2].items())
data[3] = dict((k,ctx.convert(v)) for (k,v) in data[3].items())
ctx._rs_cache[:] = data
return ctx._rs_cache[2], ctx._rs_cache[3]
#-------------------------------------------------------------------------------#
# #
# Rzeta_simul(s,k=0) #
# #
#-------------------------------------------------------------------------------#
# This function return a list with the values:
# Rzeta(sigma+it), conj(Rzeta(1-sigma+it)),Rzeta'(sigma+it), conj(Rzeta'(1-sigma+it)),
# .... , Rzeta^{(k)}(sigma+it), conj(Rzeta^{(k)}(1-sigma+it))
#
# Useful to compute the function zeta(s) and Z(w) or its derivatives.
#
def aux_M_Fp(ctx, xA, xeps4, a, xB1, xL):
# COMPUTING M NUMBER OF DERIVATIVES Fp[m] TO COMPUTE
# See II Section 3.11 equations (47) and (48)
aux1 = 126.0657606*xA/xeps4 # 126.06.. = 316/sqrt(2*pi)
aux1 = ctx.ln(aux1)
aux2 = (2*ctx.ln(ctx.pi)+ctx.ln(xB1)+ctx.ln(a))/3 -ctx.ln(2*ctx.pi)/2
m = 3*xL-3
aux3= (ctx.loggamma(m+1)-ctx.loggamma(m/3.0+2))/2 -ctx.loggamma((m+1)/2.)
while((aux1 < m*aux2+ aux3)and (m>1)):
m = m - 1
aux3 = (ctx.loggamma(m+1)-ctx.loggamma(m/3.0+2))/2 -ctx.loggamma((m+1)/2.)
xM = m
return xM
def aux_J_needed(ctx, xA, xeps4, a, xB1, xM):
# DETERMINATION OF J THE NUMBER OF TERMS NEEDED
# IN THE TAYLOR SERIES OF F.
# See II Section 3.11 equation (49))
# Only determine one
h1 = xeps4/(632*xA)
h2 = xB1*a * 126.31337419529260248 # = pi^2*e^2*sqrt(3)
h2 = h1 * ctx.power((h2/xM**2),(xM-1)/3) / xM
h3 = min(h1,h2)
return h3
def Rzeta_simul(ctx, s, der=0):
# First we take the value of ctx.prec
wpinitial = ctx.prec
# INITIALIZATION
# Take the real and imaginary part of s
t = ctx._im(s)
xsigma = ctx._re(s)
ysigma = 1 - xsigma
# Now compute several parameter that appear on the program
ctx.prec = 15
a = ctx.sqrt(t/(2*ctx.pi))
xasigma = a ** xsigma
yasigma = a ** ysigma
# We need a simple bound A1 < asigma (see II Section 3.1 and 3.3)
xA1=ctx.power(2, ctx.mag(xasigma)-1)
yA1=ctx.power(2, ctx.mag(yasigma)-1)
# We compute various epsilon's (see II end of Section 3.1)
eps = ctx.power(2, -wpinitial)
eps1 = eps/6.
xeps2 = eps * xA1/3.
yeps2 = eps * yA1/3.
# COMPUTING SOME COEFFICIENTS THAT DEPENDS
# ON sigma
# constant b and c (see I Theorem 2 formula (26) )
# coefficients A and B1 (see I Section 6.1 equation (50))
#
# here we not need high precision
ctx.prec = 15
if xsigma > 0:
xb = 2.
xc = math.pow(9,xsigma)/4.44288
# 4.44288 =(math.sqrt(2)*math.pi)
xA = math.pow(9,xsigma)
xB1 = 1
else:
xb = 2.25158 # math.sqrt( (3-2* math.log(2))*math.pi )
xc = math.pow(2,-xsigma)/4.44288
xA = math.pow(2,-xsigma)
xB1 = 1.10789 # = 2*sqrt(1-log(2))
if(ysigma > 0):
yb = 2.
yc = math.pow(9,ysigma)/4.44288
# 4.44288 =(math.sqrt(2)*math.pi)
yA = math.pow(9,ysigma)
yB1 = 1
else:
yb = 2.25158 # math.sqrt( (3-2* math.log(2))*math.pi )
yc = math.pow(2,-ysigma)/4.44288
yA = math.pow(2,-ysigma)
yB1 = 1.10789 # = 2*sqrt(1-log(2))
# COMPUTING L THE NUMBER OF TERMS NEEDED IN THE RIEMANN-SIEGEL
# CORRECTION
# See II Section 3.2
ctx.prec = 15
xL = 1
while 3*xc*ctx.gamma(xL*0.5) * ctx.power(xb*a,-xL) >= xeps2:
xL = xL+1
xL = max(2,xL)
yL = 1
while 3*yc*ctx.gamma(yL*0.5) * ctx.power(yb*a,-yL) >= yeps2:
yL = yL+1
yL = max(2,yL)
# The number L has to satify some conditions.
# If not RS can not compute Rzeta(s) with the prescribed precision
# (see II, Section 3.2 condition (20) ) and
# (II, Section 3.3 condition (22) ). Also we have added
# an additional technical condition in Section 3.17 Proposition 17
if ((3*xL >= 2*a*a/25.) or (3*xL+2+xsigma<0) or (abs(xsigma) > a/2.) or \
(3*yL >= 2*a*a/25.) or (3*yL+2+ysigma<0) or (abs(ysigma) > a/2.)):
ctx.prec = wpinitial
raise NotImplementedError("Riemann-Siegel can not compute with such precision")
# We take the maximum of the two values
L = max(xL, yL)
# INITIALIZATION (CONTINUATION)
#
# eps3 is the constant defined on (II, Section 3.5 equation (27) )
# each term of the RS correction must be computed with error <= eps3
xeps3 = xeps2/(4*xL)
yeps3 = yeps2/(4*yL)
# eps4 is defined on (II Section 3.6 equation (30) )
# each component of the formula (II Section 3.6 equation (29) )
# must be computed with error <= eps4
xeps4 = xeps3/(3*xL)
yeps4 = yeps3/(3*yL)
# COMPUTING M NUMBER OF DERIVATIVES Fp[m] TO COMPUTE
xM = aux_M_Fp(ctx, xA, xeps4, a, xB1, xL)
yM = aux_M_Fp(ctx, yA, yeps4, a, yB1, yL)
M = max(xM, yM)
# COMPUTING NUMBER OF TERMS J NEEDED
h3 = aux_J_needed(ctx, xA, xeps4, a, xB1, xM)
h4 = aux_J_needed(ctx, yA, yeps4, a, yB1, yM)
h3 = min(h3,h4)
J = 12
jvalue = (2*ctx.pi)**J / ctx.gamma(J+1)
while jvalue > h3:
J = J+1
jvalue = (2*ctx.pi)*jvalue/J
# COMPUTING eps5[m] for 1 <= m <= 21
# See II Section 10 equation (43)
# We choose the minimum of the two possibilities
eps5={}
xforeps5 = math.pi*math.pi*xB1*a
yforeps5 = math.pi*math.pi*yB1*a
for m in range(0,22):
xaux1 = math.pow(xforeps5, m/3)/(316.*xA)
yaux1 = math.pow(yforeps5, m/3)/(316.*yA)
aux1 = min(xaux1, yaux1)
aux2 = ctx.gamma(m+1)/ctx.gamma(m/3.0+0.5)
aux2 = math.sqrt(aux2)
eps5[m] = (aux1*aux2*min(xeps4,yeps4))
# COMPUTING wpfp
# See II Section 3.13 equation (59)
twenty = min(3*L-3, 21)+1
aux = 6812*J
wpfp = ctx.mag(44*J)
for m in range(0,twenty):
wpfp = max(wpfp, ctx.mag(aux*ctx.gamma(m+1)/eps5[m]))
# COMPUTING N AND p
# See II Section
ctx.prec = wpfp + ctx.mag(t)+20
a = ctx.sqrt(t/(2*ctx.pi))
N = ctx.floor(a)
p = 1-2*(a-N)
# now we get a rounded version of p
# to the precision wpfp
# this possibly is not necessary
num=ctx.floor(p*(ctx.mpf('2')**wpfp))
difference = p * (ctx.mpf('2')**wpfp)-num
if (difference < 0.5):
num = num
else:
num = num+1
p = ctx.convert(num * (ctx.mpf('2')**(-wpfp)))
# COMPUTING THE COEFFICIENTS c[n] = cc[n]
# We shall use the notation cc[n], since there is
# a constant that is called c
# See II Section 3.14
# We compute the coefficients and also save then in a
# cache. The bulk of the computation is passed to
# the function coef()
#
# eps6 is defined in II Section 3.13 equation (58)
eps6 = ctx.power(ctx.convert(2*ctx.pi), J)/(ctx.gamma(J+1)*3*J)
# Now we compute the coefficients
cc = {}
cont = {}
cont, pipowers = coef(ctx, J, eps6)
cc=cont.copy() # we need a copy since we have to change his values.
Fp={} # this is the adequate locus of this
for n in range(M, 3*L-2):
Fp[n] = 0
Fp={}
ctx.prec = wpfp
for m in range(0,M+1):
sumP = 0
for k in range(2*J-m-1,-1,-1):
sumP = (sumP * p)+ cc[k]
Fp[m] = sumP
# preparation of the new coefficients
for k in range(0,2*J-m-1):
cc[k] = (k+1)* cc[k+1]
# COMPUTING THE NUMBERS xd[u,n,k], yd[u,n,k]
# See II Section 3.17
#
# First we compute the working precisions xwpd[k]
# Se II equation (92)
xwpd={}
d1 = max(6,ctx.mag(40*L*L))
xd2 = 13+ctx.mag((1+abs(xsigma))*xA)-ctx.mag(xeps4)-1
xconst = ctx.ln(8/(ctx.pi*ctx.pi*a*a*xB1*xB1)) /2
for n in range(0,L):
xd3 = ctx.mag(ctx.sqrt(ctx.gamma(n-0.5)))-ctx.floor(n*xconst)+xd2
xwpd[n]=max(xd3,d1)
# procedure of II Section 3.17
ctx.prec = xwpd[1]+10
xpsigma = 1-(2*xsigma)
xd = {}
xd[0,0,-2]=0; xd[0,0,-1]=0; xd[0,0,0]=1; xd[0,0,1]=0
xd[0,-1,-2]=0; xd[0,-1,-1]=0; xd[0,-1,0]=1; xd[0,-1,1]=0
for n in range(1,L):
ctx.prec = xwpd[n]+10
for k in range(0,3*n//2+1):
m = 3*n-2*k
if(m!=0):
m1 = ctx.one/m
c1= m1/4
c2=(xpsigma*m1)/2
c3=-(m+1)
xd[0,n,k]=c3*xd[0,n-1,k-2]+c1*xd[0,n-1,k]+c2*xd[0,n-1,k-1]
else:
xd[0,n,k]=0
for r in range(0,k):
add=xd[0,n,r]*(ctx.mpf('1.0')*ctx.fac(2*k-2*r)/ctx.fac(k-r))
xd[0,n,k] -= ((-1)**(k-r))*add
xd[0,n,-2]=0; xd[0,n,-1]=0; xd[0,n,3*n//2+1]=0
for mu in range(-2,der+1):
for n in range(-2,L):
for k in range(-3,max(1,3*n//2+2)):
if( (mu<0)or (n<0) or(k<0)or (k>3*n//2)):
xd[mu,n,k] = 0
for mu in range(1,der+1):
for n in range(0,L):
ctx.prec = xwpd[n]+10
for k in range(0,3*n//2+1):
aux=(2*mu-2)*xd[mu-2,n-2,k-3]+2*(xsigma+n-2)*xd[mu-1,n-2,k-3]
xd[mu,n,k] = aux - xd[mu-1,n-1,k-1]
# Now we compute the working precisions ywpd[k]
# Se II equation (92)
ywpd={}
d1 = max(6,ctx.mag(40*L*L))
yd2 = 13+ctx.mag((1+abs(ysigma))*yA)-ctx.mag(yeps4)-1
yconst = ctx.ln(8/(ctx.pi*ctx.pi*a*a*yB1*yB1)) /2
for n in range(0,L):
yd3 = ctx.mag(ctx.sqrt(ctx.gamma(n-0.5)))-ctx.floor(n*yconst)+yd2
ywpd[n]=max(yd3,d1)
# procedure of II Section 3.17
ctx.prec = ywpd[1]+10
ypsigma = 1-(2*ysigma)
yd = {}
yd[0,0,-2]=0; yd[0,0,-1]=0; yd[0,0,0]=1; yd[0,0,1]=0
yd[0,-1,-2]=0; yd[0,-1,-1]=0; yd[0,-1,0]=1; yd[0,-1,1]=0
for n in range(1,L):
ctx.prec = ywpd[n]+10
for k in range(0,3*n//2+1):
m = 3*n-2*k
if(m!=0):
m1 = ctx.one/m
c1= m1/4
c2=(ypsigma*m1)/2
c3=-(m+1)
yd[0,n,k]=c3*yd[0,n-1,k-2]+c1*yd[0,n-1,k]+c2*yd[0,n-1,k-1]
else:
yd[0,n,k]=0
for r in range(0,k):
add=yd[0,n,r]*(ctx.mpf('1.0')*ctx.fac(2*k-2*r)/ctx.fac(k-r))
yd[0,n,k] -= ((-1)**(k-r))*add
yd[0,n,-2]=0; yd[0,n,-1]=0; yd[0,n,3*n//2+1]=0
for mu in range(-2,der+1):
for n in range(-2,L):
for k in range(-3,max(1,3*n//2+2)):
if( (mu<0)or (n<0) or(k<0)or (k>3*n//2)):
yd[mu,n,k] = 0
for mu in range(1,der+1):
for n in range(0,L):
ctx.prec = ywpd[n]+10
for k in range(0,3*n//2+1):
aux=(2*mu-2)*yd[mu-2,n-2,k-3]+2*(ysigma+n-2)*yd[mu-1,n-2,k-3]
yd[mu,n,k] = aux - yd[mu-1,n-1,k-1]
# COMPUTING THE COEFFICIENTS xtcoef[k,l]
# See II Section 3.9
#
# computing the needed wp
xwptcoef={}
xwpterm={}
ctx.prec = 15
c1 = ctx.mag(40*(L+2))
xc2 = ctx.mag(68*(L+2)*xA)
xc4 = ctx.mag(xB1*a*math.sqrt(ctx.pi))-1
for k in range(0,L):
xc3 = xc2 - k*xc4+ctx.mag(ctx.fac(k+0.5))/2.
xwptcoef[k] = (max(c1,xc3-ctx.mag(xeps4)+1)+1 +20)*1.5
xwpterm[k] = (max(c1,ctx.mag(L+2)+xc3-ctx.mag(xeps3)+1)+1 +20)
ywptcoef={}
ywpterm={}
ctx.prec = 15
c1 = ctx.mag(40*(L+2))
yc2 = ctx.mag(68*(L+2)*yA)
yc4 = ctx.mag(yB1*a*math.sqrt(ctx.pi))-1
for k in range(0,L):
yc3 = yc2 - k*yc4+ctx.mag(ctx.fac(k+0.5))/2.
ywptcoef[k] = ((max(c1,yc3-ctx.mag(yeps4)+1))+10)*1.5
ywpterm[k] = (max(c1,ctx.mag(L+2)+yc3-ctx.mag(yeps3)+1)+1)+10
# check of power of pi
# computing the fortcoef[mu,k,ell]
xfortcoef={}
for mu in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
xfortcoef[mu,k,ell]=0
for mu in range(0,der+1):
for k in range(0,L):
ctx.prec = xwptcoef[k]
for ell in range(0,3*k//2+1):
xfortcoef[mu,k,ell]=xd[mu,k,ell]*Fp[3*k-2*ell]/pipowers[2*k-ell]
xfortcoef[mu,k,ell]=xfortcoef[mu,k,ell]/((2*ctx.j)**ell)
def trunc_a(t):
wp = ctx.prec
ctx.prec = wp + 2
aa = ctx.sqrt(t/(2*ctx.pi))
ctx.prec = wp
return aa
# computing the tcoef[k,ell]
xtcoef={}
for mu in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
xtcoef[mu,k,ell]=0
ctx.prec = max(xwptcoef[0],ywptcoef[0])+3
aa= trunc_a(t)
la = -ctx.ln(aa)
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = xwptcoef[k]
for ell in range(0,3*k//2+1):
xtcoef[chi,k,ell] =0
for mu in range(0, chi+1):
tcoefter=ctx.binomial(chi,mu)*ctx.power(la,mu)*xfortcoef[chi-mu,k,ell]
xtcoef[chi,k,ell] += tcoefter
# COMPUTING THE COEFFICIENTS ytcoef[k,l]
# See II Section 3.9
#
# computing the needed wp
# check of power of pi
# computing the fortcoef[mu,k,ell]
yfortcoef={}
for mu in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
yfortcoef[mu,k,ell]=0
for mu in range(0,der+1):
for k in range(0,L):
ctx.prec = ywptcoef[k]
for ell in range(0,3*k//2+1):
yfortcoef[mu,k,ell]=yd[mu,k,ell]*Fp[3*k-2*ell]/pipowers[2*k-ell]
yfortcoef[mu,k,ell]=yfortcoef[mu,k,ell]/((2*ctx.j)**ell)
# computing the tcoef[k,ell]
ytcoef={}
for chi in range(0,der+1):
for k in range(0,L):
for ell in range(-2,3*k//2+1):
ytcoef[chi,k,ell]=0
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = ywptcoef[k]
for ell in range(0,3*k//2+1):
ytcoef[chi,k,ell] =0
for mu in range(0, chi+1):
tcoefter=ctx.binomial(chi,mu)*ctx.power(la,mu)*yfortcoef[chi-mu,k,ell]
ytcoef[chi,k,ell] += tcoefter
# COMPUTING tv[k,ell]
# See II Section 3.8
#
# a has a good value
ctx.prec = max(xwptcoef[0], ywptcoef[0])+2
av = {}
av[0] = 1
av[1] = av[0]/a
ctx.prec = max(xwptcoef[0],ywptcoef[0])
for k in range(2,L):
av[k] = av[k-1] * av[1]
# Computing the quotients
xtv = {}
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = xwptcoef[k]
for ell in range(0,3*k//2+1):
xtv[chi,k,ell] = xtcoef[chi,k,ell]* av[k]
# Computing the quotients
ytv = {}
for chi in range(0,der+1):
for k in range(0,L):
ctx.prec = ywptcoef[k]
for ell in range(0,3*k//2+1):
ytv[chi,k,ell] = ytcoef[chi,k,ell]* av[k]
# COMPUTING THE TERMS xterm[k]
# See II Section 3.6
xterm = {}
for chi in range(0,der+1):
for n in range(0,L):
ctx.prec = xwpterm[n]
te = 0
for k in range(0, 3*n//2+1):
te += xtv[chi,n,k]
xterm[chi,n] = te
# COMPUTING THE TERMS yterm[k]
# See II Section 3.6
yterm = {}
for chi in range(0,der+1):
for n in range(0,L):
ctx.prec = ywpterm[n]
te = 0
for k in range(0, 3*n//2+1):
te += ytv[chi,n,k]
yterm[chi,n] = te
# COMPUTING rssum
# See II Section 3.5
xrssum={}
ctx.prec=15
xrsbound = math.sqrt(ctx.pi) * xc /(xb*a)
ctx.prec=15
xwprssum = ctx.mag(4.4*((L+3)**2)*xrsbound / xeps2)
xwprssum = max(xwprssum, ctx.mag(10*(L+1)))
ctx.prec = xwprssum
for chi in range(0,der+1):
xrssum[chi] = 0
for k in range(1,L+1):
xrssum[chi] += xterm[chi,L-k]
yrssum={}
ctx.prec=15
yrsbound = math.sqrt(ctx.pi) * yc /(yb*a)
ctx.prec=15
ywprssum = ctx.mag(4.4*((L+3)**2)*yrsbound / yeps2)
ywprssum = max(ywprssum, ctx.mag(10*(L+1)))
ctx.prec = ywprssum
for chi in range(0,der+1):
yrssum[chi] = 0
for k in range(1,L+1):
yrssum[chi] += yterm[chi,L-k]
# COMPUTING S3
# See II Section 3.19
ctx.prec = 15
A2 = 2**(max(ctx.mag(abs(xrssum[0])), ctx.mag(abs(yrssum[0]))))
eps8 = eps/(3*A2)
T = t *ctx.ln(t/(2*ctx.pi))
xwps3 = 5 + ctx.mag((1+(2/eps8)*ctx.power(a,-xsigma))*T)
ywps3 = 5 + ctx.mag((1+(2/eps8)*ctx.power(a,-ysigma))*T)
ctx.prec = max(xwps3, ywps3)
tpi = t/(2*ctx.pi)
arg = (t/2)*ctx.ln(tpi)-(t/2)-ctx.pi/8
U = ctx.expj(-arg)
a = trunc_a(t)
xasigma = ctx.power(a, -xsigma)
yasigma = ctx.power(a, -ysigma)
xS3 = ((-1)**(N-1)) * xasigma * U
yS3 = ((-1)**(N-1)) * yasigma * U
# COMPUTING S1 the zetasum
# See II Section 3.18
ctx.prec = 15
xwpsum = 4+ ctx.mag((N+ctx.power(N,1-xsigma))*ctx.ln(N) /eps1)
ywpsum = 4+ ctx.mag((N+ctx.power(N,1-ysigma))*ctx.ln(N) /eps1)
wpsum = max(xwpsum, ywpsum)
ctx.prec = wpsum +10
'''
# This can be improved
xS1={}
yS1={}
for chi in range(0,der+1):
xS1[chi] = 0
yS1[chi] = 0
for n in range(1,int(N)+1):
ln = ctx.ln(n)
xexpn = ctx.exp(-ln*(xsigma+ctx.j*t))
yexpn = ctx.conj(1/(n*xexpn))
for chi in range(0,der+1):
pown = ctx.power(-ln, chi)
xterm = pown*xexpn
yterm = pown*yexpn
xS1[chi] += xterm
yS1[chi] += yterm
'''
xS1, yS1 = ctx._zetasum(s, 1, int(N)-1, range(0,der+1), True)
# END OF COMPUTATION of xrz, yrz
# See II Section 3.1
ctx.prec = 15
xabsS1 = abs(xS1[der])
xabsS2 = abs(xrssum[der] * xS3)
xwpend = max(6, wpinitial+ctx.mag(6*(3*xabsS1+7*xabsS2) ) )
ctx.prec = xwpend
xrz={}
for chi in range(0,der+1):
xrz[chi] = xS1[chi]+xrssum[chi]*xS3
ctx.prec = 15
yabsS1 = abs(yS1[der])
yabsS2 = abs(yrssum[der] * yS3)
ywpend = max(6, wpinitial+ctx.mag(6*(3*yabsS1+7*yabsS2) ) )
ctx.prec = ywpend
yrz={}
for chi in range(0,der+1):
yrz[chi] = yS1[chi]+yrssum[chi]*yS3
yrz[chi] = ctx.conj(yrz[chi])
ctx.prec = wpinitial
return xrz, yrz
def Rzeta_set(ctx, s, derivatives=[0]):
r"""
Computes several derivatives of the auxiliary function of Riemann `R(s)`.
**Definition**
The function is defined by
.. math ::
\begin{equation}
{\mathop{\mathcal R }\nolimits}(s)=
\int_{0\swarrow1}\frac{x^{-s} e^{\pi i x^2}}{e^{\pi i x}-
e^{-\pi i x}}\,dx
\end{equation}
To this function we apply the Riemann-Siegel expansion.
"""
der = max(derivatives)
# First we take the value of ctx.prec
# During the computation we will change ctx.prec, and finally we will
# restaurate the initial value
wpinitial = ctx.prec
# Take the real and imaginary part of s
t = ctx._im(s)
sigma = ctx._re(s)
# Now compute several parameter that appear on the program
ctx.prec = 15
a = ctx.sqrt(t/(2*ctx.pi)) # Careful
asigma = ctx.power(a, sigma) # Careful
# We need a simple bound A1 < asigma (see II Section 3.1 and 3.3)
A1 = ctx.power(2, ctx.mag(asigma)-1)
# We compute various epsilon's (see II end of Section 3.1)
eps = ctx.power(2, -wpinitial)
eps1 = eps/6.
eps2 = eps * A1/3.
# COMPUTING SOME COEFFICIENTS THAT DEPENDS
# ON sigma
# constant b and c (see I Theorem 2 formula (26) )
# coefficients A and B1 (see I Section 6.1 equation (50))
# here we not need high precision
ctx.prec = 15
if sigma > 0:
b = 2.
c = math.pow(9,sigma)/4.44288
# 4.44288 =(math.sqrt(2)*math.pi)
A = math.pow(9,sigma)
B1 = 1
else:
b = 2.25158 # math.sqrt( (3-2* math.log(2))*math.pi )
c = math.pow(2,-sigma)/4.44288
A = math.pow(2,-sigma)
B1 = 1.10789 # = 2*sqrt(1-log(2))
# COMPUTING L THE NUMBER OF TERMS NEEDED IN THE RIEMANN-SIEGEL
# CORRECTION
# See II Section 3.2
ctx.prec = 15
L = 1
while 3*c*ctx.gamma(L*0.5) * ctx.power(b*a,-L) >= eps2:
L = L+1
L = max(2,L)
# The number L has to satify some conditions.
# If not RS can not compute Rzeta(s) with the prescribed precision
# (see II, Section 3.2 condition (20) ) and
# (II, Section 3.3 condition (22) ). Also we have added
# an additional technical condition in Section 3.17 Proposition 17
if ((3*L >= 2*a*a/25.) or (3*L+2+sigma<0) or (abs(sigma)> a/2.)):
#print 'Error Riemann-Siegel can not compute with such precision'
ctx.prec = wpinitial
raise NotImplementedError("Riemann-Siegel can not compute with such precision")
# INITIALIZATION (CONTINUATION)
#
# eps3 is the constant defined on (II, Section 3.5 equation (27) )
# each term of the RS correction must be computed with error <= eps3
eps3 = eps2/(4*L)
# eps4 is defined on (II Section 3.6 equation (30) )
# each component of the formula (II Section 3.6 equation (29) )
# must be computed with error <= eps4
eps4 = eps3/(3*L)
# COMPUTING M. NUMBER OF DERIVATIVES Fp[m] TO COMPUTE
M = aux_M_Fp(ctx, A, eps4, a, B1, L)
Fp = {}
for n in range(M, 3*L-2):
Fp[n] = 0
# But I have not seen an instance of M != 3*L-3
#
# DETERMINATION OF J THE NUMBER OF TERMS NEEDED
# IN THE TAYLOR SERIES OF F.
# See II Section 3.11 equation (49))
h1 = eps4/(632*A)
h2 = ctx.pi*ctx.pi*B1*a *ctx.sqrt(3)*math.e*math.e
h2 = h1 * ctx.power((h2/M**2),(M-1)/3) / M
h3 = min(h1,h2)
J=12
jvalue = (2*ctx.pi)**J / ctx.gamma(J+1)
while jvalue > h3:
J = J+1
jvalue = (2*ctx.pi)*jvalue/J
# COMPUTING eps5[m] for 1 <= m <= 21
# See II Section 10 equation (43)
eps5={}
foreps5 = math.pi*math.pi*B1*a
for m in range(0,22):
aux1 = math.pow(foreps5, m/3)/(316.*A)
aux2 = ctx.gamma(m+1)/ctx.gamma(m/3.0+0.5)
aux2 = math.sqrt(aux2)
eps5[m] = aux1*aux2*eps4
# COMPUTING wpfp
# See II Section 3.13 equation (59)
twenty = min(3*L-3, 21)+1
aux = 6812*J
wpfp = ctx.mag(44*J)
for m in range(0, twenty):
wpfp = max(wpfp, ctx.mag(aux*ctx.gamma(m+1)/eps5[m]))
# COMPUTING N AND p
# See II Section
ctx.prec = wpfp + ctx.mag(t) + 20
a = ctx.sqrt(t/(2*ctx.pi))
N = ctx.floor(a)
p = 1-2*(a-N)
# now we get a rounded version of p to the precision wpfp
# this possibly is not necessary
num = ctx.floor(p*(ctx.mpf(2)**wpfp))
difference = p * (ctx.mpf(2)**wpfp)-num
if difference < 0.5:
num = num
else:
num = num+1
p = ctx.convert(num * (ctx.mpf(2)**(-wpfp)))
# COMPUTING THE COEFFICIENTS c[n] = cc[n]
# We shall use the notation cc[n], since there is
# a constant that is called c
# See II Section 3.14
# We compute the coefficients and also save then in a
# cache. The bulk of the computation is passed to
# the function coef()
#
# eps6 is defined in II Section 3.13 equation (58)
eps6 = ctx.power(2*ctx.pi, J)/(ctx.gamma(J+1)*3*J)
# Now we compute the coefficients
cc={}
cont={}
cont, pipowers = coef(ctx, J, eps6)
cc = cont.copy() # we need a copy since we have
Fp={}
for n in range(M, 3*L-2):
Fp[n] = 0
ctx.prec = wpfp
for m in range(0,M+1):
sumP = 0
for k in range(2*J-m-1,-1,-1):
sumP = (sumP * p) + cc[k]
Fp[m] = sumP
# preparation of the new coefficients
for k in range(0, 2*J-m-1):
cc[k] = (k+1) * cc[k+1]
# COMPUTING THE NUMBERS d[n,k]
# See II Section 3.17
# First we compute the working precisions wpd[k]
# Se II equation (92)
wpd = {}
d1 = max(6, ctx.mag(40*L*L))
d2 = 13+ctx.mag((1+abs(sigma))*A)-ctx.mag(eps4)-1
const = ctx.ln(8/(ctx.pi*ctx.pi*a*a*B1*B1)) /2
for n in range(0,L):
d3 = ctx.mag(ctx.sqrt(ctx.gamma(n-0.5)))-ctx.floor(n*const)+d2
wpd[n] = max(d3,d1)
# procedure of II Section 3.17
ctx.prec = wpd[1]+10
psigma = 1-(2*sigma)
d = {}
d[0,0,-2]=0; d[0,0,-1]=0; d[0,0,0]=1; d[0,0,1]=0
d[0,-1,-2]=0; d[0,-1,-1]=0; d[0,-1,0]=1; d[0,-1,1]=0
for n in range(1,L):
ctx.prec = wpd[n]+10
for k in range(0,3*n//2+1):
m = 3*n-2*k
if (m!=0):
m1 = ctx.one/m
c1 = m1/4
c2 = (psigma*m1)/2
c3 = -(m+1)
d[0,n,k] = c3*d[0,n-1,k-2]+c1*d[0,n-1,k]+c2*d[0,n-1,k-1]
else:
d[0,n,k]=0
for r in range(0,k):
add = d[0,n,r]*(ctx.one*ctx.fac(2*k-2*r)/ctx.fac(k-r))
d[0,n,k] -= ((-1)**(k-r))*add
d[0,n,-2]=0; d[0,n,-1]=0; d[0,n,3*n//2+1]=0
for mu in range(-2,der+1):
for n in range(-2,L):
for k in range(-3,max(1,3*n//2+2)):
if ((mu<0)or (n<0) or(k<0)or (k>3*n//2)):
d[mu,n,k] = 0
for mu in range(1,der+1):
for n in range(0,L):
ctx.prec = wpd[n]+10
for k in range(0,3*n//2+1):
aux=(2*mu-2)*d[mu-2,n-2,k-3]+2*(sigma+n-2)*d[mu-1,n-2,k-3]
d[mu,n,k] = aux - d[mu-1,n-1,k-1]
# COMPUTING THE COEFFICIENTS t[k,l]
# See II Section 3.9
#
# computing the needed wp
wptcoef = {}
wpterm = {}
ctx.prec = 15
c1 = ctx.mag(40*(L+2))
c2 = ctx.mag(68*(L+2)*A)
c4 = ctx.mag(B1*a*math.sqrt(ctx.pi))-1
for k in range(0,L):
c3 = c2 - k*c4+ctx.mag(ctx.fac(k+0.5))/2.
wptcoef[k] = max(c1,c3-ctx.mag(eps4)+1)+1 +10
wpterm[k] = max(c1,ctx.mag(L+2)+c3-ctx.mag(eps3)+1)+1 +10
# check of power of pi
# computing the fortcoef[mu,k,ell]
fortcoef={}
for mu in derivatives:
for k in range(0,L):
for ell in range(-2,3*k//2+1):
fortcoef[mu,k,ell]=0
for mu in derivatives:
for k in range(0,L):
ctx.prec = wptcoef[k]
for ell in range(0,3*k//2+1):
fortcoef[mu,k,ell]=d[mu,k,ell]*Fp[3*k-2*ell]/pipowers[2*k-ell]
fortcoef[mu,k,ell]=fortcoef[mu,k,ell]/((2*ctx.j)**ell)
def trunc_a(t):
wp = ctx.prec
ctx.prec = wp + 2
aa = ctx.sqrt(t/(2*ctx.pi))
ctx.prec = wp
return aa
# computing the tcoef[chi,k,ell]
tcoef={}
for chi in derivatives:
for k in range(0,L):
for ell in range(-2,3*k//2+1):
tcoef[chi,k,ell]=0
ctx.prec = wptcoef[0]+3
aa = trunc_a(t)
la = -ctx.ln(aa)
for chi in derivatives:
for k in range(0,L):
ctx.prec = wptcoef[k]
for ell in range(0,3*k//2+1):
tcoef[chi,k,ell] = 0
for mu in range(0, chi+1):
tcoefter = ctx.binomial(chi,mu) * la**mu * \
fortcoef[chi-mu,k,ell]
tcoef[chi,k,ell] += tcoefter
# COMPUTING tv[k,ell]
# See II Section 3.8
# Computing the powers av[k] = a**(-k)
ctx.prec = wptcoef[0] + 2
# a has a good value of a.
# See II Section 3.6
av = {}
av[0] = 1
av[1] = av[0]/a
ctx.prec = wptcoef[0]
for k in range(2,L):
av[k] = av[k-1] * av[1]
# Computing the quotients
tv = {}
for chi in derivatives:
for k in range(0,L):
ctx.prec = wptcoef[k]
for ell in range(0,3*k//2+1):
tv[chi,k,ell] = tcoef[chi,k,ell]* av[k]
# COMPUTING THE TERMS term[k]
# See II Section 3.6
term = {}
for chi in derivatives:
for n in range(0,L):
ctx.prec = wpterm[n]
te = 0
for k in range(0, 3*n//2+1):
te += tv[chi,n,k]
term[chi,n] = te
# COMPUTING rssum
# See II Section 3.5
rssum={}
ctx.prec=15
rsbound = math.sqrt(ctx.pi) * c /(b*a)
ctx.prec=15
wprssum = ctx.mag(4.4*((L+3)**2)*rsbound / eps2)
wprssum = max(wprssum, ctx.mag(10*(L+1)))
ctx.prec = wprssum
for chi in derivatives:
rssum[chi] = 0
for k in range(1,L+1):
rssum[chi] += term[chi,L-k]
# COMPUTING S3
# See II Section 3.19
ctx.prec = 15
A2 = 2**(ctx.mag(rssum[0]))
eps8 = eps/(3* A2)
T = t * ctx.ln(t/(2*ctx.pi))
wps3 = 5 + ctx.mag((1+(2/eps8)*ctx.power(a,-sigma))*T)
ctx.prec = wps3
tpi = t/(2*ctx.pi)
arg = (t/2)*ctx.ln(tpi)-(t/2)-ctx.pi/8
U = ctx.expj(-arg)
a = trunc_a(t)
asigma = ctx.power(a, -sigma)
S3 = ((-1)**(N-1)) * asigma * U
# COMPUTING S1 the zetasum
# See II Section 3.18
ctx.prec = 15
wpsum = 4 + ctx.mag((N+ctx.power(N,1-sigma))*ctx.ln(N)/eps1)
ctx.prec = wpsum + 10
'''
# This can be improved
S1 = {}
for chi in derivatives:
S1[chi] = 0
for n in range(1,int(N)+1):
ln = ctx.ln(n)
expn = ctx.exp(-ln*(sigma+ctx.j*t))
for chi in derivatives:
term = ctx.power(-ln, chi)*expn
S1[chi] += term
'''
S1 = ctx._zetasum(s, 1, int(N)-1, derivatives)[0]
# END OF COMPUTATION
# See II Section 3.1
ctx.prec = 15
absS1 = abs(S1[der])
absS2 = abs(rssum[der] * S3)
wpend = max(6, wpinitial + ctx.mag(6*(3*absS1+7*absS2)))
ctx.prec = wpend
rz = {}
for chi in derivatives:
rz[chi] = S1[chi]+rssum[chi]*S3
ctx.prec = wpinitial
return rz
def z_half(ctx,t,der=0):
r"""
z_half(t,der=0) Computes Z^(der)(t)
"""
s=ctx.mpf('0.5')+ctx.j*t
wpinitial = ctx.prec
ctx.prec = 15
tt = t/(2*ctx.pi)
wptheta = wpinitial +1 + ctx.mag(3*(tt**1.5)*ctx.ln(tt))
wpz = wpinitial + 1 + ctx.mag(12*tt*ctx.ln(tt))
ctx.prec = wptheta
theta = ctx.siegeltheta(t)
ctx.prec = wpz
rz = Rzeta_set(ctx,s, range(der+1))
if der > 0: ps1 = ctx._re(ctx.psi(0,s/2)/2 - ctx.ln(ctx.pi)/2)
if der > 1: ps2 = ctx._re(ctx.j*ctx.psi(1,s/2)/4)
if der > 2: ps3 = ctx._re(-ctx.psi(2,s/2)/8)
if der > 3: ps4 = ctx._re(-ctx.j*ctx.psi(3,s/2)/16)
exptheta = ctx.expj(theta)
if der == 0:
z = 2*exptheta*rz[0]
if der == 1:
zf = 2j*exptheta
z = zf*(ps1*rz[0]+rz[1])
if der == 2:
zf = 2 * exptheta
z = -zf*(2*rz[1]*ps1+rz[0]*ps1**2+rz[2]-ctx.j*rz[0]*ps2)
if der == 3:
zf = -2j*exptheta
z = 3*rz[1]*ps1**2+rz[0]*ps1**3+3*ps1*rz[2]
z = zf*(z-3j*rz[1]*ps2-3j*rz[0]*ps1*ps2+rz[3]-rz[0]*ps3)
if der == 4:
zf = 2*exptheta
z = 4*rz[1]*ps1**3+rz[0]*ps1**4+6*ps1**2*rz[2]
z = z-12j*rz[1]*ps1*ps2-6j*rz[0]*ps1**2*ps2-6j*rz[2]*ps2-3*rz[0]*ps2*ps2
z = z + 4*ps1*rz[3]-4*rz[1]*ps3-4*rz[0]*ps1*ps3+rz[4]+ctx.j*rz[0]*ps4
z = zf*z
ctx.prec = wpinitial
return ctx._re(z)
def zeta_half(ctx, s, k=0):
"""
zeta_half(s,k=0) Computes zeta^(k)(s) when Re s = 0.5
"""
wpinitial = ctx.prec
sigma = ctx._re(s)
t = ctx._im(s)
#--- compute wptheta, wpR, wpbasic ---
ctx.prec = 53
# X see II Section 3.21 (109) and (110)
if sigma > 0:
X = ctx.sqrt(abs(s))
else:
X = (2*ctx.pi)**(sigma-1) * abs(1-s)**(0.5-sigma)
# M1 see II Section 3.21 (111) and (112)
if sigma > 0:
M1 = 2*ctx.sqrt(t/(2*ctx.pi))
else:
M1 = 4 * t * X
# T see II Section 3.21 (113)
abst = abs(0.5-s)
T = 2* abst*math.log(abst)
# computing wpbasic, wptheta, wpR see II Section 3.21
wpbasic = max(6,3+ctx.mag(t))
wpbasic2 = 2+ctx.mag(2.12*M1+21.2*M1*X+1.3*M1*X*T)+wpinitial+1
wpbasic = max(wpbasic, wpbasic2)
wptheta = max(4, 3+ctx.mag(2.7*M1*X)+wpinitial+1)
wpR = 3+ctx.mag(1.1+2*X)+wpinitial+1
ctx.prec = wptheta
theta = ctx.siegeltheta(t-ctx.j*(sigma-ctx.mpf('0.5')))
if k > 0: ps1 = (ctx._re(ctx.psi(0,s/2)))/2 - ctx.ln(ctx.pi)/2
if k > 1: ps2 = -(ctx._im(ctx.psi(1,s/2)))/4
if k > 2: ps3 = -(ctx._re(ctx.psi(2,s/2)))/8
if k > 3: ps4 = (ctx._im(ctx.psi(3,s/2)))/16
ctx.prec = wpR
xrz = Rzeta_set(ctx,s,range(k+1))
yrz={}
for chi in range(0,k+1):
yrz[chi] = ctx.conj(xrz[chi])
ctx.prec = wpbasic
exptheta = ctx.expj(-2*theta)
if k==0:
zv = xrz[0]+exptheta*yrz[0]
if k==1:
zv1 = -yrz[1] - 2*yrz[0]*ps1
zv = xrz[1] + exptheta*zv1
if k==2:
zv1 = 4*yrz[1]*ps1+4*yrz[0]*(ps1**2)+yrz[2]+2j*yrz[0]*ps2
zv = xrz[2]+exptheta*zv1
if k==3:
zv1 = -12*yrz[1]*ps1**2-8*yrz[0]*ps1**3-6*yrz[2]*ps1-6j*yrz[1]*ps2
zv1 = zv1 - 12j*yrz[0]*ps1*ps2-yrz[3]+2*yrz[0]*ps3
zv = xrz[3]+exptheta*zv1
if k == 4:
zv1 = 32*yrz[1]*ps1**3 +16*yrz[0]*ps1**4+24*yrz[2]*ps1**2
zv1 = zv1 +48j*yrz[1]*ps1*ps2+48j*yrz[0]*(ps1**2)*ps2
zv1 = zv1+12j*yrz[2]*ps2-12*yrz[0]*ps2**2+8*yrz[3]*ps1-8*yrz[1]*ps3
zv1 = zv1-16*yrz[0]*ps1*ps3+yrz[4]-2j*yrz[0]*ps4
zv = xrz[4]+exptheta*zv1
ctx.prec = wpinitial
return zv
def zeta_offline(ctx, s, k=0):
"""
Computes zeta^(k)(s) off the line
"""
wpinitial = ctx.prec
sigma = ctx._re(s)
t = ctx._im(s)
#--- compute wptheta, wpR, wpbasic ---
ctx.prec = 53
# X see II Section 3.21 (109) and (110)
if sigma > 0:
X = ctx.power(abs(s), 0.5)
else:
X = ctx.power(2*ctx.pi, sigma-1)*ctx.power(abs(1-s),0.5-sigma)
# M1 see II Section 3.21 (111) and (112)
if (sigma > 0):
M1 = 2*ctx.sqrt(t/(2*ctx.pi))
else:
M1 = 4 * t * X
# M2 see II Section 3.21 (111) and (112)
if (1-sigma > 0):
M2 = 2*ctx.sqrt(t/(2*ctx.pi))
else:
M2 = 4*t*ctx.power(2*ctx.pi, -sigma)*ctx.power(abs(s),sigma-0.5)
# T see II Section 3.21 (113)
abst = abs(0.5-s)
T = 2* abst*math.log(abst)
# computing wpbasic, wptheta, wpR see II Section 3.21
wpbasic = max(6,3+ctx.mag(t))
wpbasic2 = 2+ctx.mag(2.12*M1+21.2*M2*X+1.3*M2*X*T)+wpinitial+1
wpbasic = max(wpbasic, wpbasic2)
wptheta = max(4, 3+ctx.mag(2.7*M2*X)+wpinitial+1)
wpR = 3+ctx.mag(1.1+2*X)+wpinitial+1
ctx.prec = wptheta
theta = ctx.siegeltheta(t-ctx.j*(sigma-ctx.mpf('0.5')))
s1 = s
s2 = ctx.conj(1-s1)
ctx.prec = wpR
xrz, yrz = Rzeta_simul(ctx, s, k)
if k > 0: ps1 = (ctx.psi(0,s1/2)+ctx.psi(0,(1-s1)/2))/4 - ctx.ln(ctx.pi)/2
if k > 1: ps2 = ctx.j*(ctx.psi(1,s1/2)-ctx.psi(1,(1-s1)/2))/8
if k > 2: ps3 = -(ctx.psi(2,s1/2)+ctx.psi(2,(1-s1)/2))/16
if k > 3: ps4 = -ctx.j*(ctx.psi(3,s1/2)-ctx.psi(3,(1-s1)/2))/32
ctx.prec = wpbasic
exptheta = ctx.expj(-2*theta)
if k == 0:
zv = xrz[0]+exptheta*yrz[0]
if k == 1:
zv1 = -yrz[1]-2*yrz[0]*ps1
zv = xrz[1]+exptheta*zv1
if k == 2:
zv1 = 4*yrz[1]*ps1+4*yrz[0]*(ps1**2) +yrz[2]+2j*yrz[0]*ps2
zv = xrz[2]+exptheta*zv1
if k == 3:
zv1 = -12*yrz[1]*ps1**2 -8*yrz[0]*ps1**3-6*yrz[2]*ps1-6j*yrz[1]*ps2
zv1 = zv1 - 12j*yrz[0]*ps1*ps2-yrz[3]+2*yrz[0]*ps3
zv = xrz[3]+exptheta*zv1
if k == 4:
zv1 = 32*yrz[1]*ps1**3 +16*yrz[0]*ps1**4+24*yrz[2]*ps1**2
zv1 = zv1 +48j*yrz[1]*ps1*ps2+48j*yrz[0]*(ps1**2)*ps2
zv1 = zv1+12j*yrz[2]*ps2-12*yrz[0]*ps2**2+8*yrz[3]*ps1-8*yrz[1]*ps3
zv1 = zv1-16*yrz[0]*ps1*ps3+yrz[4]-2j*yrz[0]*ps4
zv = xrz[4]+exptheta*zv1
ctx.prec = wpinitial
return zv
def z_offline(ctx, w, k=0):
r"""
Computes Z(w) and its derivatives off the line
"""
s = ctx.mpf('0.5')+ctx.j*w
s1 = s
s2 = ctx.conj(1-s1)
wpinitial = ctx.prec
ctx.prec = 35
# X see II Section 3.21 (109) and (110)
# M1 see II Section 3.21 (111) and (112)
if (ctx._re(s1) >= 0):
M1 = 2*ctx.sqrt(ctx._im(s1)/(2 * ctx.pi))
X = ctx.sqrt(abs(s1))
else:
X = (2*ctx.pi)**(ctx._re(s1)-1) * abs(1-s1)**(0.5-ctx._re(s1))
M1 = 4 * ctx._im(s1)*X
# M2 see II Section 3.21 (111) and (112)
if (ctx._re(s2) >= 0):
M2 = 2*ctx.sqrt(ctx._im(s2)/(2 * ctx.pi))
else:
M2 = 4 * ctx._im(s2)*(2*ctx.pi)**(ctx._re(s2)-1)*abs(1-s2)**(0.5-ctx._re(s2))
# T see II Section 3.21 Prop. 27
T = 2*abs(ctx.siegeltheta(w))
# defining some precisions
# see II Section 3.22 (115), (116), (117)
aux1 = ctx.sqrt(X)
aux2 = aux1*(M1+M2)
aux3 = 3 +wpinitial
wpbasic = max(6, 3+ctx.mag(T), ctx.mag(aux2*(26+2*T))+aux3)
wptheta = max(4,ctx.mag(2.04*aux2)+aux3)
wpR = ctx.mag(4*aux1)+aux3
# now the computations
ctx.prec = wptheta
theta = ctx.siegeltheta(w)
ctx.prec = wpR
xrz, yrz = Rzeta_simul(ctx,s,k)
pta = 0.25 + 0.5j*w
ptb = 0.25 - 0.5j*w
if k > 0: ps1 = 0.25*(ctx.psi(0,pta)+ctx.psi(0,ptb)) - ctx.ln(ctx.pi)/2
if k > 1: ps2 = (1j/8)*(ctx.psi(1,pta)-ctx.psi(1,ptb))
if k > 2: ps3 = (-1./16)*(ctx.psi(2,pta)+ctx.psi(2,ptb))
if k > 3: ps4 = (-1j/32)*(ctx.psi(3,pta)-ctx.psi(3,ptb))
ctx.prec = wpbasic
exptheta = ctx.expj(theta)
if k == 0:
zv = exptheta*xrz[0]+yrz[0]/exptheta
j = ctx.j
if k == 1:
zv = j*exptheta*(xrz[1]+xrz[0]*ps1)-j*(yrz[1]+yrz[0]*ps1)/exptheta
if k == 2:
zv = exptheta*(-2*xrz[1]*ps1-xrz[0]*ps1**2-xrz[2]+j*xrz[0]*ps2)
zv =zv + (-2*yrz[1]*ps1-yrz[0]*ps1**2-yrz[2]-j*yrz[0]*ps2)/exptheta
if k == 3:
zv1 = -3*xrz[1]*ps1**2-xrz[0]*ps1**3-3*xrz[2]*ps1+j*3*xrz[1]*ps2
zv1 = (zv1+ 3j*xrz[0]*ps1*ps2-xrz[3]+xrz[0]*ps3)*j*exptheta
zv2 = 3*yrz[1]*ps1**2+yrz[0]*ps1**3+3*yrz[2]*ps1+j*3*yrz[1]*ps2
zv2 = j*(zv2 + 3j*yrz[0]*ps1*ps2+ yrz[3]-yrz[0]*ps3)/exptheta
zv = zv1+zv2
if k == 4:
zv1 = 4*xrz[1]*ps1**3+xrz[0]*ps1**4 + 6*xrz[2]*ps1**2
zv1 = zv1-12j*xrz[1]*ps1*ps2-6j*xrz[0]*ps1**2*ps2-6j*xrz[2]*ps2
zv1 = zv1-3*xrz[0]*ps2*ps2+4*xrz[3]*ps1-4*xrz[1]*ps3-4*xrz[0]*ps1*ps3
zv1 = zv1+xrz[4]+j*xrz[0]*ps4
zv2 = 4*yrz[1]*ps1**3+yrz[0]*ps1**4 + 6*yrz[2]*ps1**2
zv2 = zv2+12j*yrz[1]*ps1*ps2+6j*yrz[0]*ps1**2*ps2+6j*yrz[2]*ps2
zv2 = zv2-3*yrz[0]*ps2*ps2+4*yrz[3]*ps1-4*yrz[1]*ps3-4*yrz[0]*ps1*ps3
zv2 = zv2+yrz[4]-j*yrz[0]*ps4
zv = exptheta*zv1+zv2/exptheta
ctx.prec = wpinitial
return zv
@defun
def rs_zeta(ctx, s, derivative=0, **kwargs):
if derivative > 4:
raise NotImplementedError
s = ctx.convert(s)
re = ctx._re(s); im = ctx._im(s)
if im < 0:
z = ctx.conj(ctx.rs_zeta(ctx.conj(s), derivative))
return z
critical_line = (re == 0.5)
if critical_line:
return zeta_half(ctx, s, derivative)
else:
return zeta_offline(ctx, s, derivative)
@defun
def rs_z(ctx, w, derivative=0):
w = ctx.convert(w)
re = ctx._re(w); im = ctx._im(w)
if re < 0:
return rs_z(ctx, -w, derivative)
critical_line = (im == 0)
if critical_line :
return z_half(ctx, w, derivative)
else:
return z_offline(ctx, w, derivative)
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